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class="vector-toc-numb">3.1</span> <span>Linear isometry</span> </div> </a> <ul id="toc-Linear_isometry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Manifold" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Manifold"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Manifold</span> </div> </a> <button aria-controls="toc-Manifold-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>Toggle Manifold subsection</span> </button> <ul id="toc-Manifold-sublist" class="vector-toc-list"> <li id="toc-Definition_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-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="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <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="Toggle the table of contents" > <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">Toggle the table of contents</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">Isometry</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="Go to an article in another language. Available in 36 languages" > <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-36" 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">36 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%82%D8%A7%D9%8A%D8%B3" title="تقايس – Arabic" lang="ar" hreflang="ar" data-title="تقايس" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Isometria" title="Isometria – Catalan" lang="ca" hreflang="ca" data-title="Isometria" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Изометри (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Изометри (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Izometrick%C3%A9_zobrazen%C3%AD" title="Izometrické zobrazení – Czech" lang="cs" hreflang="cs" data-title="Izometrické zobrazení" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Isometreg" title="Isometreg – Welsh" lang="cy" hreflang="cy" data-title="Isometreg" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Isometrie" title="Isometrie – German" lang="de" hreflang="de" data-title="Isometrie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Isomeetria" title="Isomeetria – Estonian" lang="et" hreflang="et" data-title="Isomeetria" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Isometr%C3%ADa" title="Isometría – Spanish" lang="es" hreflang="es" data-title="Isometría" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Izometrio" title="Izometrio – Esperanto" lang="eo" hreflang="eo" data-title="Izometrio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Isometria" title="Isometria – Basque" lang="eu" hreflang="eu" data-title="Isometria" data-language-autonym="Euskara" data-language-local-name="Basque" 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%A7%DB%8C%D8%B2%D9%88%D9%85%D8%AA%D8%B1%DB%8C" title="ایزومتری – Persian" lang="fa" hreflang="fa" data-title="ایزومتری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Isom%C3%A9trie" title="Isométrie – French" lang="fr" hreflang="fr" data-title="Isométrie" data-language-autonym="Français" data-language-local-name="French" 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/Isometr%C3%ADa" title="Isometría – Galician" lang="gl" hreflang="gl" data-title="Isometría" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%93%B1%EA%B1%B0%EB%A6%AC%EB%B3%80%ED%99%98" title="등거리변환 – Korean" lang="ko" hreflang="ko" data-title="등거리변환" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Isometri" title="Isometri – Indonesian" lang="id" hreflang="id" data-title="Isometri" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" 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/Isometria" title="Isometria – Italian" lang="it" hreflang="it" data-title="Isometria" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%96%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94" title="איזומטריה – Hebrew" lang="he" hreflang="he" data-title="איזומטריה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%9A%D0%BE%D0%B7%D2%93%D0%B0%D0%BB%D1%8B%D1%81_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Қозғалыс (математика) – Kazakh" lang="kk" hreflang="kk" data-title="Қозғалыс (математика)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Egybev%C3%A1g%C3%B3s%C3%A1gi_transzform%C3%A1ci%C3%B3" title="Egybevágósági transzformáció – Hungarian" lang="hu" hreflang="hu" data-title="Egybevágósági transzformáció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Isometrie_(wiskunde)" title="Isometrie (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Isometrie (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%AD%89%E9%95%B7%E5%86%99%E5%83%8F" title="等長写像 – Japanese" lang="ja" hreflang="ja" data-title="等長写像" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Isometri" title="Isometri – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Isometri" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Isometri" title="Isometri – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Isometri" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%86%E0%A8%87%E0%A8%B8%E0%A9%8B%E0%A8%AE%E0%A9%80%E0%A8%9F%E0%A8%B0%E0%A9%80" title="ਆਇਸੋਮੀਟਰੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਆਇਸੋਮੀਟਰੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" 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/Izometria" title="Izometria – Polish" lang="pl" hreflang="pl" data-title="Izometria" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Isometria_(geometria)" title="Isometria (geometria) – Portuguese" lang="pt" hreflang="pt" data-title="Isometria (geometria)" data-language-autonym="Português" data-language-local-name="Portuguese" 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/Izometrie" title="Izometrie – Romanian" lang="ro" hreflang="ro" data-title="Izometrie" data-language-autonym="Română" data-language-local-name="Romanian" 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%98%D0%B7%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Изометрия (математика) – Russian" lang="ru" hreflang="ru" data-title="Изометрия (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Isometry" title="Isometry – Simple English" lang="en-simple" hreflang="en-simple" data-title="Isometry" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Togi_premik" title="Togi premik – Slovenian" lang="sl" hreflang="sl" data-title="Togi premik" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Isometria" title="Isometria – Finnish" lang="fi" hreflang="fi" data-title="Isometria" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Isometri" title="Isometri – Swedish" lang="sv" hreflang="sv" data-title="Isometri" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%AE%E0%AE%B5%E0%AE%B3%E0%AE%B5%E0%AF%88_%E0%AE%89%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="சமவளவை உருமாற்றம் – Tamil" lang="ta" hreflang="ta" data-title="சமவளவை உருமாற்றம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%B7%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ізометрія (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Ізометрія (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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/Ph%C3%A9p_%C4%91%E1%BA%B3ng_c%E1%BB%B1" title="Phép đẳng cự – Vietnamese" lang="vi" hreflang="vi" data-title="Phép đẳng cự" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%AD%89%E8%B7%9D%E5%90%8C%E6%9E%84" title="等距同构 – Chinese" lang="zh" hreflang="zh" data-title="等距同构" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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/Q740207#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</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="Namespaces"> <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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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Distance-preserving mathematical transformation</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about distance-preserving functions. For other mathematical uses, see <a href="/wiki/Isometry_(disambiguation)" class="mw-disambig" title="Isometry (disambiguation)">isometry (disambiguation)</a>. For non-mathematical uses, see <a href="/wiki/Isometric_(disambiguation)" class="mw-redirect mw-disambig" title="Isometric (disambiguation)">Isometric</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Isometric_projection" title="Isometric projection">Isometric projection</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.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 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Merge_from plainlinks metadata ambox ambox-move" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Mergefrom.svg/50px-Mergefrom.svg.png" decoding="async" width="50" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Mergefrom.svg/75px-Mergefrom.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Mergefrom.svg/100px-Mergefrom.svg.png 2x" data-file-width="50" data-file-height="20" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">It has been suggested that <i><a href="/wiki/Isometry_(disambiguation)" class="mw-disambig" title="Isometry (disambiguation)">Isometry (disambiguation)</a></i> be <a href="/wiki/Wikipedia:Merging" title="Wikipedia:Merging">merged</a> into this article. (<a href="/wiki/Talk:Isometry#Proposed_merge_of_Isometry_(disambiguation)_into_Isometry" title="Talk:Isometry">Discuss</a>)<small><i> Proposed since November 2024.</i></small></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Academ_Reflections_with_parallel_axis_on_wallpaper.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Academ_Reflections_with_parallel_axis_on_wallpaper.svg/310px-Academ_Reflections_with_parallel_axis_on_wallpaper.svg.png" decoding="async" width="310" height="310" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Academ_Reflections_with_parallel_axis_on_wallpaper.svg/465px-Academ_Reflections_with_parallel_axis_on_wallpaper.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Academ_Reflections_with_parallel_axis_on_wallpaper.svg/620px-Academ_Reflections_with_parallel_axis_on_wallpaper.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>A <a href="/wiki/Function_composition" title="Function composition">composition</a> of two <a href="/wiki/Euclidean_group#Direct_and_indirect_isometries" title="Euclidean group">opposite</a> isometries is a direct isometry. <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">A reflection</a> in a line is an opposite isometry, like <span class="texhtml"><i>R</i><sub> 1</sub></span> or <span class="texhtml"><i>R</i><sub> 2</sub></span> on the image. <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">Translation</a> <span class="texhtml"><i>T</i></span> is a direct isometry: <a href="/wiki/Rigid_body" title="Rigid body">a rigid motion</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>In mathematics, an <b>isometry</b> (or <b>congruence</b>, or <b>congruent transformation</b>) is a <a href="/wiki/Distance" title="Distance">distance</a>-preserving transformation between <a href="/wiki/Metric_space" title="Metric space">metric spaces</a>, usually assumed to be <a href="/wiki/Bijection" title="Bijection">bijective</a>.