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href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=zh.wikipedia.org&uselang=zh"><span>资助维基百科</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E6%B5%81%E5%BD%A2" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E6%B5%81%E5%BD%A2" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-简介" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#简介"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>简介</span> </div> </a> <ul id="toc-简介-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-引例" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#引例"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>引例</span> </div> </a> <button aria-controls="toc-引例-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关引例子章节</span> </button> <ul id="toc-引例-sublist" class="vector-toc-list"> <li id="toc-圆" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#圆"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>圆</span> </div> </a> <ul id="toc-圆-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-其他曲线" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#其他曲线"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>其他曲线</span> </div> </a> <ul id="toc-其他曲线-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-定义" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#定义"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>定义</span> </div> </a> <button aria-controls="toc-定义-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关定义子章节</span> </button> <ul id="toc-定义-sublist" class="vector-toc-list"> <li id="toc-坐标卡" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#坐标卡"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>坐标卡</span> </div> </a> <ul id="toc-坐标卡-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-图册" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#图册"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>图册</span> </div> </a> <ul id="toc-图册-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-坐标变换映射" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#坐标变换映射"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>坐标变换映射</span> </div> </a> <ul id="toc-坐标变换映射-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-附加结构" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#附加结构"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>附加结构</span> </div> </a> <ul id="toc-附加结构-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-构造" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#构造"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>构造</span> </div> </a> <button aria-controls="toc-构造-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关构造子章节</span> </button> <ul id="toc-构造-sublist" class="vector-toc-list"> <li id="toc-图册构造" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#图册构造"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>图册构造</span> </div> </a> <ul id="toc-图册构造-sublist" class="vector-toc-list"> <li id="toc-带图册的球面" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#带图册的球面"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>带图册的球面</span> </div> </a> <ul id="toc-带图册的球面-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-贴补" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#贴补"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>贴补</span> </div> </a> <ul id="toc-贴补-sublist" class="vector-toc-list"> <li id="toc-内在和外在的观点" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#内在和外在的观点"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>内在和外在的观点</span> </div> </a> <ul id="toc-内在和外在的观点-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-作为贴补的n维球面" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#作为贴补的n维球面"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.2</span> <span>作为贴补的<i>n</i>维球面</span> </div> </a> <ul id="toc-作为贴补的n维球面-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-函数的零点" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#函数的零点"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>函数的零点</span> </div> </a> <ul id="toc-函数的零点-sublist" class="vector-toc-list"> <li id="toc-作为一个函数零点的n维球面" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#作为一个函数零点的n维球面"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>作为一个函数零点的<i>n</i>维球面</span> </div> </a> <ul id="toc-作为一个函数零点的n维球面-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-黏合一个流形上的不同点" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#黏合一个流形上的不同点"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>黏合一个流形上的不同点</span> </div> </a> <ul id="toc-黏合一个流形上的不同点-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-直积" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#直积"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>直积</span> </div> </a> <ul id="toc-直积-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-沿边界粘合" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#沿边界粘合"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>沿边界粘合</span> </div> </a> <ul id="toc-沿边界粘合-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-拓扑流形" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#拓扑流形"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>拓扑流形</span> </div> </a> <ul id="toc-拓扑流形-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-微分流形" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#微分流形"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>微分流形</span> </div> </a> <ul id="toc-微分流形-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-可定向性" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#可定向性"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>可定向性</span> </div> </a> <button aria-controls="toc-可定向性-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关可定向性子章节</span> </button> <ul id="toc-可定向性-sublist" class="vector-toc-list"> <li id="toc-莫比乌斯带" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#莫比乌斯带"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>莫比乌斯带</span> </div> </a> <ul id="toc-莫比乌斯带-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-克莱因瓶" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#克莱因瓶"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>克莱因瓶</span> </div> </a> <ul id="toc-克莱因瓶-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-实射影平面" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#实射影平面"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>实射影平面</span> </div> </a> <ul id="toc-实射影平面-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-豪斯多夫假设" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#豪斯多夫假设"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>豪斯多夫假设</span> </div> </a> <button aria-controls="toc-豪斯多夫假设-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关豪斯多夫假设子章节</span> </button> <ul id="toc-豪斯多夫假设-sublist" class="vector-toc-list"> <li id="toc-两个原点的线" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#两个原点的线"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>两个原点的线</span> </div> </a> <ul id="toc-两个原点的线-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-流形的其他类型和推广" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#流形的其他类型和推广"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>流形的其他类型和推广</span> </div> </a> <ul id="toc-流形的其他类型和推广-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-历史" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#历史"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>历史</span> </div> </a> <ul id="toc-历史-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-相关条目" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#相关条目"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>相关条目</span> </div> </a> <ul id="toc-相关条目-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>参考文献</span> </div> </a> <button aria-controls="toc-参考文献-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关参考文献子章节</span> </button> <ul id="toc-参考文献-sublist" class="vector-toc-list"> <li id="toc-註釋" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#註釋"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.1</span> <span>註釋</span> </div> </a> <ul id="toc-註釋-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-一般參考" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#一般參考"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.2</span> <span>一般參考</span> </div> </a> <ul id="toc-一般參考-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-外部連結" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#外部連結"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>外部連結</span> </div> </a> <ul id="toc-外部連結-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">开关目录</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">流形</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。55种语言可用" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-55" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">55种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Mannigfaltigkeit" title="Mannigfaltigkeit – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Mannigfaltigkeit" data-language-autonym="Alemannisch" data-language-local-name="瑞士德语" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%B9%D8%AF%D8%AF_%D8%B4%D8%B9%D8%A8" title="متعدد شعب – 阿拉伯语" lang="ar" hreflang="ar" data-title="متعدد شعب" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87oxobrazl%C4%B1" title="Çoxobrazlı – 阿塞拜疆语" lang="az" hreflang="az" data-title="Çoxobrazlı" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Многообразие – 保加利亚语" lang="bg" hreflang="bg" data-title="Многообразие" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B9%E0%A7%81%E0%A6%AD%E0%A6%BE%E0%A6%81%E0%A6%9C" title="বহুভাঁজ – 孟加拉语" lang="bn" hreflang="bn" data-title="বহুভাঁজ" data-language-autonym="বাংলা" data-language-local-name="孟加拉语" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Varietat_(matem%C3%A0tiques)" title="Varietat (matemàtiques) – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Varietat (matemàtiques)" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Varieta_(matematika)" title="Varieta (matematika) – 捷克语" lang="cs" hreflang="cs" data-title="Varieta (matematika)" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9D%D1%83%D0%BC%D0%B0%D0%B9%D1%81%C4%83%D0%BD%D0%B0%D1%80%D0%BB%C4%83%D1%85" title="Нумайсăнарлăх – 楚瓦什语" lang="cv" hreflang="cv" data-title="Нумайсăнарлăх" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Mangfoldighed_(matematik)" title="Mangfoldighed (matematik) – 丹麦语" lang="da" hreflang="da" data-title="Mangfoldighed (matematik)" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mannigfaltigkeit" title="Mannigfaltigkeit – 德语" lang="de" hreflang="de" data-title="Mannigfaltigkeit" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BF%CE%BB%CE%BB%CE%B1%CF%80%CE%BB%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Πολλαπλότητα – 希腊语" lang="el" hreflang="el" data-title="Πολλαπλότητα" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Manifold" title="Manifold – 英语" lang="en" hreflang="en" data-title="Manifold" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Sterna%C4%B5o" title="Sternaĵo – 世界语" lang="eo" hreflang="eo" data-title="Sternaĵo" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Variedad_(matem%C3%A1ticas)" title="Variedad (matemáticas) – 西班牙语" lang="es" hreflang="es" data-title="Variedad (matemáticas)" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Muutkond" title="Muutkond – 爱沙尼亚语" lang="et" hreflang="et" data-title="Muutkond" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Barietate_(matematika)" title="Barietate (matematika) – 巴斯克语" lang="eu" hreflang="eu" data-title="Barietate (matematika)" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%86%DB%8C%D9%81%D9%84%D8%AF_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="منیفلد (هندسه) – 波斯语" lang="fa" hreflang="fa" data-title="منیفلد (هندسه)" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Monisto" title="Monisto – 芬兰语" lang="fi" hreflang="fi" data-title="Monisto" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_(g%C3%A9om%C3%A9trie)" title="Variété (géométrie) – 法语" lang="fr" hreflang="fr" data-title="Variété (géométrie)" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Variedade_(xeometr%C3%ADa)" title="Variedade (xeometría) – 加利西亚语" lang="gl" hreflang="gl" data-title="Variedade (xeometría)" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%A8%D7%99%D7%A2%D7%94" title="יריעה – 希伯来语" lang="he" hreflang="he" data-title="יריעה" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Mnogostrukost" title="Mnogostrukost – 克罗地亚语" lang="hr" hreflang="hr" data-title="Mnogostrukost" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sokas%C3%A1g_(matematika)" title="Sokaság (matematika) – 匈牙利语" lang="hu" hreflang="hu" data-title="Sokaság (matematika)" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D5%A6%D5%B4%D5%A1%D5%B1%D6%87%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Բազմաձևություն – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Բազմաձևություն" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lipatan_(matematika)" title="Lipatan (matematika) – 印度尼西亚语" lang="id" hreflang="id" data-title="Lipatan (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Variet%C3%A0_(geometria)" title="Varietà (geometria) – 意大利语" lang="it" hreflang="it" data-title="Varietà (geometria)" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%9A%E6%A7%98%E4%BD%93" title="多様体 – 日语" lang="ja" hreflang="ja" data-title="多様体" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%EC%96%91%EC%B2%B4" title="다양체 – 韩语" lang="ko" hreflang="ko" data-title="다양체" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Mannifalt" title="Mannifalt – 卢森堡语" lang="lb" hreflang="lb" data-title="Mannifalt" data-language-autonym="Lëtzebuergesch" data-language-local-name="卢森堡语" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Daugdara" title="Daugdara – 立陶宛语" lang="lt" hreflang="lt" data-title="Daugdara" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vari%C3%ABteit_(wiskunde)" title="Variëteit (wiskunde) – 荷兰语" lang="nl" hreflang="nl" data-title="Variëteit (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Mangfald_i_matematikk" title="Mangfald i matematikk – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Mangfald i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Mangfoldighet" title="Mangfoldighet – 书面挪威语" lang="nb" hreflang="nb" data-title="Mangfoldighet" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AE%E0%A9%88%E0%A8%A8%E0%A9%80%E0%A8%AB%E0%A9%8B%E0%A8%B2%E0%A8%A1" title="ਮੈਨੀਫੋਲਡ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਮੈਨੀਫੋਲਡ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普语" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozmaito%C5%9B%C4%87" title="Rozmaitość – 波兰语" lang="pl" hreflang="pl" data-title="Rozmaitość" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Variet%C3%A0_topol%C3%B2gica" title="Varietà topològica – 皮埃蒙特文" lang="pms" hreflang="pms" data-title="Varietà topològica" data-language-autonym="Piemontèis" data-language-local-name="皮埃蒙特文" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%DB%8C%D9%86%DB%8C%D9%81%D9%88%D9%84%DA%88" title="مینیفولڈ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="مینیفولڈ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Variedade_(matem%C3%A1tica)" title="Variedade (matemática) – 葡萄牙语" lang="pt" hreflang="pt" data-title="Variedade (matemática)" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Varietate_(geometrie)" title="Varietate (geometrie) – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Varietate (geometrie)" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Многообразие – 俄语" lang="ru" hreflang="ru" data-title="Многообразие" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Mnogostrukost" title="Mnogostrukost – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Mnogostrukost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Manifold" title="Manifold – Simple English" lang="en-simple" hreflang="en-simple" data-title="Manifold" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Varieta_(matematika)" title="Varieta (matematika) – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Varieta (matematika)" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Mnogoterost" title="Mnogoterost – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Mnogoterost" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Durthi" title="Durthi – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Durthi" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%81%D1%82%D1%80%D1%83%D0%BA%D0%BE%D1%81%D1%82" title="Многострукост – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Многострукост" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/M%C3%A5ngfald_(matematik)" title="Mångfald (matematik) – 瑞典语" lang="sv" hreflang="sv" data-title="Mångfald (matematik)" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%A1%E0%B8%99%E0%B8%B4%E0%B9%82%E0%B8%9F%E0%B8%A5%E0%B8%94%E0%B9%8C" title="แมนิโฟลด์ – 泰语" lang="th" hreflang="th" data-title="แมนิโฟลด์" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Manipoldo" title="Manipoldo – 他加禄语" lang="tl" hreflang="tl" data-title="Manipoldo" data-language-autonym="Tagalog" data-language-local-name="他加禄语" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87ok_katl%C4%B1" title="Çok katlı – 土耳其语" lang="tr" hreflang="tr" data-title="Çok katlı" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%B2%D0%B8%D0%B4" title="Многовид – 乌克兰语" lang="uk" hreflang="uk" data-title="Многовид" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90a_t%E1%BA%A1p" title="Đa tạp – 越南语" lang="vi" hreflang="vi" data-title="Đa tạp" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%9C%D7%90%D7%9B%D7%98%D7%A2" title="פלאכטע – 意第绪语" lang="yi" hreflang="yi" data-title="פלאכטע" data-language-autonym="ייִדיש" data-language-local-name="意第绪语" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/To-i%C5%AB%E2%81%BF-th%C3%A9" title="To-iūⁿ-thé – 闽南语" lang="nan" hreflang="nan" data-title="To-iūⁿ-thé" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="闽南语" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B5%81%E5%BD%A2" title="流形 – 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class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li 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class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r83732972">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 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src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目已<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源">列出參考資料</a>，但<a href="/wiki/Help:%E8%84%9A%E6%B3%A8" title="Help:脚注">文內引註</a>不足，<b>部分內容的來源仍然不明</b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2014年1月31日</span>)</i></small><span class="hide-when-compact"><br /><small>请加上合适的文內引註加以<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit">改善</a>。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83732972"><table class="box-Refimprove plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Tango-nosources.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/45px-Tango-nosources.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/68px-Tango-nosources.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/90px-Tango-nosources.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目<b>需要补充更多<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源">来源</a></b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2013年6月24日</span>)</i></small><span class="hide-when-compact"><br /><small>请协助補充多方面<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">可靠来源</a>以<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit">改善这篇条目</a>，<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AF%81" class="mw-redirect" title="Wikipedia:可供查证">无法查证</a>的内容可能會因為<a href="/wiki/Template:Fact" class="mw-redirect" title="Template:Fact">异议提出</a>而被移除。<br />致使用者：请搜索一下条目的标题（来源搜索：<span class="plainlinks"><a rel="nofollow" class="external text" href="//www.google.com/search?&as_eq=wikipedia&q=%22%E6%B5%81%E5%BD%A2%22">"流形"</a> — <a rel="nofollow" class="external text" href="//www.google.com/search?q=%22%E6%B5%81%E5%BD%A2%22">网页</a>、<a rel="nofollow" class="external text" href="//www.google.com/search?tbm=nws&q=&as_src=-newswire+-wire+-presswire+-PR+-press+-release+-wikipedia&q=%22%E6%B5%81%E5%BD%A2%22">新闻</a>、<a rel="nofollow" class="external text" href="//books.google.com/books?&as_brr=0&as_pub=-icon&q=%22%E6%B5%81%E5%BD%A2%22">书籍</a>、<a rel="nofollow" class="external text" href="//scholar.google.com/scholar?&q=%22%E6%B5%81%E5%BD%A2%22">学术</a>、<a rel="nofollow" class="external text" href="//www.google.com/search?tbm=isch&safe=off&q=%22%E6%B5%81%E5%BD%A2%22">图像</a></span>），以检查网络上是否存在该主题的更多可靠来源（<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">判定指引</a>）。