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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-定义" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#定义"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>定义</span> </div> </a> <ul id="toc-定义-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-闭曲面分类" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#闭曲面分类"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>闭曲面分类</span> </div> </a> <ul id="toc-闭曲面分类-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-紧曲面" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#紧曲面"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>紧曲面</span> </div> </a> <ul id="toc-紧曲面-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-在R3中的嵌入" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#在R3中的嵌入"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>在R<sup>3</sup>中的嵌入</span> </div> </a> <ul id="toc-在R3中的嵌入-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-微分几何" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#微分几何"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>微分几何</span> </div> </a> <ul id="toc-微分几何-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-一些模型" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#一些模型"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>一些模型</span> </div> </a> <ul id="toc-一些模型-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-基本多边形" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#基本多边形"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>基本多边形</span> </div> </a> <ul id="toc-基本多边形-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-曲面的连通和" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#曲面的连通和"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>曲面的连通和</span> </div> </a> <ul id="toc-曲面的连通和-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-代数曲面" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#代数曲面"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>代数曲面</span> </div> </a> <ul id="toc-代数曲面-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参看" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参看"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>参看</span> </div> </a> <ul id="toc-参看-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部链接" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部链接"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>外部链接</span> </div> </a> <ul id="toc-外部链接-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">曲面</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。71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-71" 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">71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-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Oppervlak" title="Oppervlak – 南非荷兰语" lang="af" hreflang="af" data-title="Oppervlak" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Fl%C3%A4che_(Topologie)" title="Fläche (Topologie) – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Fläche (Topologie)" 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-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Superficie" title="Superficie – 阿拉贡语" lang="an" hreflang="an" data-title="Superficie" data-language-autonym="Aragonés" data-language-local-name="阿拉贡语" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B3%D8%B7%D8%AD" title="سطح – 阿拉伯语" lang="ar" hreflang="ar" data-title="سطح" 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-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Superficie" title="Superficie – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Superficie" data-language-autonym="Asturianu" data-language-local-name="阿斯图里亚斯语" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/S%C9%99th" title="Səth – 阿塞拜疆语" lang="az" hreflang="az" data-title="Səth" 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-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D1%8F" title="Паверхня – 