<sup id="cite_ref-CoxeterIsometryDef_3-0" class="reference"><a href="#cite_note-CoxeterIsometryDef-3"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> The word isometry is derived from the <a href="/wiki/Ancient_Greek" title="Ancient Greek">Ancient Greek</a>: ἴσος <i>isos</i> meaning "equal", and μέτρον <i>metron</i> meaning "measure". If the transformation is from a metric space to itself, it is a kind of <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformation</a> known as a <a href="/wiki/Motion_(geometry)" title="Motion (geometry)">motion</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a <a href="/wiki/Transformation_(geometry)" class="mw-redirect" title="Transformation (geometry)">transformation</a> which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, two geometric figures are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> if they are related by an isometry;<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> the isometry that relates them is either a rigid motion (translation or rotation), or a <a href="/wiki/Function_composition" title="Function composition">composition</a> of a rigid motion and a <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a>. </p><p>Isometries are often used in constructions where one space is <a href="/wiki/Embedding" title="Embedding">embedded</a> in another space. For instance, the <a href="/wiki/Complete_space#Completion" class="mw-redirect" title="Complete space">completion</a> of a metric space <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> involves an isometry from <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> into <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"> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo>,</mo> </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/25107a19b72535114c13007143129e6d150d70c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.83ex; height:2.843ex;" alt="{\displaystyle M',}" /></span> a <a href="/wiki/Quotient_set" class="mw-redirect" title="Quotient set">quotient set</a> of the space of <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a> on <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> <mo>.</mo> </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/b865c33e30eb83000cd6387517c66dbbf3c3df9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.089ex; height:2.176ex;" alt="{\displaystyle M.}" /></span> The original space <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> is thus isometrically <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> to a subspace of a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</a>, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a <a href="/wiki/Closed_set" title="Closed set">closed subset</a> of some <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> and that every complete metric space is isometrically isomorphic to a closed subset of some <a href="/wiki/Banach_space" title="Banach space">Banach space</a>. </p><p>An isometric surjective linear operator on a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> is called a <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operator</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></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 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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /></span> be <a href="/wiki/Metric_space" title="Metric space">metric spaces</a> with metrics (e.g., distances) <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="{\textstyle d_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/638fb5e5478ba7260bb0204e389ee23000b2792e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.841ex; height:2.509ex;" alt="{\textstyle d_{X}}" /></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="{\textstyle d_{Y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d_{Y}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83c2601116c53d1f93b46898262141d76ad6f8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.342ex; height:2.509ex;" alt="{\textstyle d_{Y}.}" /></span> A <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">map</a> <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="{\textstyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de4d6ef0299af482ee990efbedcd412ff2fae8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\textstyle f\colon X\to Y}" /></span> is called an <b>isometry</b> or <b>distance-preserving map</b> if for any <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,b\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5774e455790a994edd7fe7fb9225c92a974a1295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.082ex; height:2.509ex;" alt="{\displaystyle a,b\in X}" /></span>, </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 d_{X}(a,b)=d_{Y}\!\left(f(a),f(b)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{X}(a,b)=d_{Y}\!