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Triangles_(spherical_geometry).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/300px-Triangles_%28spherical_geometry%29.jpg" decoding="async" width="300" height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/450px-Triangles_%28spherical_geometry%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/600px-Triangles_%28spherical_geometry%29.jpg 2x" data-file-width="2489" data-file-height="2048" /></a><figcaption><a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a>（<a href="/wiki/%E7%90%83_(%E6%95%B0%E5%AD%A6)" title="球 (数学)">球</a>的表面）為二維的流形，由於它能夠由一群二維的圖形來表示。</figcaption></figure> <p>在数学中，<b>流形</b>（英語：<span lang="en">manifold</span>）是可以“局部<a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间">欧几里得空间</a>化”的一个<a href="/wiki/%E6%8B%93%E6%89%91%E7%A9%BA%E9%97%B4" title="拓扑空间">拓扑空间</a>，即在此拓扑空间中，每个点附近“局部类似于欧氏空间”。更精确地说，<b>n维流形</b>（<i>n</i>-manifold），简称<b>n流形</b>，是一个拓扑空间，其性质是每个点都有一个<a href="/wiki/%E9%82%BB%E5%9F%9F" title="邻域">邻域</a>，该邻域<a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a>于n维欧氏空间的一个<a href="/wiki/%E5%BC%80%E9%9B%86" title="开集">开集</a>。 </p><p>流形是欧几里得空间中的曲线、曲面等概念的推广。欧几里得空间就是最简单的流形的实例。地球表面这样的球面则是一个稍微复杂的例子。一般的流形可以通过把许多平直的片折弯并粘连而成。 </p><p>流形在数学中用于描述几何形体，它们为研究形体的可微性提供了一个自然的平台。<a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%A6" title="物理学">物理学</a>上，<a href="/wiki/%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6" title="经典力学">经典力学</a>的<a href="/wiki/%E7%9B%B8%E7%A9%BA%E9%97%B4" class="mw-redirect" title="相空间">相空间</a>和构造<a href="/wiki/%E5%B9%BF%E4%B9%89%E7%9B%B8%E5%AF%B9%E8%AE%BA" class="mw-redirect" title="广义相对论">广义相对论</a>的<a href="/wiki/%E6%97%B6%E7%A9%BA" title="时空">时空</a>模型的<a href="/wiki/%E5%9B%9B%E7%BB%B4" class="mw-redirect" title="四维">四维</a>伪<a href="/wiki/%E9%BB%8E%E6%9B%BC%E6%B5%81%E5%BD%A2" title="黎曼流形">黎曼流形</a>都是流形的实例。<a href="/wiki/%E4%BD%8D%E5%BD%A2%E7%A9%BA%E9%97%B4" title="位形空间">位形空间</a>中也可以定义流形。<a href="/wiki/%E7%8E%AF%E9%9D%A2" title="环面">环面</a>就是双摆的位形空间。 </p><p>一般可以把几何形体的拓扑结构看作是完全“柔软”的，因为所有变形（同胚）会保持拓扑结构不变；而把<a href="/wiki/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95" title="解析几何">解析几何</a>结构看作是“硬”的，因为整体的结构都是固定的。例如，当一个<a href="/wiki/%E5%A4%9A%E9%A1%B9%E5%BC%8F" class="mw-redirect" title="多项式">多项式</a>在 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span> 区间的取值确定了，则其在整个实数范围的值都被固定，可见局部的变动会导致全局的变化。光滑流形可以看作是介于两者之间的模型：其无穷小的结构是“硬”的，而整体结构则是“柔软”的。这也许是中文译名“流形”的原因（整体的形态可以流动）。该译名由著名数学家和数学教育学家<a href="/wiki/%E6%B1%9F%E6%B3%BD%E6%B6%B5" title="江泽涵">江泽涵</a>引入。这样，流形的硬度使它能够容纳微分结构，而它的软度使得它可以作为很多需要独立的局部扰动的数学和物理的模型。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="简介"><span id=".E7.AE.80.E4.BB.8B"></span>简介</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=1" title="编辑章节：简介"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Triangles_(spherical_geometry).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/200px-Triangles_%28spherical_geometry%29.jpg" decoding="async" width="200" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/300px-Triangles_%28spherical_geometry%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/400px-Triangles_%28spherical_geometry%29.jpg 2x" data-file-width="2489" data-file-height="2048" /></a><figcaption>理想化的地球是一个流形。越近看就越近似于平面（“大三角形”是曲边的，但右下角<b>非常小</b>的三角形就和平面上一样了）。</figcaption></figure> <p>流形可以视为近看起来像<a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间">欧几里得空间</a>或其他相对简单的空间的物体<sup id="cite_ref-gva_1-0" class="reference"><a href="#cite_note-gva-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">:<span>1</span></sup>。例如，人们曾经以为地球是平的。这是因为相对于地球来说人類实在太小，平常看到的地面是地球表面微小的一部分。所以，尽管知道地球实际上差不多是一个圆球，如果只需要考虑其中微小的一部分上发生的事情，比如测量操场跑道的长度或进行房地产交易时，仍然把地面看成一个平面。一个理想的数学上的<a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球</a>面在足够小的区域上的特性就像一个<a href="/wiki/%E5%B9%B3%E9%9D%A2_(%E6%95%B0%E5%AD%A6)" title="平面 (数学)">平面</a>，这表明它是一个流形<sup id="cite_ref-baz_2-0" class="reference"><a href="#cite_note-baz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">:<span>283</span></sup>。但是球面和平面的整体结构是完全不同的：如果在球面上沿一个固定方向走，最终会回到起点，而在一个平面上，可以一直走下去。 </p><p>回到地球的例子。像旅行的时候，会用平面的<a href="/wiki/%E5%9C%B0%E5%9B%BE" title="地图">地图</a>来指示方位。如果将整个地球的各个地区的地图合订成一本地图集，那么在观看各个地区的地图后，就可以在脑海中“拼接”出整个地球的景貌。为了能让阅读者顺利从一张地图接到下一张，相邻的地图之间会有重叠的部分，以便在脑海里“粘合”两张图。类似地，在数学中，也可以用一系列“地图”（称为<b>坐标图</b>或<b>坐标卡</b>）组成的“地图集”（atlas, 亦称为<b>图册</b>）来描述一个流形<sup id="cite_ref-baz_2-1" class="reference"><a href="#cite_note-baz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">:<span>283</span></sup>。而“地图”之间重叠的部分在不同的地图里如何变换，则描述了不同“地图”的相互关系。 </p><p>描述一个流形往往需要不止一个“地图”，因为一般来说流形并不是真正的欧几里得空间。举例来说，地球就没法用一张平面的地图来合适地描绘。 </p><p>流形要求局部“看起来像”简单的空间，这不是一个简单的要求。例如，在球上吊一根线，这个整体就不是一个流形。包含了线和球连接的那一点的附近区域一定不是简单的：既不是线也不是面，无论这个区域有多小。 </p><p>流形有很多种。最简单的是<a href="/wiki/%E6%8B%93%E6%89%91%E6%B5%81%E5%BD%A2" title="拓扑流形">拓扑流形</a>，它们局部看来像<a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间">欧几里得空间</a>。其他的种类包含了它们在使用中所需要的额外的结构。例如，一个<a href="/wiki/%E5%BE%AE%E5%88%86%E6%B5%81%E5%BD%A2" title="微分流形">微分流形</a>不仅支持<a href="/wiki/%E6%8B%93%E6%89%91%E5%AD%A6" title="拓扑学">拓扑</a>，而且要支持<a href="/wiki/%E5%BE%AE%E7%A7%AF%E5%88%86%E5%AD%A6" title="微积分学">微积分</a>。<a href="/wiki/%E9%BB%8E%E6%9B%BC%E6%B5%81%E5%BD%A2" title="黎曼流形">黎曼流形</a>的思想导致了<a href="/wiki/%E5%BB%A3%E7%BE%A9%E7%9B%B8%E5%B0%8D%E8%AB%96" title="廣義相對論">广义相对论</a>的数学基础，使得人们能够用<a href="/wiki/%E6%9B%B2%E7%8E%87" title="曲率">曲率</a>来描述<a href="/wiki/%E6%97%B6%E7%A9%BA" title="时空">时空</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="引例"><span id=".E5.BC.95.E4.BE.8B"></span>引例</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=2" title="编辑章节：引例"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="圆"><span id=".E5.9C.86"></span>圆</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=3" title="编辑章节：圆"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_with_overlapping_manifold_charts.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle_with_overlapping_manifold_charts.png/220px-Circle_with_overlapping_manifold_charts.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle_with_overlapping_manifold_charts.png/330px-Circle_with_overlapping_manifold_charts.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle_with_overlapping_manifold_charts.png/440px-Circle_with_overlapping_manifold_charts.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>四张图分别把圆的一部分映射到一个开区间，它们合在一起覆盖了整个圆。</figcaption></figure> <p><a href="/wiki/%E5%9C%86" title="圆">圆</a>是除欧几里得空间外的拓扑流形的一个简单例子。考虑一个半径为1，圆心在原点的圆。若<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>是圆上的点的坐标，则有<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1}"></span>。 </p><p>局部看来，圆像一条线，而线是一维的。换句话说，只要一个坐标就可以在局部描述一个圆。例如，圆的上半部，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>坐标大于零的部分（右图中黄色的部分），任何一点都可以用<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>坐标确定。投影映射： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {top} }:(x,y)\mapsto x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>:</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {top} }:(x,y)\mapsto x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cba2f3afea33dc5175b1a2d5efb03762e13ed46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.59ex; height:3.009ex;" alt="{\displaystyle \phi _{\mathrm {top} }:(x,y)\mapsto x\,}"></span></dd></dl> <p>把上半圆映射到开区间<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e120a3bd60fc89b495dd7ef6039465b7e6a703b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,1)}"></span>。反过来，给定一个<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,{\sqrt {1-x^{2}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,{\sqrt {1-x^{2}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e424d1534c7cdf1d316fd295e3aacd9cd2aa3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.883ex; height:3.509ex;" alt="{\displaystyle (x,{\sqrt {1-x^{2}}})}"></span>就是上半圆的一点： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {top} }^{(-1)}:x\mapsto (x,{\sqrt {1-x^{2}}})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {top} }^{(-1)}:x\mapsto (x,{\sqrt {1-x^{2}}})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eea390dee28f67c7d5b0a34657089a420b8f50a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.149ex; height:3.843ex;" alt="{\displaystyle \phi _{\mathrm {top} }^{(-1)}:x\mapsto (x,{\sqrt {1-x^{2}}})\,}"></span></dd></dl> <p>这样的一个映射<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {top} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {top} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9134d27bb7734279a3781070b523eb7b009f9154" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.993ex; height:2.843ex;" alt="{\displaystyle \phi _{\mathrm {top} }}"></span>就是一个<b>坐标卡</b>（<span lang="en">chart</span>）。它的作用，就是告诉读者“地图”上的一点对应着实际中的哪一点。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {top} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {top} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9134d27bb7734279a3781070b523eb7b009f9154" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.993ex; height:2.843ex;" alt="{\displaystyle \phi _{\mathrm {top} }}"></span>和它的逆映射都是<a href="/wiki/%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0" title="连续函数">连续函数</a>甚至是<a href="/wiki/%E5%85%89%E6%BB%91%E5%87%BD%E6%95%B0" title="光滑函数">光滑函数</a>，这样的映射也叫做一个（微分）<a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a><sup id="cite_ref-gva_1-1" class="reference"><a href="#cite_note-gva-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">:<span>4</span></sup>。类似的，也可以为圆的下半部（红），左半部（蓝），右半部（绿）建立坐标卡。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\phi _{\mathrm {bottom} }(x,y)&=x\\\phi _{\mathrm {left} }(x,y)&=y\\\phi _{\mathrm {right} }(x,y)&=y.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\phi _{\mathrm {bottom} }(x,y)&=x\\\phi _{\mathrm {left} }(x,y)&=y\\\phi _{\mathrm {right} }(x,y)&=y.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a33c3f2b0725f4396dccaafa3a4524ee30a1b9a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.805ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\phi _{\mathrm {bottom} }(x,y)&=x\\\phi _{\mathrm {left} }(x,y)&=y\\\phi _{\mathrm {right} }(x,y)&=y.\end{aligned}}}"></span></dd></dl> <p>这四个部分合起来覆盖了整个圆，这四个坐标卡就组成了该圆的一个<b>图册</b>（<span lang="en">atlas</span>）。 </p><p>注意圆上部和右部的重叠部分，也就是位于圆上<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>坐标大于0的四分之一圆弧。