白俄罗斯语" lang="be" hreflang="be" data-title="Паверхня" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D1%8A%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82" title="Повърхност – 保加利亚语" lang="bg" hreflang="bg" data-title="Повърхност" 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%A4%E0%A6%B2_(%E0%A6%9F%E0%A6%AA%E0%A7%8B%E0%A6%B2%E0%A6%9C%E0%A6%BF)" title="তল (টপোলজি) – 孟加拉语" lang="bn" hreflang="bn" data-title="তল (টপোলজি)" 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-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Povr%C5%A1" title="Površ – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Površ" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Superf%C3%ADcie_(matem%C3%A0tiques)" title="Superfície (matemàtiques) – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Superfície (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-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D9%88%D9%88" title="ڕوو – 中库尔德语" lang="ckb" hreflang="ckb" data-title="ڕوو" data-language-autonym="کوردی" data-language-local-name="中库尔德语" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Plocha" title="Plocha – 捷克语" lang="cs" hreflang="cs" data-title="Plocha" 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/%C3%87%D0%B8%D0%B9" title="Çий – 楚瓦什语" lang="cv" hreflang="cv" data-title="Çий" 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-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Surface_(topology)" title="Surface (topology) – 英语" lang="en" hreflang="en" data-title="Surface (topology)" 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/Surfaco" title="Surfaco – 世界语" lang="eo" hreflang="eo" data-title="Surfaco" 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/Superficie_(topolog%C3%ADa)" title="Superficie (topología) – 西班牙语" lang="es" hreflang="es" data-title="Superficie (topología)" 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/Pind" title="Pind – 爱沙尼亚语" lang="et" hreflang="et" data-title="Pind" 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/Gainazal" title="Gainazal – 巴斯克语" lang="eu" hreflang="eu" data-title="Gainazal" 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/%D8%B1%D9%88%DB%8C%D9%87_(%D8%AA%D9%88%D9%BE%D9%88%D9%84%D9%88%DA%98%DB%8C)" title="رویه (توپولوژی) – 波斯语" lang="fa" hreflang="fa" data-title="رویه (توپولوژی)" 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/Pinta_(geometria)" title="Pinta (geometria) – 芬兰语" lang="fi" hreflang="fi" data-title="Pinta (geometria)" 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 badge-Q70893996 mw-list-item" title=""><a href="https://fr.wikipedia.org/wiki/Surface_(g%C3%A9om%C3%A9trie)" title="Surface (géométrie) – 法语" lang="fr" hreflang="fr" data-title="Surface (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-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Superficie" title="Superficie – 弗留利语" lang="fur" hreflang="fur" data-title="Superficie" data-language-autonym="Furlan" data-language-local-name="弗留利语" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Dromchla_(toipeola%C3%ADocht)" title="Dromchla (toipeolaíocht) – 爱尔兰语" lang="ga" hreflang="ga" data-title="Dromchla (toipeolaíocht)" data-language-autonym="Gaeilge" data-language-local-name="爱尔兰语" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Superficie" title="Superficie – 加利西亚语" lang="gl" hreflang="gl" data-title="Superficie" 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%9E%D7%A9%D7%98%D7%97_(%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94)" title="משטח (טופולוגיה) – 希伯来语" lang="he" hreflang="he" data-title="משטח (טופולוגיה)" 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-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%83%E0%A4%B7%E0%A5%8D%E0%A4%9F" title="पृष्ट – 印地语" lang="hi" hreflang="hi" data-title="पृष्ट" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Ploha_(geometrija)" title="Ploha (geometrija) – 克罗地亚语" lang="hr" hreflang="hr" data-title="Ploha (geometrija)" 