\left(f(a),f(b)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47678a339079d37fdf82fab611a41158dcf829c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.986ex; height:2.843ex;" alt="{\displaystyle d_{X}(a,b)=d_{Y}\!\left(f(a),f(b)\right).}" /></span><sup id="cite_ref-Beckman-Quarles-1953_6-0" class="reference"><a href="#cite_note-Beckman-Quarles-1953-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup></dd></dl> <p>An isometry is automatically <a href="/wiki/Injective_function" title="Injective function">injective</a>;<sup id="cite_ref-CoxeterIsometryDef_3-1" class="reference"><a href="#cite_note-CoxeterIsometryDef-3"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> otherwise two distinct points, <i>a</i> and <i>b</i>, could be mapped to the same point, thereby contradicting the coincidence axiom of the metric <i>d</i>, i.e., <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 d(a,b)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(a,b)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377648ce1bee22b8cc977d99a36410a1c693dbe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.547ex; height:2.843ex;" alt="{\displaystyle d(a,b)=0}" /></span> if and only 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 a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}" /></span>. This proof is similar to the proof that an <a href="/wiki/Order_embedding" title="Order embedding">order embedding</a> between <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered sets</a> is injective. Clearly, every isometry between metric spaces is a <a href="/wiki/Topological_embedding" class="mw-redirect" title="Topological embedding">topological embedding</a>. </p><p>A <b>global isometry</b>, <b>isometric isomorphism</b> or <b>congruence mapping</b> is a <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> isometry. Like any other bijection, a global isometry has a <a href="/wiki/Function_inverse" class="mw-redirect" title="Function inverse">function inverse</a>. The inverse of a global isometry is also a global isometry. </p><p>Two metric spaces <i>X</i> and <i>Y</i> are called <b>isometric</b> if there is a bijective isometry from <i>X</i> to <i>Y</i>. The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of bijective isometries from a metric space to itself forms a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> with respect to <a href="/wiki/Function_composition" title="Function composition">function composition</a>, called the <b><a href="/wiki/Isometry_group" title="Isometry group">isometry group</a></b>. </p><p>There is also the weaker notion of <i>path isometry</i> or <i>arcwise isometry</i>: </p><p>A <b>path isometry</b> or <b>arcwise isometry</b> is a map which preserves the <a href="/wiki/Arc_length#Definition" title="Arc length">lengths of curves</a>; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This term is often abridged to simply <i>isometry</i>, so one should take care to determine from context which type is intended. </p> <dl><dt>Examples</dt></dl> <ul><li>Any <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a>, <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> and <a href="/wiki/Rotation" title="Rotation">rotation</a> is a global isometry on <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a>. See also <a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean group</a> and <a href="/wiki/Euclidean_space#Isometries" title="Euclidean space">Euclidean space § Isometries</a>.</li> <li>The map <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|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto |x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f7cb7bcc9c126b2d18614065cfeeb20f8e5514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.567ex; height:2.843ex;" alt="{\displaystyle x\mapsto |x|}" /></span> in <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /></span> is a <i>path isometry</i> but not a (general) isometry. Note that unlike an isometry, this path isometry does not need to be injective.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Isometries_between_normed_spaces">Isometries between normed spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=3" title="Edit section: Isometries between normed spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following theorem is due to Mazur and Ulam. </p> <dl><dd><b>Definition</b>:<sup id="cite_ref-FOOTNOTENariciBeckenstein2011275–339_10-0" class="reference"><a href="#cite_note-FOOTNOTENariciBeckenstein2011275–339-10"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The <b>midpoint</b> of two elements <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> in a vector space is the vector <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>(<i>x</i> + <i>y</i>)</span>.