两个坐标卡<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {top} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {top} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9134d27bb7734279a3781070b523eb7b009f9154" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.993ex; height:2.843ex;" alt="{\displaystyle \phi _{\mathrm {top} }}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {right} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {right} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08f9b2d58117f9be0dad1d9c7601c5668240641e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.095ex; height:2.843ex;" alt="{\displaystyle \phi _{\mathrm {right} }}"></span>都将这部分双射到区间<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>。这样就有一个从<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>到它自己的双射<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>：首先取<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>上面一点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>（黄色线段右半部分的点）黄色坐标卡的逆映射到达圆上的对应点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,{\sqrt {1-a^{2}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,{\sqrt {1-a^{2}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/203934c6faa256b9cd87f8fe6148386f1949d09c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.684ex; height:3.509ex;" alt="{\displaystyle (a,{\sqrt {1-a^{2}}})}"></span>，再通过绿色坐标卡映射到<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>上: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(a)=\phi _{\mathrm {right} }\left(\phi _{\mathrm {top} }^{(-1)}(a)\right)=\phi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)={\sqrt {1-a^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(a)=\phi _{\mathrm {right} }\left(\phi _{\mathrm {top} }^{(-1)}(a)\right)=\phi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)={\sqrt {1-a^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763f7a737c755cbc79373a0dcdef8e3220fafc8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.655ex; height:4.843ex;" alt="{\displaystyle T(a)=\phi _{\mathrm {right} }\left(\phi _{\mathrm {top} }^{(-1)}(a)\right)=\phi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)={\sqrt {1-a^{2}}}.}"></span></dd></dl> <p>映射<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>称为<b>坐标变换映射</b>（<span lang="en">transition map</span>），它告诉读者一张“地图”上的点是如何对应到另一张“地图”上的相应的点，说明了两张地图之间的关系<sup id="cite_ref-gva_1-2" class="reference"><a href="#cite_note-gva-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">:<span>5</span></sup>。 </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_manifold_chart_from_slope.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Circle_manifold_chart_from_slope.png/220px-Circle_manifold_chart_from_slope.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Circle_manifold_chart_from_slope.png/330px-Circle_manifold_chart_from_slope.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Circle_manifold_chart_from_slope.png/440px-Circle_manifold_chart_from_slope.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>圆流形基于斜率的坐标卡集，每个图覆盖除了一点之外的所有点。</figcaption></figure> <p>上，下，左，右四个坐标卡表明圓是一个流形，但它们不是唯一可以描述圆形的图册。坐标卡除了可以是几何投影，也可以是别的映射，而坐标卡的数量也可以不是四个，只要能够覆盖整个圆就行了。考虑以下两个坐标卡 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {minus} }(x,y)=s={y \over {1+x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {minus} }(x,y)=s={y \over {1+x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a79fb33ecb8089229a745f38c6a9ac024797a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.704ex; height:5.009ex;" alt="{\displaystyle \phi _{\mathrm {minus} }(x,y)=s={y \over {1+x}}}"></span> 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle \phi _{\mathrm {plus} }(x,y)=t={y \over {1-x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle \phi _{\mathrm {plus} }(x,y)=t={y \over {1-x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45f83e91c9f3aeb29773cf3fcb4828237b4fb7e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.731ex; height:5.009ex;" alt="{\displaystyle \displaystyle \phi _{\mathrm {plus} }(x,y)=t={y \over {1-x}}.}"></span></dd></dl> <p>这里<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>是过点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span>和固定点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f5f5f03e9f4380e41314d8c5d0129861c8ecf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,0)}"></span>之直线的斜率。比如右图中，点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-0.28,0.96)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>0.28</mn> <mo>,</mo> <mn>0.96</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-0.28,0.96)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d0f6484d9150ff6f90cdb7840e79b0bb1407d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.92ex; height:2.843ex;" alt="{\displaystyle (-0.28,0.96)}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f5f5f03e9f4380e41314d8c5d0129861c8ecf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,0)}"></span>确定的直线（右图黄色直线）斜率是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1{1 \over 3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1{1 \over 3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a9831408ff271afe7adc60366768ebc00635b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle 1{1 \over 3}}"></span>；点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0.6,-0.8)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0.6</mn> <mo>,</mo> <mo>−</mo> <mn>0.8</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0.6,-0.8)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49da8d29cc8bf6011c9883aaeb4fedaa6925f19b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.595ex; height:2.843ex;" alt="{\displaystyle (0.6,-0.8)}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f5f5f03e9f4380e41314d8c5d0129861c8ecf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,0)}"></span>确定的直线（右图红色直线）的斜率则是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{1 \over 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{1 \over 2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e44c5706264d03c62f8b02cbcce08ed7b0d4258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.807ex; height:5.176ex;" alt="{\displaystyle -{1 \over 2}}"></span>。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {minus} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {minus} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca5b208e705945830db880744caeb9d57aa51f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.92ex; height:2.509ex;" alt="{\displaystyle \phi _{\mathrm {minus} }}"></span>可以把圆上面除了点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f5f5f03e9f4380e41314d8c5d0129861c8ecf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,0)}"></span>以外的点一一映射到实数轴<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.299ex; height:2.843ex;" alt="{\displaystyle (-\infty ,\infty )}"></span>上。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {plus} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {plus} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa45508711a51c21bd70cfdb2758a055431fdab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.551ex; height:2.843ex;" alt="{\displaystyle \phi _{\mathrm {plus} }}"></span>则是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {minus} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {minus} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca5b208e705945830db880744caeb9d57aa51f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.92ex; height:2.509ex;" alt="{\displaystyle \phi _{\mathrm {minus} }}"></span>关于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>轴的镜像对称（也就是左右对称），固定点是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (+1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (+1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed562c3a4207a1b2450e3261f51a79213a8487d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (+1,0)}"></span>。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {minus} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {minus} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca5b208e705945830db880744caeb9d57aa51f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.92ex; height:2.509ex;" alt="{\displaystyle \phi _{\mathrm {minus} }}"></span>的逆映射为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={{1-s^{2}} \over {1+s^{2}}},\qquad y={{2s} \over {1+s^{2}}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={{1-s^{2}} \over {1+s^{2}}},\qquad y={{2s} \over {1+s^{2}}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47903e93aa2cf688ec0846bfc0db1ba93166a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.975ex; height:6.176ex;" alt="{\displaystyle x={{1-s^{2}} \over {1+s^{2}}},\qquad y={{2s} \over {1+s^{2}}};}"></span></dd></dl> <p>很容易确认<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1}"></span>对于所有斜率值<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>成立。这两个坐标卡提供了圆的又一个图册，其变换映射为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={1 \over s}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t={1 \over s}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6688452c7c1a69fa57b30b444748375c46b82f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.584ex; height:5.176ex;" alt="{\displaystyle t={1 \over s}.}"></span></dd></dl> <p>注意每个坐标卡都缺了一点，对于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>是点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f5f5f03e9f4380e41314d8c5d0129861c8ecf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,0)}"></span>，对于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>是点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (+1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (+1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed562c3a4207a1b2450e3261f51a79213a8487d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (+1,0)}"></span>，所以每个坐标卡不能独自覆盖整个圆。利用拓扑学的工具可以证明，没有单个的坐标卡可以覆盖整个圆；在这个简单的例子裡，已经可以看到流形拥有多个坐标卡的灵活性<sup id="cite_ref-gva_1-3" class="reference"><a href="#cite_note-gva-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">:<span>4</span></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="其他曲线"><span id=".E5.85.B6.E4.BB.96.E6.9B.B2.E7.BA.BF"></span>其他曲线</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=4" title="编辑章节：其他曲线"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Conics_and_cubic.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Conics_and_cubic.png/220px-Conics_and_cubic.