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/Felsz%C3%ADn" title="Felszín – 匈牙利语" lang="hu" hreflang="hu" data-title="Felszín" 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/%D5%84%D5%A1%D5%AF%D5%A5%D6%80%D6%87%D5%B8%D6%82%D5%B5%D5%A9" title="Մակերևույթ – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Մակերևույթ" 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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Superficie" title="Superficie – 国际语" lang="ia" hreflang="ia" data-title="Superficie" data-language-autonym="Interlingua" data-language-local-name="国际语" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Permukaan_(topologi)" title="Permukaan (topologi) – 印度尼西亚语" lang="id" hreflang="id" data-title="Permukaan (topologi)" 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-inh mw-list-item"><a href="https://inh.wikipedia.org/wiki/%D0%A2%D3%80%D0%B5%D1%85%D0%B5" title="ТӀехе – 印古什语" lang="inh" hreflang="inh" data-title="ТӀехе" data-language-autonym="ГӀалгӀай" data-language-local-name="印古什语" class="interlanguage-link-target"><span>ГӀалгӀай</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Surfaco" title="Surfaco – 伊多语" lang="io" hreflang="io" data-title="Surfaco" data-language-autonym="Ido" data-language-local-name="伊多语" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Yfirbor%C3%B0" title="Yfirborð – 冰岛语" lang="is" hreflang="is" data-title="Yfirborð" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Superficie" title="Superficie – 意大利语" lang="it" hreflang="it" data-title="Superficie" 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/%E6%9B%B2%E9%9D%A2" title="曲面 – 日语" lang="ja" hreflang="ja" data-title="曲面" 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-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%B5%D1%82_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Бет (геометрия) – 哈萨克语" lang="kk" hreflang="kk" data-title="Бет (геометрия)" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%A1%EB%A9%B4" title="곡면 – 韩语" lang="ko" hreflang="ko" data-title="곡면" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%91%D0%B5%D1%82_(%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Бет (Геометрия) – 柯尔克孜语" lang="ky" hreflang="ky" data-title="Бет (Геометрия)" data-language-autonym="Кыргызча" data-language-local-name="柯尔克孜语" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Superfi%C3%A7ie_(matematica)" title="Superfiçie (matematica) – 利古里亚语" lang="lij" hreflang="lij" data-title="Superfiçie (matematica)" data-language-autonym="Ligure" data-language-local-name="利古里亚语" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pavir%C5%A1ius" title="Paviršius – 立陶宛语" lang="lt" hreflang="lt" data-title="Paviršius" 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-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Virsma" title="Virsma – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Virsma" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%86%E0%A4%A1" title="आड – 马拉地语" lang="mr" hreflang="mr" data-title="आड" data-language-autonym="मराठी" data-language-local-name="马拉地语" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Oppervlak_(topologie)" title="Oppervlak (topologie) – 荷兰语" lang="nl" hreflang="nl" data-title="Oppervlak (topologie)" 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/Flate" title="Flate – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Flate" 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/Flate" title="Flate – 书面挪威语" lang="nb" hreflang="nb" data-title="Flate" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Superf%C3%ADcia_(matematicas)" title="Superfícia (matematicas) – 奥克语" lang="oc" hreflang="oc" data-title="Superfícia (matematicas)" data-language-autonym="Occitan" data-language-local-name="奥克语" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Powierzchnia" title="Powierzchnia – 波兰语" lang="pl" hreflang="pl" data-title="Powierzchnia" 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/Surfassa" title="Surfassa – 皮埃蒙特文" lang="pms" hreflang="pms" data-title="Surfassa" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Superf%C3%ADcie" title="Superfície – 葡萄牙语" lang="pt" hreflang="pt" data-title="Superfície" 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/Suprafa%C8%9B%C4%83" title="Suprafață – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Suprafață" 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-rsk mw-list-item"><a href="https://rsk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%86" title="Поверхносц – Pannonian Rusyn" lang="rsk" hreflang="rsk" data-title="Поверхносц" data-language-autonym="Руски" data-language-local-name="Pannonian Rusyn" class="interlanguage-link-target"><span>Руски</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Поверхность – 俄语" lang="ru" hreflang="ru" data-title="Поверхность" 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/Povr%C5%A1" title="Površ – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Površ" 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/Surface" title="Surface – Simple English" lang="en-simple" hreflang="en-simple" data-title="Surface" 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/Povrch" title="Povrch – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Povrch" 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/Ploskev" title="Ploskev – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Ploskev" 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-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Chiso_(Chiumbwa)" title="Chiso (Chiumbwa) – 绍纳语" lang="sn" hreflang="sn" data-title="Chiso (Chiumbwa)" data-language-autonym="ChiShona" data-language-local-name="绍纳语" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Oogo_(dhul)" title="Oogo (dhul) – 索马里语" lang="so" hreflang="so" data-title="Oogo (dhul)" data-language-autonym="Soomaaliga" data-language-local-name="索马里语" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D1%80%D1%88" title="Површ – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Површ" 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/Yta" title="Yta – 瑞典语" lang="sv" hreflang="sv" data-title="Yta" 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-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%89%E0%B0%AA%E0%B0%B0%E0%B0%BF%E0%B0%A4%E0%B0%B2%E0%B0%82" title="ఉపరితలం – 泰卢固语" lang="te" hreflang="te" data-title="ఉపరితలం" data-language-autonym="తెలుగు" data-language-local-name="泰卢固语" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Y%C3%BCzey" title="Yüzey – 土耳其语" lang="tr" hreflang="tr" data-title="Yüzey" 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%9F%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D1%8F" title="Поверхня – 乌克兰语" lang="uk" hreflang="uk" data-title="Поверхня" 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-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B3%D8%B7%D8%AD" title="سطح – 乌尔都语" lang="ur" hreflang="ur" data-title="سطح" data-language-autonym="اردو" data-language-local-name="乌尔都语" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Sirt" title="Sirt – 乌兹别克语" lang="uz" hreflang="uz" data-title="Sirt" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Superficie" title="Superficie – 威尼斯语" lang="vec" hreflang="vec" data-title="Superficie" data-language-autonym="Vèneto" data-language-local-name="威尼斯语" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%E1%BA%B7t_(t%C3%B4_p%C3%B4)" title="Mặt (tô pô) – 越南语" lang="vi" hreflang="vi" data-title="Mặt (tô pô)" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%9B%B2%E9%9D%A2" title="曲面 – 吴语" lang="wuu" hreflang="wuu" data-title="曲面" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%9B%B2%E9%9D%A2" title="曲面 – 粤语" lang="yue" hreflang="yue" data-title="曲面" data-language-autonym="粵語" data-language-local-name="粤语" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q484298#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon 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<div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav 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 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.ambox-protection{border-left-color:#a2a9b1!