</dd></dl> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem<sup id="cite_ref-FOOTNOTENariciBeckenstein2011275–339_10-1" class="reference"><a href="#cite_note-FOOTNOTENariciBeckenstein2011275–339-10"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEWilansky201321–26_11-0" class="reference"><a href="#cite_note-FOOTNOTEWilansky201321–26-11"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret">—</span>Let <span class="texhtml"><i>A</i> : <i>X</i> → <i>Y</i></span> be a surjective isometry between <a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">normed spaces</a> that maps 0 to 0 (<a href="/wiki/Stefan_Banach" title="Stefan Banach">Stefan Banach</a> called such maps <b>rotations</b>) where note that <span class="texhtml mvar" style="font-style:italic;">A</span> is <i>not</i> assumed to be a <i>linear</i> isometry. Then <span class="texhtml mvar" style="font-style:italic;">A</span> maps midpoints to midpoints and is linear as a map over the real numbers <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /></span>. If <span class="texhtml mvar" style="font-style:italic;">X</span> and <span class="texhtml mvar" style="font-style:italic;">Y</span> are complex vector spaces then <span class="texhtml mvar" style="font-style:italic;">A</span> may fail to be linear as a map over <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /></span>. </p> </div> <div class="mw-heading mw-heading3"><h3 id="Linear_isometry">Linear isometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=4" title="Edit section: Linear isometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given two <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector spaces</a> <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" /></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 W,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e52f6de64b3b96ce1db5651329f808cca7d3b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.082ex; height:2.509ex;" alt="{\displaystyle W,}" /></span> a <b>linear isometry</b> is a <a href="/wiki/Linear_map" title="Linear map">linear map</a> <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:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b4b6380b2f4700cea977536a2af33d51efb89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.517ex; height:2.176ex;" alt="{\displaystyle A:V\to W}" /></span> that preserves the norms: </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 \|Av\|_{W}=\|v\|_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>A</mi> <mi>v</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|Av\|_{W}=\|v\|_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/146fac3ddc73a50d2e7c0a822b929324cd4c9b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.197ex; height:2.843ex;" alt="{\displaystyle \|Av\|_{W}=\|v\|_{V}}" /></span></dd></dl> <p>for all <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 v\in V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13375e380c5699070639c00dcc62c3d91b05c7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.402ex; height:2.176ex;" alt="{\displaystyle v\in V.}" /></span><sup id="cite_ref-Thomsen_2017_p125_12-0" class="reference"><a href="#cite_note-Thomsen_2017_p125-12"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a>. </p><p>In an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>, the above definition reduces to </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 \langle v,v\rangle _{V}=\langle Av,Av\rangle _{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo>,</mo> <mi>v</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mi>v</mi> <mo>,</mo> <mi>A</mi> <mi>v</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v,v\rangle _{V}=\langle Av,Av\rangle _{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1fde7e5a43fc2f367de8613966267901372a91e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.232ex; height:2.843ex;" alt="{\displaystyle \langle v,v\rangle _{V}=\langle Av,Av\rangle _{W}}" /></span></dd></dl> <p>for all <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 v\in V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0415927601750a7c5a284675cdf3b50c6481c294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.402ex; height:2.509ex;" alt="{\displaystyle v\in V,}" /></span> which is equivalent to saying that <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^{\dagger }A=\operatorname {Id} _{V}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mi>A</mi> <mo>=</mo> <msub> <mi>Id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\dagger }A=\operatorname {Id} _{V}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f955ac014227a7091dd4a08b6530357b58fdc03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.822ex; height:3.009ex;" alt="{\displaystyle A^{\dagger }A=\operatorname {Id} _{V}.}" /></span> This also implies that isometries preserve inner products, as </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 \langle Au,Av\rangle _{W}=\langle u,A^{\dagger }Av\rangle _{V}=\langle u,v\rangle _{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mi>u</mi> <mo>,</mo> <mi>A</mi> <mi>v</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mi>A</mi> <mi>v</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle Au,Av\rangle _{W}=\langle u,A^{\dagger }Av\rangle _{V}=\langle u,v\rangle _{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50cf21fbb48a565a0ca0a03b2d24abb807fc683f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.979ex; height:3.176ex;" alt="{\displaystyle \langle Au,Av\rangle _{W}=\langle u,A^{\dagger }Av\rangle _{V}=\langle u,v\rangle _{V}}" /></span>.