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Conics_and_cubic.png/330px-Conics_and_cubic.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Conics_and_cubic.png/440px-Conics_and_cubic.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>从代数曲线来的四个流形: <span style="color:#bc1e47">■</span> 圆, <span style="color:#fec200">■</span> 抛物线, <span style="color:#0081cd">■</span> 双曲线, <span style="color:#009246">■</span> 三次曲线.</figcaption></figure> <p>流形不必<a href="/wiki/%E8%BF%9E%E9%80%9A%E7%A9%BA%E9%97%B4" title="连通空间">连通</a>（整个只有一片）；一对分离的圆可以是一个流形。它们不必是闭的，所以不带两个端点的线段也是流形。它们也不必有限，这样<a href="/wiki/%E6%8A%9B%E7%89%A9%E7%BA%BF" title="抛物线">抛物线</a>也是一个流形。 </p> <div class="mw-heading mw-heading2"><h2 id="定义"><span id=".E5.AE.9A.E4.B9.89"></span>定义</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=5" title="编辑章节：定义"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>拓扑流形的数学定义可表述为<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>： </p> <dl><dd>设<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span></i>是<a href="/wiki/%E8%B1%AA%E6%96%AF%E5%A4%9A%E5%A4%AB%E7%A9%BA%E9%97%B4" title="豪斯多夫空间">豪斯多夫空间</a>，若对任意 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df57d73e9532bb93a1439890bcddbc2806f5859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}"></span>，都有包含 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 的<a href="/wiki/%E9%82%BB%E5%9F%9F" title="邻域">邻域</a> <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc13aedb28dbebb2c1f22b8f82d9d36a3af8a9d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.76ex; height:2.509ex;" alt="{\displaystyle U_{x}}"></span></i> ，<a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a>于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a87a024931038d1858dc22e8a194e5978c3412e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}"></span>（<i>m</i>维<a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间">欧几里得空间</a>）的某个<a href="/wiki/%E5%BC%80%E9%9B%86" title="开集">开集</a>，則称<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span></i>是个<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span></i> 维（<b>拓扑</b>）<b>流形</b> 。</dd></dl> <div class="mw-heading mw-heading3"><h3 id="坐标卡"><span id=".E5.9D.90.E6.A0.87.E5.8D.A1"></span>坐标卡</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=6" title="编辑章节：坐标卡"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一个流形的一个<b>坐标映射</b>（coordinate map），<b>坐标卡</b>（coordinate chart），或简称<b>卡</b>是一个在流形的一个开子集和一个简单空间之间的双射，使得该映射及其逆都保持所要的结构。对于拓扑流形，该简单空间是某个<a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间">欧几里得空间</a><b>R</b><sup><i>n</i></sup>（或<b>R</b><sup><i>n</i></sup>的某个开子集）而一般感兴趣的是其拓扑结构。这个结构被<a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a>（homeomorphism）保持，也就是可逆的在两个方向都连续的映射。例如上节提到的映射<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathrm {top} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\mathrm {top} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9134d27bb7734279a3781070b523eb7b009f9154" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.993ex; height:2.843ex;" alt="{\displaystyle \phi _{\mathrm {top} }}"></span>是圆圈的一个卡。卡对于计算极其重要，因为它使得计算可以在简单空间进行，再把结果传回流形。 </p> <div class="mw-heading mw-heading3"><h3 id="图册"><span id=".E5.9B.BE.E5.86.8C"></span>图册</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=7" title="编辑章节：图册"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r85100532">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%9B%BE%E5%86%8C" title="图册">图册</a></div> <p>多数流形的表述需要多于一个的卡（只有最简单的流形只用一个卡）。覆盖流形的一个特定的卡的集合称为一个<b>图册</b>（atlas）。图册不是唯一的，因为所有流形可以被不同的卡的组合用很多方式覆盖。 </p><p>包含所有和给定图册相一致的卡的图册称为<b>极大图册</b>(maximal atlas)。不像普通的图册，极大图册是唯一的。虽然可能在定义中有用，但这个对象非常抽象，通常不直接使用（例如，在计算中）。 </p> <div class="mw-heading mw-heading3"><h3 id="坐标变换映射"><span id=".E5.9D.90.E6.A0.87.E5.8F.98.E6.8D.A2.E6.98.A0.E5.B0.84"></span>坐标变换映射</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=8" title="编辑章节：坐标变换映射"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>图册中的卡通常会互相重叠，而流形中的一个点可能会被好几个图所表示。如果两个图重叠，它们的部分会表示流形的同一个区域。这些部分之间的关联代表流形上同一点的坐标点的映射，譬如上面圆圈例子中的映射<i>T</i>，称为<b>坐标变换</b>，<b>变换函数</b>，或者<b>转换映射</b>。 </p> <div class="mw-heading mw-heading3"><h3 id="附加结构"><span id=".E9.99.84.E5.8A.A0.E7.BB.93.E6.9E.84"></span>附加结构</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=9" title="编辑章节：附加结构"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>图册也可用于定义流形上的附加结构。结构首先在每个卡上分别定义。如果所有变换映射和这个结构相容，该结构就可以转到流形上。 </p><p>这是微分流形的标准定义方式。如果图册的变换映射对于一个拓扑流形保持<b>R</b><sup><i>n</i></sup>自然的微分结构（也就是说，如果它们是<a href="/wiki/%E5%BE%AE%E5%88%86%E5%90%8C%E8%83%9A" title="微分同胚">微分同胚</a>），该微分结构就传到了流形上并把它变成微分流形。 </p><p>通常，流形的结构依赖于图册，但有时不同的图册给出相同的结构。这样的图册称为<b>相容</b>的。 </p> <div class="mw-heading mw-heading2"><h2 id="构造"><span id=".E6.9E.84.E9.80.A0"></span>构造</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=10" title="编辑章节：构造"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一个流形可以以不同方式构造，每个方式强调了流形的一个方面，因而导致了不同的观点。 </p> <div class="mw-heading mw-heading3"><h3 id="图册构造"><span id=".E5.9B.BE.E5.86.8C.E6.9E.84.E9.80.A0"></span>图册构造</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=11" title="编辑章节：图册构造"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sphere_with_chart.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Sphere_with_chart.png/220px-Sphere_with_chart.png" decoding="async" width="220" height="355" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Sphere_with_chart.png/330px-Sphere_with_chart.png 1.5x, //upload.wikimedia.org/wikipedia/commons/f/f7/Sphere_with_chart.png 2x" data-file-width="366" data-file-height="591" /></a><figcaption>该坐标图把球面有正<i>z</i>坐标的部分映射到一个圆盘。</figcaption></figure> <p>构造一个流形的一个简单方法是在上面的例子中的圆圈的构造方法。首先，确认<b>R</b><sup>2</sup>的一个子集，然后覆盖这个子集的图册被构造出来。<i>流形</i>的概念历史上就是从这样的构造发展出来的。这裡有另一个例子，把这个方法应用在球面的构造上: </p><p><br /> </p> <div class="mw-heading mw-heading4"><h4 id="带图册的球面"><span id=".E5.B8.A6.E5.9B.BE.E5.86.8C.E7.9A.84.E7.90.83.E9.9D.A2"></span>带图册的球面</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=12" title="编辑章节：带图册的球面"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a>的表面可以几乎和圆圈一样的方法来处理。可把球面视作<b>R</b><sup>3</sup>的子集: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{(x,y,z)\in \mathbf {R} ^{3}|x^{2}+y^{2}+z^{2}=1\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{(x,y,z)\in \mathbf {R} ^{3}|x^{2}+y^{2}+z^{2}=1\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0383bb273530bfbd12038582722333f7262da266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.25ex; height:3.176ex;" alt="{\displaystyle S=\{(x,y,z)\in \mathbf {R} ^{3}|x^{2}+y^{2}+z^{2}=1\}.}"></span></dd></dl> <p>球面是二维的，所以每个坐标图将映射球面的一部分到一个<b>R</b><sup>2</sup>的开子集。例如考虑北半球，它是带正<i>z</i>坐标的部分。（在右图中它着红色）定义如下的函数χ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (x,y,z)=(x,y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (x,y,z)=(x,y),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/401d81dee89e32487fee6981e078ebebb40baab6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.979ex; height:2.843ex;" alt="{\displaystyle \chi (x,y,z)=(x,y),}"></span></dd></dl> <p>把北半球映射到开<a href="/wiki/%E5%8D%95%E4%BD%8D%E5%9C%86%E7%9B%98" title="单位圆盘">单位圆盘</a>，通过把它投影到（<i>x</i>, <i>y</i>）平面。类似的坐标图对南半球也存在。和投影到（<i>x</i>, <i>z</i>）平面的两个坐标图以及投影到（<i>y</i>, <i>z</i>）平面的两个坐标图一起，就得到了一个覆盖整个球面的含6个坐标图的图册。 </p><p>这可以很容易地扩展到高维的球面。 </p> <div class="mw-heading mw-heading3"><h3 id="贴补"><span id=".E8.B4.B4.E8.A1.A5"></span>贴补</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=13" title="编辑章节：贴补"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>流形可以通过把碎片以一种相容的方式粘合来构造，使得碎片成为互相覆盖的坐标图。这种构造对于任何流形都是可行的，所以经常作为流形的表述，特别是微分和黎曼流形。它集中于图册的构造，把流形作为坐标图所自然的提供的贴片，因为不涉及外部的空间，这导致了流形的内在的观点。 </p><p>这裡，流形通过给定图册来构造，图册通过定义转换映射来得到。因此，流形上的一个点可以看成是通过变换映射映到同一个点的坐标点的<a href="/wiki/%E7%AD%89%E4%BB%B7%E7%B1%BB" title="等价类">等价类</a>。坐标图把等价类映射到一个贴片上的点。通常会对变换映射有很强的一致性要求。对于拓扑流形，它们被要求为<a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a>；如果它们也是<a href="/wiki/%E5%BE%AE%E5%88%86%E5%90%8C%E8%83%9A" title="微分同胚">微分同胚</a>，最后得到的流形就是微分流形。 </p><p>这可以通过变换映射圆圈例子的第二部分中的<i>t</i> = <sup>1</sup>⁄<sub><i>s</i></sub>来解释。从直线的两个拷贝开始。第一个拷贝用坐标<i>s</i>，第二个拷贝用<i>t</i>。现在，通过把第二个拷贝上的点<i>t</i>和第一个拷贝上的点<sup>1</sup>⁄<sub><i>s</i></sub>作为同一个点来粘合起来(点<i>t</i> = 0不和任何第一个拷贝上的点认同)。这就给出了一个圆圈。 </p> <div class="mw-heading mw-heading4"><h4 id="内在和外在的观点"><span id=".E5.86.85.E5.9C.A8.E5.92.8C.E5.A4.96.E5.9C.A8.E7.9A.84.E8.A7.82.E7.82.B9"></span>内在和外在的观点</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=14" title="编辑章节：内在和外在的观点"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>第一种构造和这种构造非常相似，但是他们代表了相当不同的观点。在第一种构造中，流形被视为<a href="/wiki/%E5%B5%8C%E5%85%A5_(%E6%95%B0%E5%AD%A6)" title="嵌入 (数学)">嵌入</a>到某个欧几里得空间中。这是<i>外在的观点</i>。当一个流形用这种方式来看的时候，它很容易通过直觉从欧几里得空间得到附加的结构。例如，在欧几里得空间，很明显某个点的一个向量是否和穿过该点的曲面 <a href="/wiki/%E7%9B%B8%E5%88%87" class="mw-redirect" title="相切">相切</a>或者<a href="/wiki/%E5%9E%82%E7%9B%B4" title="垂直">垂直</a>。 </p><p>贴补构造不用任何嵌入，只是简单把流形看作拓扑空间本身。这个抽象的观点称为<i>内在的观点</i>。这使得什么是切向量更难以想象。但是它表达了流形的本质，在计算上来讲，这可以避免使用更高的维数，例如只要二维而不是三维就可以作球面上的计算。 </p> <div class="mw-heading mw-heading4"><h4 id="作为贴补的n维球面"><span id=".E4.BD.9C.E4.B8.BA.E8.B4.B4.E8.A1.A5.E7.9A.84n.E7.BB.B4.E7.90.83.E9.9D.A2"></span>作为贴补的<i>n</i>维球面</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=15" title="编辑章节：作为贴补的n维球面"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><a href="/wiki/N%E7%BB%B4%E7%90%83%E9%9D%A2" title="N维球面"><i>n</i>维球面</a><b>S</b><sup><i>n</i></sup>可以通过粘合<b>R</b><sup><i>n</i></sup>的两个拷贝来构造。他们之间的变换函数定义为 <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}\setminus \{0\}\to \mathbf {R} ^{n}\setminus \{0\}:x\mapsto x/\|x\|^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}\setminus \{0\}\to \mathbf {R} ^{n}\setminus \{0\}:x\mapsto x/\|x\|^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6105fb5f5ed3e72349df72971f4baf22fcf8d721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.15ex; height:3.176ex;" alt="{\displaystyle \mathbf {R} ^{n}\setminus \{0\}\to \mathbf {R} ^{n}\setminus \{0\}:x\mapsto x/\|x\|^{2}.}"></span></dd></dl></dd> <dd>这个函数是它自身的逆，因而可以在两个方向使用。因为变换映射是一个<a href="/wiki/%E5%85%89%E6%BB%91%E5%87%BD%E6%95%B0" title="光滑函数">光滑函数</a>，这个图册定义了一个光滑流形。</dd> <dd>如果取<i>n</i> = 1，就得到了上面圆圈的例子。</dd></dl> <div class="mw-heading mw-heading3"><h3 id="函数的零点"><span id=".E5.87.BD.E6.95.B0.E7.9A.84.E9.9B.B6.E7.82.B9"></span>函数的零点</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=16" title="编辑章节：函数的零点"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>很多流形可以定义为某个函数的零点集。