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-os .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-Expand plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Antistub.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Antistub.svg/44px-Antistub.svg.png" decoding="async" width="44" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Antistub.svg/66px-Antistub.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Antistub.svg/88px-Antistub.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目<b>需要擴充。</b><span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2016年7月25日</span>)</i></small><span class="hide-when-compact"><br /><small>请協助<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit">改善这篇條目</a>，更進一步的信息可能會在<a href="/wiki/Talk:%E6%9B%B2%E9%9D%A2" title="Talk:曲面">討論頁</a>或<a href="/wiki/Wikipedia:%E6%89%A9%E5%85%85%E8%AF%B7%E6%B1%82" title="Wikipedia:扩充请求">扩充请求</a>中找到。请在擴充條目後將此模板移除。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Saddle_pt.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Saddle_pt.jpg/300px-Saddle_pt.jpg" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Saddle_pt.jpg/450px-Saddle_pt.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Saddle_pt.jpg/600px-Saddle_pt.jpg 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>显示了X-, Y-, Z-轮廓的一个开曲面。</figcaption></figure> <p><b><style data-mw-deduplicate="TemplateStyles:r80905552">.mw-parser-output ruby.zy{text-align:justify;text-justify:none}.mw-parser-output ruby.zy>rp{user-select:none}.mw-parser-output ruby.zy>rt{font-feature-settings:"ruby"1;font-size:90%;user-select:none}.mw-parser-output ruby.zy>rt+rt{padding-left:0.15em}</style><ruby class="zy center"><rb>曲</rb><rp>（</rp><rt lang="zh-Latn-pinyin">qū</rt><rp>）</rp></ruby>面</b>（curved surface，surface）是实体的边缘或空间中的曲线依某特定条件运动所形成的轨迹，例如球面、圆柱面、圆锥面等。 </p><p>在<a href="/wiki/%E6%95%B0%E5%AD%A6" title="数学">数学</a>（<a href="/wiki/%E6%8B%93%E6%89%91%E5%AD%A6" title="拓扑学">拓扑学</a>）中，一个<b>曲面</b>（surface）是一个二维<a href="/wiki/%E6%B5%81%E5%BD%A2" title="流形">流形</a>。三维空间中的例子有三维实心物体的边界。<a href="/wiki/%E6%B5%81%E4%BD%93" title="流体">流体</a>的表面，例如<a href="/wiki/%E9%9B%A8%E6%BB%B4" class="mw-redirect" title="雨滴">雨滴</a>或<a href="/wiki/%E8%82%A5%E7%9A%82%E6%B3%A1" title="肥皂泡">肥皂泡</a>是一种理想化的曲面。关于<a href="/wiki/%E9%9B%AA%E8%8A%B1" title="雪花">雪花</a>的表面，它有很多精细的结构，超越了这个简单的数学定义。关于实际的曲面的资料，请参看<a href="/wiki/%E8%A1%A8%E9%9D%A2%E5%BC%A0%E5%8A%9B" title="表面张力">表面张力</a>，<a href="/wiki/%E8%A1%A8%E9%9D%A2%E5%8C%96%E5%AD%B8" class="mw-redirect" title="表面化學">表面化学</a>，<a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2%E8%83%BD%E9%87%8F&action=edit&redlink=1" class="new" title="曲面能量（页面不存在）">曲面能量</a>。 </p> <meta property="mw:PageProp/toc" /> <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%9B%B2%E9%9D%A2&action=edit&section=1" title="编辑章节：定义"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在下文中，所有曲面视为<a href="/w/index.php?title=%E7%AC%AC%E4%BA%8C%E5%8F%AF%E6%95%B0&action=edit&redlink=1" class="new" title="第二可数（页面不存在）">第二可数</a>2-维流形。 </p><p>更精确一点的讲：一个<b>拓扑（带边界）曲面</b>是一个<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>，其中每点有一个开<a href="/wiki/%E9%82%BB%E5%9F%9F" title="邻域">邻域</a><a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a>于或者一个<i>E<sup>2</sup></i>的<a href="/wiki/%E5%BC%80%E9%9B%86" title="开集">开子集</a>或者<i>E<sup>2</sup></i>的闭的一半的开子集。 </p><p>有一个同胚于<i>E<sup>n</sup></i>的开子集的点的集合称为流形的<b>内部</b>；它总是非空的。内部的补集称为<b>边界</b>；它是一个（<i>1</i>）流形，或者说闭曲线的并集。 </p><p>无边界的曲面称为<b>闭</b>的，如果它是<a href="/wiki/%E7%B4%A7%E9%9B%86" class="mw-redirect" title="紧集">紧</a>的，否则称为<b>开</b>。 </p> <div class="mw-heading mw-heading2"><h2 id="闭曲面分类"><span id=".E9.97.AD.E6.9B.B2.E9.9D.A2.E5.88.86.E7.B1.BB"></span>闭曲面分类</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit&section=2" title="编辑章节：闭曲面分类"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>闭（<a href="/wiki/%E7%B4%A7" class="mw-redirect" title="紧">紧</a>无<a href="/wiki/%E8%BE%B9%E7%95%8C" title="边界">边界</a>）连通曲面有完整的分类，同类的曲面至多相差一个同胚。所有这种曲面属于下面三个无穷多的集合之一： </p> <ul><li><a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a>加上<i>n</i>个柄（称为<i>n-环</i>）。这些是<a href="/wiki/%E6%AC%A7%E6%8B%89%E7%A4%BA%E6%80%A7%E6%95%B0" title="欧拉示性数">欧拉示性数</a>为2-2<i>n</i>的<a href="/wiki/%E5%8F%AF%E5%AE%9A%E5%90%91" class="mw-redirect" title="可定向">可定向</a>曲面，也称为亏格（genus）为<i>n</i>的曲面。</li> <li>带<i>n</i>个柄的<a href="/wiki/%E5%B0%84%E5%BD%B1%E5%B9%B3%E9%9D%A2" title="射影平面">射影平面</a>（Projective plane）。这些是欧拉特征数为1-2<i>n</i>的不可定向曲面。</li> <li>带<i>n</i>个柄的<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>。这些是欧拉特征数为-2<i>n</i>的不可定向曲面。</li></ul> <p>所以<a href="/wiki/%E6%AC%A7%E6%8B%89%E7%A4%BA%E6%80%A7%E6%95%B0" title="欧拉示性数">欧拉示性数</a>和<a href="/wiki/%E5%8F%AF%E5%AE%9A%E5%90%91%E6%80%A7" title="可定向性">可定向性</a>描述了一个紧曲面除了可能的<a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a>（若曲面光滑则为<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>）. </p> <div class="mw-heading mw-heading2"><h2 id="紧曲面"><span id=".E7.B4.A7.E6.9B.B2.E9.9D.A2"></span>紧曲面</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit&section=3" title="编辑章节：紧曲面"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>带边界紧曲面就是有一个或多个开<a href="/wiki/%E5%9C%86%E7%9B%98" title="圆盘">圆盘</a>被取掉的曲面，而且这些圆盘的闭包互不相交。 </p> <div class="mw-heading mw-heading2"><h2 id="在R3中的嵌入"><span id=".E5.9C.A8R3.E4.B8.AD.E7.9A.84.E5.B5.8C.E5.85.A5"></span>在R<sup>3</sup>中的嵌入</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit&section=4" title="编辑章节：在R3中的嵌入"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一个紧曲面可以嵌入到<b>R</b><sup>3</sup>，只要它可定向或有非空边界。<a href="/w/index.php?title=Whitney%E5%B5%8C%E5%85%A5%E5%AE%9A%E7%90%86&action=edit&redlink=1" class="new" title="Whitney嵌入定理（页面不存在）">Whitney嵌入定理</a>的结果表明任何曲面可以嵌入<b>R</b><sup>4</sup>. </p> <div class="mw-heading mw-heading2"><h2 id="微分几何"><span id=".E5.BE.AE.E5.88.86.E5.87.A0.E4.BD.95"></span>微分几何</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit&section=5" title="编辑章节：微分几何"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>曲面在<i>n</i>维的嵌入的简单回顾和这样一个曲面的面积的计算可以在<a href="/wiki/%E4%BD%93%E7%A7%AF%E5%BD%A2%E5%BC%8F" title="体积形式">体积形式</a>条目中找到。<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>的度量性质在条目<a href="/wiki/%E5%BA%9E%E5%8A%A0%E8%8E%B1%E5%BA%A6%E9%87%8F" title="庞加莱度量">庞加莱度量</a>中有简单介绍。 </p> <div class="mw-heading mw-heading2"><h2 id="一些模型"><span id=".E4.B8.80.E4.BA.9B.E6.A8.A1.E5.9E.8B"></span>一些模型</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit&section=6" title="编辑章节：一些模型"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>把下面这些的边贴起来可以得到一些模型： </p> <pre> * * B B v v v ^ *>>>>>* *>>>>>* v v v ^ v v v v A v v A A v ^ A A v v A A v v A v v v ^ v v v v v v v ^ *<<<<<* *>>>>>* * * B B </pre> <pre> <a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a> <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> <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> <a href="/wiki/%E7%8E%AF%E9%9D%A2" title="环面">环面</a> (打了孔的莫比乌斯带) (面包圈) </pre> <div class="mw-heading mw-heading2"><h2 id="基本多边形"><span id=".E5.9F.BA.E6.9C.AC.E5.A4.9A.E8.BE.B9.E5.BD.A2"></span>基本多边形</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit&section=7" title="编辑章节：基本多边形"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>每个闭曲面可以从一个偶数边可定向多边形通过将边成对等同构造出来，该多边形称为<b><a href="/wiki/%E5%9F%BA%E6%9C%AC%E5%A4%9A%E8%BE%B9%E5%BD%A2" title="基本多边形">基本多边形</a></b>。 </p><p>这个构造可以用一串长度2n的包含n个不同符号的字符串表示，每个符号出现两次，可以带+1或-1指数。指数-1表示该边的方向和基本多边形的方向相反。 </p><p>上面的模型可作如下描述： </p> <ul><li>球面：<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 AA^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AA^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c325d9ca0ff68c692e09962f86053602766821e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.819ex; height:2.676ex;" alt="{\displaystyle AA^{-1}}"></span></li> <li>射影平面：<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 AA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4bbbeba3c9275c1b8612ed282f8f1347c9629b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.486ex; height:2.176ex;" alt="{\displaystyle AA}"></span></li> <li>克莱因瓶：<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 ABA^{-1}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABA^{-1}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/117b6784024382aecdc67aedeaef6a2632e9cde8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.347ex; height:2.676ex;" alt="{\displaystyle ABA^{-1}B}"></span></li> <li>环面：<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 ABA^{-1}B^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABA^{-1}B^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6caa444b9e9549d0ae674d9805da32d35f3993f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.68ex; height:2.676ex;" alt="{\displaystyle ABA^{-1}B^{-1}}"></span></li></ul> <p><i>（细节请见</i><a href="/wiki/%E5%9F%BA%E6%9C%AC%E5%A4%9A%E8%BE%B9%E5%BD%A2" title="基本多边形">基本多边形</a><i>。）</i> </p> <div class="mw-heading mw-heading2"><h2 id="曲面的连通和"><span id=".E6.9B.B2.E9.9D.A2.E7.9A.84.E8.BF.9E.E9.80.9A.E5.92.8C"></span>曲面的连通和</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit&section=8" title="编辑章节：曲面的连通和"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>给定两个曲面M和M'，他们的<b>连通和(connected sum)</b> M # M' 可以通过在每个曲面上除去一个圆盘再把他们在新的边界分量上粘起来。 </p><p>我们采用下面的记号。 </p> <ul><li>球：S</li> <li>环：T</li> <li>克莱因瓶：K</li> <li>射影平面：P</li></ul> <p>一些结果： </p> <ul><li>S # S = S</li> <li>S # M = M</li> <li>P # P = K</li> <li>P # K = P # T</li></ul> <p>我们用一些缩略记法：nM = M # M # ... # M（n次）以及 0M = S. </p><p>闭曲面可以分类如下： </p> <ul><li>gT（g-叠环）：亏格为g的可定向曲面。</li> <li>gP（g-叠射影平面）：亏格为g的不可定向曲面。</li></ul> <div class="mw-heading mw-heading2"><h2 id="代数曲面"><span id=".E4.BB.A3.E6.95.B0.E6.9B.B2.E9.9D.A2"></span>代数曲面</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit&section=9" title="编辑章节：代数曲面"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>曲面的概念和<i><a href="/w/index.php?title=%E4%BB%A3%E6%95%B0%E6%9B%B2%E9%9D%A2&action=edit&redlink=1" class="new" title="代数曲面（页面不存在）">代数曲面</a></i>不同。一个非奇异复射影<a href="/wiki/%E4%BB%A3%E6%95%B0%E6%9B%B2%E7%BA%BF" class="mw-redirect" title="代数曲线">代数曲线</a>是一个光滑曲面。<a href="/wiki/%E8%A4%87%E6%95%B8_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="複數 (數學)">複數</a>域上的代数曲面作为流形考虑时维度是4。 </p> <div class="mw-heading mw-heading2"><h2 id="参看"><span id=".E5.8F.82.E7.9C.8B"></span>参看</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit&section=10" title="编辑章节：参看"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%9E%81%E5%B0%8F%E6%9B%B2%E9%9D%A2" title="极小曲面">极小曲面</a></li> <li><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> <li><a href="/w/index.php?title=%E4%BB%A3%E6%95%B0%E6%9B%B2%E9%9D%A2&action=edit&redlink=1" class="new" title="代数曲面（页面不存在）">代数曲面</a></li> <li><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></li> <li><a href="/wiki/%E7%8E%AF%E9%9D%A2" title="环面">环面</a></li> <li><a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a></li> <li><a href="/wiki/%E5%9C%86%E6%9F%B1" class="mw-redirect" title="圆柱">圆柱</a></li> <li><a href="/wiki/%E5%B0%84%E5%BD%B1%E5%B9%B3%E9%9D%A2" title="射影平面">射影平面</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="外部链接"><span id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5"></span>外部链接</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%9B%B2%E9%9D%A2&action=edit&section=11" title="编辑章节：外部链接"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://xahlee.org/surface/gallery.html">数学曲面画廊，60 ~ 曲面和提供实时转动观看功能的Java Applet</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20050829024319/http://xahlee.org/surface/gallery.html">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 data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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