</dd></dl> <p>Linear isometries are not always <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operators</a>, though, as those require additionally that <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 V=W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/740038d36bd79466d6938d73b83fe737161fa1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.321ex; height:2.176ex;" alt="{\displaystyle V=W}" /></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 AA^{\dagger }=\operatorname {Id} _{V}}"> <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> </mrow> </msup> <mo>=</mo> <msub> <mi>Id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AA^{\dagger }=\operatorname {Id} _{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72f180ab5fd9ce72f27ffb791383974099fdc48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.175ex; height:3.009ex;" alt="{\displaystyle AA^{\dagger }=\operatorname {Id} _{V}}" /></span> (i.e. the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> and <a href="/wiki/Codomain" title="Codomain">codomain</a> coincide 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 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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> defines a <a href="/wiki/Unitary_operator" title="Unitary operator">coisometry</a>). </p><p>By the <a href="/wiki/Mazur%E2%80%93Ulam_theorem" title="Mazur–Ulam theorem">Mazur–Ulam theorem</a>, any isometry of normed vector spaces over <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /></span> is <a href="/wiki/Affine_transformation" title="Affine transformation">affine</a>. </p><p>A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a <a href="/wiki/Conformal_linear_transformation" title="Conformal linear transformation">conformal linear transformation</a>. </p> <dl><dt>Examples</dt></dl> <ul><li>A <a href="/wiki/Linear_map" title="Linear map">linear map</a> from <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 {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}" /></span> to itself is an isometry (for the <a href="/wiki/Dot_product" title="Dot product">dot product</a>) if and only if its matrix is <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Zhang-Zha-2004_15-0" class="reference"><a href="#cite_note-Zhang-Zha-2004-15"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Zhang-Wang-2006_16-0" class="reference"><a href="#cite_note-Zhang-Wang-2006-16"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Manifold">Manifold</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=5" title="Edit section: Manifold"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An isometry of a <a href="/wiki/Manifold" title="Manifold">manifold</a> is any (smooth) mapping of that manifold into itself, or into another manifold that preserves the notion of distance between points. The definition of an isometry requires the notion of a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> on the manifold; a manifold with a (positive-definite) metric is a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, one with an indefinite metric is a <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a>. Thus, isometries are studied in <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>. </p><p>A <b>local isometry</b> from one (<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo</a>-)<a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> to another is a map which <a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">pulls back</a> the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> on the second manifold to the metric tensor on the first. When such a map is also a <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a>, such a map is called an <b>isometry</b> (or <b>isometric isomorphism</b>), and provides a notion of <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> ("sameness") in the <a href="/wiki/Category_theory" title="Category theory">category</a> <b>Rm</b> of Riemannian manifolds. </p> <div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=6" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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 R=(M,g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=(M,g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c90da685e18113ccfa7feef0306f880be45d9c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.264ex; height:2.843ex;" alt="{\displaystyle R=(M,g)}" /></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 R'=(M',g')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R'=(M',g')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e9827b737cc8adaebfea7e288034902802cc10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.377ex; height:3.009ex;" alt="{\displaystyle R'=(M',g')}" /></span> be two (pseudo-)Riemannian manifolds, and let <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 f:R\to R'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <msup> <mi>R</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:R\to R'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600595876ed2f302630e40f0cc9ffd51fcdc5fa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.043ex; height:2.843ex;" alt="{\displaystyle f:R\to R'}" /></span> be a diffeomorphism. Then <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span> is called an <b>isometry</b> (or <b>isometric isomorphism</b>) if </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 g=f^{*}g',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=f^{*}g',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b93dbecfbc8aae3e2a396c9bf80477a741322cfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.039ex; height:2.843ex;" alt="{\displaystyle g=f^{*}g',}" /></span></dd></dl> <p>where <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 