这个构造自然的把流形嵌入一个欧几里得空间，因而导向一个外在的观点。这很形象，但不幸的是不是每个流形都可以这样表示。 </p><p>如果一个<a href="/wiki/%E5%AF%BC%E6%95%B0" title="导数">可微函数</a>的<a href="/wiki/%E9%9B%85%E5%8F%AF%E6%AF%94%E7%9F%A9%E9%98%B5" title="雅可比矩阵">雅可比矩阵</a>在函数为0的每一点是<a href="/wiki/%E6%BB%A1%E7%A7%A9" class="mw-redirect" title="满秩">满秩</a>的，则根据<a href="/wiki/%E9%9A%90%E5%87%BD%E6%95%B0%E5%AE%9A%E7%90%86" title="隐函数定理">隐函数定理</a>，每个这样的点周围存在一个为0的邻域微分同胚于一个欧几里得空间。因此零点集是一个流形。 </p> <div class="mw-heading mw-heading4"><h4 id="作为一个函数零点的n维球面"><span id=".E4.BD.9C.E4.B8.BA.E4.B8.80.E4.B8.AA.E5.87.BD.E6.95.B0.E9.9B.B6.E7.82.B9.E7.9A.84n.E7.BB.B4.E7.90.83.E9.9D.A2"></span>作为一个函数零点的<i>n</i>维球面</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=17" title="编辑章节：作为一个函数零点的n维球面"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><i>n</i>维<a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a><b>S</b><sup><i>n</i></sup>经常定义为 <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} ^{n}:=\{x\in \mathbf {R} ^{n+1}:\|x\|=1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} ^{n}:=\{x\in \mathbf {R} ^{n+1}:\|x\|=1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d4a2b1a37d8cea92581ce2d07eda8ce90cb08a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.119ex; height:3.176ex;" alt="{\displaystyle \mathbf {S} ^{n}:=\{x\in \mathbf {R} ^{n+1}:\|x\|=1\}}"></span></dd></dl></dd> <dd>这等价为如下函数的零点 <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \|x\|-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto \|x\|-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1c535ff37046eaa1bc34c0c78bec7f676e1adf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.248ex; height:2.843ex;" alt="{\displaystyle x\mapsto \|x\|-1.}"></span></dd></dl></dd></dl> <dl><dd>这个函数的雅可比矩阵是 <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}x_{1}&\ldots &x_{n+1}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}x_{1}&\ldots &x_{n+1}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de0007c59048ed1f364a6279777ee5926de6452f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.591ex; margin-bottom: -0.247ex; width:16.446ex; height:2.843ex;" alt="{\displaystyle {\begin{bmatrix}x_{1}&\ldots &x_{n+1}\end{bmatrix}}}"></span>，</dd></dl></dd> <dd>它的秩对于除了原点的所有点为1(对于1×<i>n</i>矩阵就是满秩的)。这证明<i>n</i>维球面是一个微分流形。</dd></dl> <div class="mw-heading mw-heading3"><h3 id="黏合一个流形上的不同点"><span id=".E9.BB.8F.E5.90.88.E4.B8.80.E4.B8.AA.E6.B5.81.E5.BD.A2.E4.B8.8A.E7.9A.84.E4.B8.8D.E5.90.8C.E7.82.B9"></span>黏合一个流形上的不同点</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=18" title="编辑章节：黏合一个流形上的不同点"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>可以把流形上的不同点定义为相同。这可以视为把不同的点粘合为同一个点。這構造出的结果不保證是流形，但在有些情况下會是流形。 </p><p>这些情况下，黏合过程是用<a href="/wiki/%E7%BE%A4" title="群">群</a>来完成的，这是<a href="/wiki/%E7%BE%A4%E4%BD%9C%E7%94%A8" title="群作用">作用</a>在流形上的群。两个点被视为同一个如果一个能被该群的一个元素移动到另一个上面。如果<i>M</i>是该流形而<i>G</i>是该群，结果空间称为<a href="/wiki/%E5%95%86%E7%A9%BA%E9%97%B4" title="商空间">商空间</a>，并记为<i>M</i>/<i>G</i>。可以通过黏合点来构造的流形包括<a href="/wiki/%E7%8E%AF%E9%9D%A2" title="环面">环面</a>和<a href="/wiki/%E5%AE%9E%E5%B0%84%E5%BD%B1%E7%A9%BA%E9%97%B4" title="实射影空间">实射影空间</a>（分别从一个平面和一个球面开始）。舉例來說，假設 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}=\{(x,y)\in \mathbb {R} ^{2}\mid x^{2}+y^{2}=1\},H=\{e,g\},e,g:S^{2}\mapsto S^{2},e(x)=x,g(x)=-x,\,\forall x\in S^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∣</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>H</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>e</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">↦</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}=\{(x,y)\in \mathbb {R} ^{2}\mid x^{2}+y^{2}=1\},H=\{e,g\},e,g:S^{2}\mapsto S^{2},e(x)=x,g(x)=-x,\,\forall x\in S^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4d3661ef85ae6f3ea5935a8a012f3b4fdae084" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:89.932ex; height:3.176ex;" aria-hidden="true" alt="{\displaystyle S^{2}=\{(x,y)\in \mathbb {R} ^{2}\mid x^{2}+y^{2}=1\},H=\{e,g\},e,g:S^{2}\mapsto S^{2},e(x)=x,g(x)=-x,\,\forall x\in S^{2}}"></span>，則二維實射影空間 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} P^{2}=S^{2}/H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} P^{2}=S^{2}/H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2dcb23c3a36e7559827bc559137373329bf2ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:13.454ex; height:3.176ex;" aria-hidden="true" alt="{\displaystyle \mathbb {R} P^{2}=S^{2}/H}"></span>。 </p> <div class="mw-heading mw-heading3"><h3 id="直积"><span id=".E7.9B.B4.E7.A7.AF"></span>直积</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=19" title="编辑章节：直积"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>流形的<a href="/wiki/%E7%AC%9B%E5%8D%A1%E5%B0%94%E7%A7%AF" class="mw-redirect" title="笛卡尔积">直积</a>也是流形。但不是每个流形都是一个积。 </p><p>积流形的维数是其因子的维数之和。其拓扑是<a href="/wiki/%E4%B9%98%E7%A7%AF%E7%A9%BA%E9%97%B4" class="mw-redirect" title="乘积空间">乘积拓扑</a>，而坐标图的直积是积流形的坐标图。这样，积流形的图册可以用其因子的图册构造。如果这些图册定义了因子上的微分结构，相应的积图册定义了积流形上的一个微分结构。因子上定义的其他结构也可以同样处理。如果一个因子有一个边界，积流形也有边界。直积可以用来构造环面和有限<a href="/wiki/%E5%9C%86%E6%9F%B1%E9%9D%A2" class="mw-redirect" title="圆柱面">圆柱面</a>，例如，分别定义它们为<b>S</b><sup>1</sup> × <b>S</b><sup>1</sup>和<b>S</b><sup>1</sup> × [0, 1]。 </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Red_cylinder.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Red_cylinder.png/220px-Red_cylinder.png" decoding="async" width="220" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Red_cylinder.png/330px-Red_cylinder.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Red_cylinder.png/440px-Red_cylinder.png 2x" data-file-width="711" data-file-height="641" /></a><figcaption>有限圆柱面是带边界的流形。</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="沿边界粘合"><span id=".E6.B2.BF.E8.BE.B9.E7.95.8C.E7.B2.98.E5.90.88"></span>沿边界粘合</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=20" title="编辑章节：沿边界粘合"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>两个带边界的流形可以沿着边界粘合。如果用正确的方式完成，结果也是流形。类似的，一个流形的两个边界也可以粘合起来。 </p><p>形式化的，粘合可以定义为两个边界的一个双射。两个点被认同为一个，如果它们互相映射到对方。对于一个拓扑流形，这个双射必须是同胚，否则结果就不是拓扑流形。类似的，对于一个微分流形，它必须是微分同胚。对于其它流形，其他的结构必须被这个双射所保持。 </p><p>有限的圆柱面可以作为一个流形构造，先从一个长条<b>R</b> × [0, 1]开始，然后把对边通过适当的微分同胚粘合起来。<a href="/wiki/%E5%85%8B%E8%8E%B1%E5%9B%A0%E7%93%B6" title="克莱因瓶">克莱因瓶</a>可以由一个带孔的球面和一个<a href="/wiki/%E8%8E%AB%E6%AF%94%E4%B9%8C%E6%96%AF%E5%B8%A6" title="莫比乌斯带">莫比乌斯带</a>沿着各自的圆形边界粘合起来得到。 </p> <div class="mw-heading mw-heading2"><h2 id="拓扑流形"><span id=".E6.8B.93.E6.89.91.E6.B5.81.E5.BD.A2"></span>拓扑流形</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=21" title="编辑章节：拓扑流形"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r85100532"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E6%8B%93%E6%89%91%E6%B5%81%E5%BD%A2" title="拓扑流形">拓扑流形</a></div> <p>最容易定义的流形是拓扑流形，它局部看起来象一些「普通」的<a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间">欧几里得空间</a><b>R</b><sup><i>n</i></sup>。形式化地讲，一个拓扑流形是一个<a href="/w/index.php?title=%E5%B1%80%E9%83%A8%E5%90%8C%E8%83%9A&action=edit&redlink=1" class="new" title="局部同胚（页面不存在）">局部同胚</a>于一个欧几里得空间的<a href="/wiki/%E6%8B%93%E6%89%91%E7%A9%BA%E9%97%B4" title="拓扑空间">拓扑空间</a>。这表示每个点有一个<a href="/wiki/%E9%82%BB%E5%9F%9F" title="邻域">邻域</a>，它有一个<a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a>（<a href="/wiki/%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0" title="连续函数">连续</a>双射其逆也连续）将它映射到<b>R</b><sup><i>n</i></sup>。这些同胚是流形的坐标图。 </p><p>通常附加的技术性假设被加在该拓扑空间上，以排除病态的情形。可以根据需要要求空间是<a href="/wiki/%E8%B4%B9%E5%88%A9%E5%85%8B%E6%96%AF%C2%B7%E8%B1%AA%E6%96%AF%E5%A4%9A%E5%A4%AB" title="费利克斯·豪斯多夫">豪斯多夫</a>的并且<a href="/wiki/%E7%AC%AC%E4%BA%8C%E5%8F%AF%E6%95%B0%E7%A9%BA%E9%97%B4" class="mw-redirect" title="第二可数空间">第二可数</a>。这表示下面所述的有两个原点的直线不是拓扑流形，因为它不是豪斯朵夫的。 </p><p>流形在某一点的<i>维数</i>就是该点映射到的欧几里得空间图的维数（定义中的数字<i>n</i>）。<a href="/wiki/%E8%BF%9E%E9%80%9A%E7%A9%BA%E9%97%B4" title="连通空间">连通</a>流形中的所有点有相同的维数。有些作者要求拓扑流形的所有的图映射到同一欧几里得空间。这种情况下，拓扑空间有一个拓扑不变量，也就是它的维数。其他作者允许拓扑流形的不交并有不同的维数。 </p> <div class="mw-heading mw-heading2"><h2 id="微分流形"><span id=".E5.BE.AE.E5.88.86.E6.B5.81.E5.BD.A2"></span>微分流形</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=22" title="编辑章节：微分流形"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r85100532"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%BE%AE%E5%88%86%E6%B5%81%E5%BD%A2" title="微分流形">微分流形</a></div> <p>很容易定义拓扑流形，但是很难在它们上面工作。对于多数应用，拓扑流形的一种，<b>微分流形</b>比较好用。如果流形上的局部坐标图以某种形式相容，就可以在该流形上讨论方向，切空间，和可微函数。特别是，可以在微分流形上应用「<a href="/wiki/%E5%BE%AE%E7%A7%AF%E5%88%86%E5%AD%A6" title="微积分学">微积分</a>」。 </p> <div class="mw-heading mw-heading2"><h2 id="可定向性"><span id=".E5.8F.AF.E5.AE.9A.E5.90.91.E6.80.A7"></span>可定向性</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=23" title="编辑章节：可定向性"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>考虑一个拓扑流形，其坐标图映射到<b>R</b><sup><i>n</i></sup>。给定一个<b>R</b><sup><i>n</i></sup>的<a href="/wiki/%E6%9C%89%E5%BA%8F%E5%9F%BA" class="mw-redirect" title="有序基">有序基</a>，坐标图就给它所覆盖的流形的一片引入了一个方向，可以视为或者右手或者左手的。重叠的坐标图不要求在方向上一致，这给了流形一个重要的自由度。对于某些流形，譬如球面，可以选取一些坐标图使得重叠区域在"手性"上一致；这些流形称为"可定向"的。对于其它的流形，这不可能做到。后面这种可能性容易被忽视，因为任何在三维空间中（不自交的）嵌入的闭曲面都是可定向的。 </p><p>考虑三个例子: (1)<a href="/wiki/%E8%8E%AB%E6%AF%94%E4%B9%8C%E6%96%AF%E5%B8%A6" title="莫比乌斯带">莫比乌斯带</a>，它是有边界的流形，（2）<a href="/wiki/%E5%85%8B%E8%8E%B1%E5%9B%A0%E7%93%B6" title="克莱因瓶">克莱因瓶</a>，它在三维空间必须自交，以及（3）<a href="/wiki/%E5%AE%9E%E5%B0%84%E5%BD%B1%E5%B9%B3%E9%9D%A2" title="实射影平面">实射影平面</a>，它很自然的出现在<a href="/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6" title="几何学">几何学</a>中。 </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:MobiusStrip-01.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/MobiusStrip-01.png/150px-MobiusStrip-01.png" decoding="async" width="150" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/MobiusStrip-01.png/225px-MobiusStrip-01.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e7/MobiusStrip-01.png 2x" data-file-width="273" data-file-height="216" /></a><figcaption>莫比乌斯带</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="莫比乌斯带"><span id=".E8.8E.AB.E6.AF.94.E4.B9.8C.E6.96.AF.E5.B8.A6"></span>莫比乌斯带</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=24" title="编辑章节：莫比乌斯带"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r85100532"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E8%8E%AB%E6%AF%94%E4%B9%8C%E6%96%AF%E5%B8%A6" title="莫比乌斯带">莫比乌斯带</a></div> <p>从一个竖着的无限圆柱面开始，这是一个无边界的流形。在高和低的地方各剪一刀，产生两个圆形边界，和它们之间的一个圆形的带子。这是一个带边界的可定向流形，若在它上面动一个小"手术"。把带子剪开，使得它能展开成一个矩形，但把两头捏住。把其中一头转180°，把内面翻倒朝外，然后把两头无缝的粘回来。现在有了一个永久半翻转的带子，就是莫比乌斯带。它的边界不再是一对圆圈，而是（拓扑上）单个圆圈；曾经是"内面"的现在和"外面"并了起来，使得它只有"单"面。（在<a href="/wiki/%E6%89%93%E5%8D%B0%E6%9C%BA" title="打印机">打印机</a>的<a href="/wiki/%E8%89%B2%E5%B8%A6" title="色带">色带</a>中有这种左扭带的应用。） </p> <div class="mw-heading mw-heading3"><h3 id="克莱因瓶"><span id=".E5.85.8B.E8.8E.B1.E5.9B.A0.E7.93.B6"></span>克莱因瓶</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=25" title="编辑章节：克莱因瓶"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r85100532"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%85%8B%E8%8E%B1%E5%9B%A0%E7%93%B6" title="克莱因瓶">克莱因瓶</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:KleinBottle-01.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/KleinBottle-01.png/120px-KleinBottle-01.png" decoding="async" width="120" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/KleinBottle-01.png/180px-KleinBottle-01.png 1.5x, //upload.wikimedia.org/wikipedia/commons/4/46/KleinBottle-01.png 2x" data-file-width="240" data-file-height="300" /></a><figcaption>嵌入到三维空间的<a href="/wiki/%E5%85%8B%E8%8E%B1%E5%9B%A0%E7%93%B6" title="克莱因瓶">克莱因瓶</a>。</figcaption></figure> <p>取两个莫比乌斯带；每个都以一个圈为边界。把每个圈拉成一个圆圈，并把带子变成<a href="/w/index.php?title=%E4%BA%A4%E5%8F%89%E5%B8%BD&action=edit&redlink=1" class="new" title="交叉帽（页面不存在）">交叉帽</a>（cross-cap）。（注意这在三维空间物理上是不可能的；克莱因瓶不能放到三维空间中，就像莫比乌斯带（或者球面）不能放在平面上一样。实际建造一个克莱因瓶必須在至少四维的空间进行）把圆圈粘合起来会产生一个新的闭合流形，没有边界的克莱因瓶。把曲面闭合起来并不能改变不可定向性，它只是移除了边界。这样克莱因瓶就成了一个不能分辨内外的闭合曲面。 </p> <div class="mw-heading mw-heading3"><h3 id="实射影平面"><span id=".E5.AE.9E.E5.B0.84.E5.BD.B1.E5.B9.B3.E9.9D.A2"></span>实射影平面</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=26" title="编辑章节：实射影平面"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>从圆心为原点的球面开始。穿过原点的每条直线在两个相对的点穿透球面。虽然在物理上不能这么做，但在数学上可以把相对点黏合为同一点。这样产生的闭合曲面是实射影平面，為一个不可定向曲面。它有一些等价的表述和构造，但是这个方法揭示了它的名字:所有给定的穿过原点的直线射影到该"平面"的一个"点"。 </p> <div class="mw-heading mw-heading2"><h2 id="豪斯多夫假设"><span id=".E8.B1.AA.E6.96.AF.E5.A4.9A.E5.A4.AB.E5.81.87.E8.AE.BE"></span>豪斯多夫假设</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=27" title="编辑章节：豪斯多夫假设"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="两个原点的线"><span id=".E4.B8.A4.E4.B8.AA.E5.8E.9F.E7.82.B9.E7.9A.84.E7.BA.BF"></span>两个原点的线</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=28" title="编辑章节：两个原点的线"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>这里给出一个空间的例子，它满足拓扑流形所有的条件，除了它不是<a href="/wiki/%E8%B1%AA%E6%96%AF%E5%A4%9A%E5%A4%AB%E7%A9%BA%E9%97%B4" title="豪斯多夫空间">豪斯多夫空间</a>（Hausdorff space）。取两个<b>R</b>的拷贝，把它们写作 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} \times \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} \times \{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574372ecf72fef09e5c2db081a3f6b62fddaf1cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.331ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} \times \{0\}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} \times \{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} \times \{1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb555a623a32ab5a16b8ff10be8c089b2643f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.331ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} \times \{1\}}"></span>,</dd></dl> <p>并定义如下<a href="/wiki/%E7%AD%89%E4%BB%B7%E5%85%B3%E7%B3%BB" title="等价关系">等价关系</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,0)\sim (x,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,0)\sim (x,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5137036e22ddd7b99321b7b2cd4cad27934978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.769ex; height:2.843ex;" alt="{\displaystyle (x,0)\sim (x,1)}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}"></span>.</dd></dl> <p>从这个等价关系得到的<a href="/wiki/%E5%95%86%E7%A9%BA%E9%97%B4" title="商空间">商空间</a> <b>L</b> 是一个象实直线那样的空间，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x,0)\mid x\in \mathbb {R} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(x,0)\mid x\in \mathbb {R} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb860f4c8b44a37deb3346b533722b72448e8bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:15.446ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle \{(x,0)\mid x\in \mathbb {R} \}}"></span>，再加上額外的一點 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (0,1)}"></span>。特别的是，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" aria-hidden="true" alt="{\displaystyle L}"></span> 上的 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (0,0)}"></span> 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (0,1)}"></span> 這兩點不能被不相交的开集所分离，所以<b>L</b>不是豪斯多夫的。它是一个拓扑流形，但不是豪斯多夫拓扑流形。 </p><p>经常，拓扑流形被定义为必须是豪斯多夫的，在这个定义下，上面的例子不是流形。 </p> <div class="mw-heading mw-heading2"><h2 id="流形的其他类型和推广"><span id=".E6.B5.81.E5.BD.A2.E7.9A.84.E5.85.B6.E4.BB.96.E7.B1.BB.E5.9E.8B.E5.92.8C.E6.8E.A8.E5.B9.BF"></span>流形的其他类型和推广</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=29" title="编辑章节：流形的其他类型和推广"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>要在流形上研究<a href="/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6" title="几何学">几何</a>，通常必须用附加的结构来装饰这些空间，例如上面的微分流形所加入的微分结构。根据所需要的不同的几何，有许多其它的可能性: </p> <ul><li><b>复流形</b>: <a href="/wiki/%E5%A4%8D%E6%B5%81%E5%BD%A2" title="复流形">复流形</a>是建模在<b>C</b><sup><i>n</i></sup>上的流形，在<a href="/wiki/%E5%9D%90%E6%A0%87%E5%9B%BE" class="mw-redirect" title="坐标图">坐标图</a>的重叠处以<a href="/wiki/%E5%85%A8%E7%BA%AF%E5%87%BD%E6%95%B0" title="全纯函数">全纯函数</a>为变换函数。这些流形是复几何研究的基本对象。一个一维复流形称为<a href="/wiki/%E9%BB%8E%E6%9B%BC%E6%9B%B2%E9%9D%A2" title="黎曼曲面">黎曼曲面</a>。</li></ul> <ul><li><b>巴拿赫和Fréchet流形</b>:要允许无穷维，可以考虑<a href="/w/index.php?title=%E5%B7%B4%E6%8B%BF%E8%B5%AB%E6%B5%81%E5%BD%A2&action=edit&redlink=1" class="new" title="巴拿赫流形（页面不存在）">巴拿赫流形</a>，它局部同胚于<a href="/wiki/%E5%B7%B4%E6%8B%BF%E8%B5%AB%E7%A9%BA%E9%97%B4" title="巴拿赫空间">巴拿赫空间</a>。类似的，Fréchet流形局部同胚于<a href="/w/index.php?title=Fr%C3%A9chet_space&action=edit&redlink=1" class="new" title="Fréchet space（页面不存在）">Fréchet space</a>。</li></ul> <ul><li><b>轨形（Orbifolds）</b>:一个<a href="/wiki/%E8%BD%A8%E5%BD%A2" title="轨形">轨形</a>是流形的推广，允许某种"<a href="/wiki/%E5%A5%87%E7%95%B0%E9%BB%9E" class="mw-redirect mw-disambig" title="奇異點">奇异点</a>"在其拓扑中存在。大致来讲，它是局部看起来像一些简单空间（例如，<a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间">欧几里得空间</a>）通过各种<a href="/wiki/%E6%9C%89%E9%99%90%E7%BE%A4" title="有限群">有限群</a>的<a href="/wiki/%E7%BE%A4%E4%BD%9C%E7%94%A8" title="群作用">群作用</a>的商。奇点对应于群作用的不动点，而作用必须在某种意义下相容。</li></ul> <ul><li><b>代数簇和概形（Algebraic varieties and schemes）</b>:一个<a href="/wiki/%E4%BB%A3%E6%95%B0%E7%B0%87" title="代数簇">代数簇</a>是几个仿射代数簇粘起来得到的，仿射代数簇是在代数封闭的域上多项式的零点集。类似的，<a href="/wiki/%E6%A6%82%E5%BD%A2" title="概形">概形</a>是仿射概形粘起来得到的，而仿射概形是代数簇的一个推广。二者都和流形相关，但都使用<a href="/wiki/%E5%B1%82_(%E6%95%B0%E5%AD%A6)" title="层 (数学)">层</a>而非坐标图集来构造。</li></ul> <div class="mw-heading mw-heading2"><h2 id="历史"><span id=".E5.8E.86.E5.8F.B2"></span>历史</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=30" title="编辑章节：历史"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>第一个清楚地把<a href="/wiki/%E6%9B%B2%E7%BA%BF" title="曲线">曲线</a>和<a href="/wiki/%E6%9B%B2%E9%9D%A2" title="曲面">曲面</a>本身构想为空间的可能是<a href="/wiki/%E5%8D%A1%E7%88%BE%C2%B7%E5%BC%97%E9%87%8C%E5%BE%B7%E9%87%8C%E5%B8%8C%C2%B7%E9%AB%98%E6%96%AF" title="卡爾·弗里德里希·高斯">高斯</a>，他以他的“<a href="/wiki/%E7%BB%9D%E5%A6%99%E5%AE%9A%E7%90%86" class="mw-redirect" title="绝妙定理">绝妙定理</a>”（Theorema egregium）建立了内在的<a href="/wiki/%E5%BE%AE%E5%88%86%E5%87%A0%E4%BD%95" title="微分几何">微分几何</a>。 </p><p><a href="/wiki/%E6%B3%A2%E6%81%A9%E5%93%88%E5%BE%B7%C2%B7%E9%BB%8E%E6%9B%BC" class="mw-redirect" title="波恩哈德·黎曼">黎曼</a>是第一个广泛的展开真正需要把流形推广到高维的工作的人。<i>流形</i>的名字来自<a href="/wiki/%E6%B3%A2%E6%81%A9%E5%93%88%E5%BE%B7%C2%B7%E9%BB%8E%E6%9B%BC" class="mw-redirect" title="波恩哈德·黎曼">黎曼</a>原来的<a href="/wiki/%E5%BE%B7%E8%AF%AD" title="德语">德语</a>术语<i>Mannigfaltigkeit</i>，<a href="/w/index.php?title=William_Kingdon_Clifford&action=edit&redlink=1" class="new" title="William Kingdon Clifford（页面不存在）">William Kingdon Clifford</a>把它翻译为"manifoldness"（多层）。在他的哥廷根就职演说中，黎曼表明一个属性可以取的所有值组成一个<i>Mannigfaltigkeit</i>。他根据值的变化连续与否对<i>stetige Mannigfaltigkeit</i>和<i>离散</i> [<i><a href="/wiki/Sic" title="Sic">sic</a></i>] <i>Mannigfaltigkeit</i>（<i>连续流形</i> 和<i>不连续流形</i>）作了区分。作为<i>stetige Mannigfaltikeiten</i>的例子，他提到了物体颜色和在空间中的位置，以及一个空间形体的可能形状。他把一个<i>n fach ausgedehnte Mannigfaltigkeit</i>（<i>n次扩展的</i>或<i>n-维流形</i>）构造为一个连续的<i>（n-1）fach ausgedehnte Mannigfaltigkeiten</i>堆。黎曼直觉上的<i>Mannigfaltigkeit</i>概念发展为今天形式化的流形。<a href="/wiki/%E9%BB%8E%E6%9B%BC%E6%B5%81%E5%BD%A2" title="黎曼流形">黎曼流形</a>和<a href="/wiki/%E9%BB%8E%E6%9B%BC%E6%9B%B2%E9%9D%A2" title="黎曼曲面">黎曼曲面</a>就是以他的名字命名的。 </p><p><a href="/w/index.php?title=%E4%BA%A4%E6%8D%A2%E7%B0%87&action=edit&redlink=1" class="new" title="交换簇（页面不存在）">交换簇</a>的概念在黎曼的时代已经被隐含地作为<a href="/wiki/%E5%A4%8D%E6%B5%81%E5%BD%A2" title="复流形">复流形</a>使用。从几何方面考虑，<a href="/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E5%8A%9B%E5%AD%A6" title="拉格朗日力学">拉格朗日力学</a>和<a href="/wiki/%E5%93%88%E5%AF%86%E9%A1%BF%E5%8A%9B%E5%AD%A6" title="哈密顿力学">哈密尔顿力学</a>的本质也是流形理论。 </p><p><a href="/wiki/%E5%BA%9E%E5%8A%A0%E8%8E%B1" class="mw-redirect" title="庞加莱">庞加莱</a>研究了三维流形，并提出著名的<a href="/wiki/%E5%BA%9E%E5%8A%A0%E8%8E%B1%E7%8C%9C%E6%83%B3" title="庞加莱猜想">庞加莱猜想</a>：所有闭简单连通的三维流形同胚于3维球吗？这个猜想已被<a href="/wiki/%E6%A0%BC%E9%87%8C%E6%88%88%E9%87%8C%C2%B7%E4%BD%A9%E9%9B%B7%E5%B0%94%E6%9B%BC" title="格里戈里·佩雷尔曼">Grigori Perelman</a>解决。 </p><p><a href="/wiki/%E8%B5%AB%E5%B0%94%E6%9B%BC%C2%B7%E5%A4%96%E5%B0%94" title="赫尔曼·外尔">赫尔曼·外尔</a>于1912年给出了微分流形的一个内在的定义。1930年代，该课题基础性方面的工作被<a href="/wiki/Hassler_Whitney" class="mw-redirect" title="Hassler Whitney">Hassler Whitney</a>等人厘清，使得从19世纪下半叶起开始发展起来的相关的直觉知识变得更精确，并通过<a href="/wiki/%E5%BE%AE%E5%88%86%E5%87%A0%E4%BD%95" title="微分几何">微分几何</a>和<a href="/wiki/%E6%9D%8E%E7%BE%A4" title="李群">李群</a>使微分流形的理论得到进一步的发展。 </p> <div class="mw-heading mw-heading2"><h2 id="相关条目"><span id=".E7.9B.B8.E5.85.B3.E6.9D.A1.E7.9B.AE"></span>相关条目</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=31" title="编辑章节：相关条目"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r80540462">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .refbegin-100{font-size:100%}</style><div class="refbegin columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2;"> <ul><li><a href="/wiki/%E5%9D%90%E6%A0%87%E8%BD%AC%E6%8D%A2" title="坐标转换">坐标转换</a></li> <li><a href="/wiki/%E8%B1%AA%E6%96%AF%E5%A4%9A%E5%A4%AB%E7%A9%BA%E9%97%B4" title="豪斯多夫空间">豪斯多夫空间</a></li> <li><a href="/wiki/%E5%9D%90%E6%A0%87%E9%82%BB%E5%9F%9F" title="坐标邻域">坐标邻域</a></li> <li><a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间">欧几里得空间</a></li> <li><a href="/wiki/%E4%BB%A3%E6%95%B0%E7%B0%87" title="代数簇">代数簇</a>（algebraic variety，或称代数多样体）</li> <li><a href="/wiki/%E6%A6%82%E5%BD%A2" title="概形">概形</a></li> <li><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="流形列表（页面不存在）">流形列表</a></li> <li><a href="/wiki/%E6%9B%B2%E9%9D%A2" title="曲面">曲面</a></li> <li><a href="/wiki/3-%E6%B5%81%E5%BD%A2" title="3-流形">3-流形</a></li> <li><a href="/w/index.php?title=4-%E6%B5%81%E5%BD%A2&action=edit&redlink=1" class="new" title="4-流形（页面不存在）">4-流形</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=32" title="编辑章节：参考文献"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="註釋"><span id=".E8.A8.BB.E9.87.8B"></span>註釋</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=33" title="编辑章节：註釋"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-gva-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-gva_1-0"><sup><b>1.0</b></sup></a> <a href="#cite_ref-gva_1-1"><sup><b>1.1</b></sup></a> <a href="#cite_ref-gva_1-2"><sup><b>1.2</b></sup></a> <a href="#cite_ref-gva_1-3"><sup><b>1.3</b></sup></a></span> <span class="reference-text"><cite class="citation book">Guillemin, Victor and Anton Pollack,. <span></span><i>Differential Topology</i><span></span>. Prentice-Hall. 1974. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0132126052" title="Special:网络书源/0132126052"><span title="国际标准书号">ISBN</span> 0132126052</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%B5%81%E5%BD%A2&rft.au=Guillemin%2C+Victor+and+Anton+Pollack%2C&rft.btitle=Differential+Topology&rft.date=1974&rft.genre=book&rft.isbn=0132126052&rft.pub=Prentice-Hall&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-baz-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-baz_2-0"><sup><b>2.0</b></sup></a> <a href="#cite_ref-baz_2-1"><sup><b>2.1</b></sup></a></span> <span class="reference-text"><cite class="citation book">B.A.卓里奇著，蒋铎、钱佩玲、周美珂、邝荣雨译. <a rel="nofollow" class="external text" href="https://archive.org/details/shuxuefenxi0002zhuo">《数学分析》第二卷</a>. 高等教育出版社. 2006. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-7-040-20257-1" title="Special:网络书源/978-7-040-20257-1"><span title="国际标准书号">ISBN</span> 978-7-040-20257-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%B5%81%E5%BD%A2&rft.au=B.A.%E5%8D%93%E9%87%8C%E5%A5%87+%E8%91%97%EF%BC%8C%E8%92%8B%E9%93%8E%E3%80%81%E9%92%B1%E4%BD%A9%E7%8E%B2%E3%80%81%E5%91%A8%E7%BE%8E%E7%8F%82%E3%80%81%E9%82%9D%E8%8D%A3%E9%9B%A8+%E8%AF%91&rft.btitle=%E3%80%8A%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90%E3%80%8B%E7%AC%AC%E4%BA%8C%E5%8D%B7&rft.date=2006&rft.genre=book&rft.isbn=978-7-040-20257-1&rft.pub=%E9%AB%98%E7%AD%89%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fshuxuefenxi0002zhuo&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>第175-183页.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation book">Chern, S.S; Weihuan Chen, Kai Shue Lam. <a rel="nofollow" class="external text" href="https://archive.org/details/lecturesondiffer0000cher">Lectures on Differential Geometry</a>. Singapore: World Scientific. 2000.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%B5%81%E5%BD%A2&rft.au=Chern%2C+S.S&rft.btitle=Lectures+on+Differential+Geometry&rft.date=2000&rft.genre=book&rft.place=Singapore&rft.pub=World+Scientific&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flecturesondiffer0000cher&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">引文使用过时参数<code style="color:inherit; border:inherit; padding:inherit;">coauthors</code> (<a href="/wiki/Help:%E5%BC%95%E6%96%87%E6%A0%BC%E5%BC%8F1%E9%94%99%E8%AF%AF#deprecated_params" title="Help:引文格式1错误">帮助</a>)</span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="一般參考"><span id=".E4.B8.80.E8.88.AC.E5.8F.83.E8.80.83"></span>一般參考</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=34" title="编辑章节：一般參考"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r80540462"><div class="refbegin columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2;"> <ul><li>Guillemin, Victor and Anton Pollack, <i>Differential Topology</i>, Prentice-Hall (1974) <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9780132126052" class="internal mw-magiclink-isbn">ISBN 978-0-13-212605-2</a>. This text was inspired by Milnor, and is commonly used for undergraduate courses.</li> <li>Hirsch, Morris, <i>Differential Topology</i>, Springer (1997) <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9780387901480" class="internal mw-magiclink-isbn">ISBN 978-0-387-90148-0</a>. Hirsch gives the most complete account with historical insights and excellent, but difficult problems. This is the best reference for those wishing to have a deep understanding of the subject.</li> <li><a href="/wiki/%E7%BE%85%E6%AF%94%E6%81%A9%C2%B7%E5%8D%A1%E6%AF%94" title="羅比恩·卡比">Kirby, Robion C</a>.; Siebenmann, Laurence C. <i>Foundational Essays on Topological Manifolds. Smoothings, and Triangulations</i>. Princeton, New Jersey: Princeton University Press (1977). <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9780691081908" class="internal mw-magiclink-isbn">ISBN 978-0-691-08190-8</a>. A detailed study of the <a href="/w/index.php?title=Category_theory&action=edit&redlink=1" class="new" title="Category theory（页面不存在）">category</a> of topological manifolds.</li> <li>Lee, John M. <i>Introduction to Topological Manifolds</i>, Springer-Verlag, New York (2000). <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9780387987590" class="internal mw-magiclink-isbn">ISBN 978-0-387-98759-0</a>. <i>Introduction to Smooth Manifolds</i>, Springer-Verlag, New York (2003). <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9780387954950" class="internal mw-magiclink-isbn">ISBN 978-0-387-95495-0</a>. Graduate-level textbooks on topological and smooth manifolds.</li> <li><a href="/wiki/%E7%BA%A6%E7%BF%B0%C2%B7%E7%B1%B3%E5%B0%94%E8%AF%BA" title="约翰·米尔诺">Milnor, John</a>, <i>Topology from the Differentiable Viewpoint</i>, Princeton University Press, (revised, 1997) <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9780691048338" class="internal mw-magiclink-isbn">ISBN 978-0-691-04833-8</a>. This short text may be the best math book ever written.</li> <li><a href="/wiki/Michael_Spivak" class="mw-redirect" title="Michael Spivak">Spivak, Michael</a>, <i>Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus</i>. HarperCollins Publishers (1965). <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9780805390216" class="internal mw-magiclink-isbn">ISBN 978-0-8053-9021-6</a>. This is the standard text used in most graduate courses.</li> <li><a href="/wiki/%E6%B3%A2%E6%81%A9%E5%93%88%E5%BE%B7%C2%B7%E9%BB%8E%E6%9B%BC" class="mw-redirect" title="波恩哈德·黎曼"> Riemann, Bernhard</a>, <a rel="nofollow" class="external text" href="http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/"><i>Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse</i></a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20210301150940/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）. The 1851 doctoral thesis in which "manifold" (<i>Mannigfaltigkeit</i>) first appears.</li> <li><a href="/wiki/%E6%B3%A2%E6%81%A9%E5%93%88%E5%BE%B7%C2%B7%E9%BB%8E%E6%9B%BC" class="mw-redirect" title="波恩哈德·黎曼"> Riemann, Bernhard</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160318034045/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/"><i>On the Hypotheses which lie at the Bases of Geometry</i></a>. The famous Göttingen inaugural lecture (Habilitationsschrift) of 1854.</li> <li><a href="/w/index.php?title=Richard_W._Sharpe&action=edit&redlink=1" class="new" title="Richard W. Sharpe（页面不存在）">Sharpe, R.W.</a>, <i>Differential Geometry:Cartan's Generalization of Klein's Erlangen Program</i>, Springer-Verlag, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9780387947327" class="internal mw-magiclink-isbn">ISBN 978-0-387-94732-7</a>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="外部連結"><span id=".E5.A4.96.E9.83.A8.E9.80.A3.E7.B5.90"></span>外部連結</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B5%81%E5%BD%A2&action=edit&section=35" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFHazewinkel2001" class="citation">Hazewinkel, Michiel (编), <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=p/m062210">Manifold</a>, <a href="/wiki/%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6" title="数学百科全书">数学百科全书</a>, <a href="/wiki/Springer_Science%2BBusiness_Media" class="mw-redirect" title="Springer Science+Business Media">Springer</a>, 2001, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-1-55608-010-4" title="Special:网络书源/978-1-55608-010-4"><span title="国际标准书号">ISBN</span> 978-1-55608-010-4</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%B5%81%E5%BD%A2&rft.atitle=Manifold&rft.aufirst=Michiel&rft.aulast=Hazewinkel&rft.btitle=%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6&rft.date=2001&rft.genre=bookitem&rft.isbn=978-1-55608-010-4&rft.pub=Springer&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3Dp%2Fm062210&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://www.dimensions-math.org/Dim_E.htm">Dimensions-math.org</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20081102233426/http://www.dimensions-math.org/Dim_E.htm">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） (A film explaining and visualizing manifolds up to fourth dimension.)</li> <li>The <a rel="nofollow" class="external text" href="http://www.map.mpim-bonn.mpg.de">manifold atlas</a> Portuguese Web Archive的<a rel="nofollow" class="external text" href="http://arquivo.pt/wayback/20160523161822/http%3A//www.map.mpim%2Dbonn.mpg.de/Main_Page">存檔</a>，存档日期2016-05-23 project of the <a rel="nofollow" class="external text" href="http://www.mpim-bonn.mpg.de">Max Planck Institute for Mathematics in Bonn</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/19970220112102/http://www.mpim-bonn.mpg.de/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li></ul> <div class="navbox-styles"><style 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