f^{*}g'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>g</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}g'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5156840c2c3951489df9146e3a78f9d096ed8b35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.178ex; height:2.843ex;" alt="{\displaystyle f^{*}g'}" /></span> denotes the <a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">pullback</a> of the rank (0, 2) metric tensor <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 g'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a53c0df5d85b36e3fd327c74db998f679f4f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.803ex; height:2.843ex;" alt="{\displaystyle g'}" /></span> by <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span>. Equivalently, in terms of the <a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">pushforward</a> <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 f_{*},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f032de5d9887b0123dd59758b4c617b6f35e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.509ex;" alt="{\displaystyle f_{*},}" /></span> we have that for any two vector fields <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 v,w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>,</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v,w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6425c6e94fa47976601cb44d7564b5d04dcfbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.826ex; height:2.009ex;" alt="{\displaystyle v,w}" /></span> on <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.e. sections of the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> <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 \mathrm {T} M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {T} M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90396560b7491143b445eb6d5263de21a938e387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.12ex; height:2.176ex;" alt="{\displaystyle \mathrm {T} M}" /></span>), </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 g(v,w)=g'\left(f_{*}v,f_{*}w\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>v</mi> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>w</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(v,w)=g'\left(f_{*}v,f_{*}w\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad21fdbb7c57812aa81addcfdeef4dac1056c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.095ex; height:3.009ex;" alt="{\displaystyle g(v,w)=g'\left(f_{*}v,f_{*}w\right).}" /></span></dd></dl> <p>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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span> is a <a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">local diffeomorphism</a> such that <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 g=f^{*}g',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=f^{*}g',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b93dbecfbc8aae3e2a396c9bf80477a741322cfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.039ex; height:2.843ex;" alt="{\displaystyle g=f^{*}g',}" /></span> then <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span> is called a <b>local isometry</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=7" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A collection of isometries typically form a group, the <a href="/wiki/Isometry_group" title="Isometry group">isometry group</a>. When the group is a <a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous group</a>, the <a href="/wiki/Lie_group" title="Lie group">infinitesimal generators</a> of the group are the <a href="/wiki/Killing_vector_field" title="Killing vector field">Killing vector fields</a>. </p><p>The <a href="/wiki/Myers%E2%80%93Steenrod_theorem" title="Myers–Steenrod theorem">Myers–Steenrod theorem</a> states that every isometry between two connected Riemannian manifolds is smooth (differentiable). A second form of this theorem states that the isometry group of a Riemannian manifold is a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>. </p><p><a href="/wiki/Symmetric_space" title="Symmetric space">Symmetric spaces</a> are important examples of <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifolds</a> that have isometries defined at every point. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=8" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Given a positive real number ε, an <b>ε-isometry</b> or <b>almost isometry</b> (also called a <b><a href="/wiki/Felix_Hausdorff" title="Felix Hausdorff">Hausdorff</a> approximation</b>) is a map <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 f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}" /></span> between metric spaces such that <ol><li>for <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,x'\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,x'\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1785fb75b6170e9a3e43dde32bd67398b5cd5f19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.199ex; height:2.843ex;" alt="{\displaystyle x,x'\in X}" /></span> one has <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 |d_{Y}(f(x),f(x'))-d_{X}(x,x')|<\varepsilon ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |d_{Y}(f(x),f(x'))-d_{X}(x,x')|<\varepsilon ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0d6c86ea1bd66704456d586557697ae7386ad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.05ex; height:3.009ex;" alt="{\displaystyle |d_{Y}(f(x),f(x'))-d_{X}(x,x')|<\varepsilon ,}" /></span> and</li> <li>for any point <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\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" /></span> there exists a point <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" /></span> with <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 d_{Y}(y,f(x))<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{Y}(y,f(x))<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e345fbd11e6078a46cae5af1020c33a20c4698e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.293ex; height:2.843ex;" alt="{\displaystyle d_{Y}(y,f(x))<\varepsilon }" /></span></li></ol></li></ul> <dl><dd>That is, an <span class="texhtml mvar" style="font-style:italic;">ε</span>-isometry preserves distances to within <span class="texhtml mvar" style="font-style:italic;">ε</span> and leaves no element of the codomain further than <span class="texhtml mvar" style="font-style:italic;">ε</span> away from the image of an element of the domain. Note that <span class="texhtml mvar" style="font-style:italic;">ε</span>-isometries are not assumed to be <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>.</dd></dl> <ul><li>The <b><a href="/wiki/Restricted_isometry_property" title="Restricted isometry property">restricted isometry property</a></b> characterizes nearly isometric matrices for sparse vectors.</li> <li><b><a href="/wiki/Quasi-isometry" title="Quasi-isometry">Quasi-isometry</a></b> is yet another useful generalization.</li> <li>One may also define an element in an abstract unital C*-algebra to be an isometry: <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 a\in {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cff173d29707f3fc722c04ac5b4aa00863c4d04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.739ex; height:2.176ex;" alt="{\displaystyle a\in {\mathfrak {A}}}" /></span> is an isometry if and only 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 a^{*}\cdot a=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{*}\cdot a=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf3e7a92bbdec22ef0c42be11d47223caf35ee7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.101ex; height:2.343ex;" alt="{\displaystyle a^{*}\cdot a=1.}" /></span></dd></dl></li></ul> <dl><dd>Note that as mentioned in the introduction this is not necessarily a unitary element because one does not in general have that left inverse is a right inverse.</dd></dl> <ul><li>On a <a href="/wiki/Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean space</a>, the term <i>isometry</i> means a linear bijection preserving magnitude. See also <a href="/wiki/Quadratic_form#Quadratic_spaces" title="Quadratic form">Quadratic spaces</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 15em;"> <ul><li><a href="/wiki/Beckman%E2%80%93Quarles_theorem" title="Beckman–Quarles theorem">Beckman–Quarles theorem</a></li> <li><a href="/wiki/Conformal_map" title="Conformal map">Conformal map</a> – Mathematical function that preserves angles</li> <li><a href="/wiki/Dual_norm#The_double_dual_of_a_normed_linear_space" title="Dual norm">The second dual of a Banach space as an isometric isomorphism</a></li> <li><a href="/wiki/Euclidean_plane_isometry" title="Euclidean plane isometry">Euclidean plane isometry</a></li> <li><a href="/wiki/Flat_(geometry)" title="Flat (geometry)">Flat (geometry)</a></li> <li><a href="/wiki/Homeomorphism_group" title="Homeomorphism group">Homeomorphism group</a></li> <li><a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">Involution</a></li> <li><a href="/wiki/Isometry_group" title="Isometry group">Isometry group</a></li> <li><a href="/wiki/Motion_(geometry)" title="Motion (geometry)">Motion (geometry)</a></li> <li><a href="/wiki/Myers%E2%80%93Steenrod_theorem" title="Myers–Steenrod theorem">Myers–Steenrod theorem</a></li> <li><a href="/wiki/Orthogonal_group#3D_isometries_that_leave_the_origin_fixed" title="Orthogonal group">3D isometries that leave the origin fixed</a></li> <li><a href="/wiki/Partial_isometry" title="Partial isometry">Partial isometry</a></li> <li><a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">Scaling (geometry)</a></li> <li><a href="/wiki/Semidefinite_embedding" title="Semidefinite embedding">Semidefinite embedding</a></li> <li><a href="/wiki/Space_group" title="Space group">Space group</a></li> <li><a href="/wiki/Symmetry_in_mathematics" title="Symmetry in mathematics">Symmetry in mathematics</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=10" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-CoxeterIsometryDef-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-CoxeterIsometryDef_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-CoxeterIsometryDef_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">"We shall find it convenient to use the word <i>transformation</i> in the special sense of a one-to-one correspondence <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 P\to P'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <msup> <mi>P</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\to P'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ca4aa2128fdfa9f025f102bd6f4590313c7d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.866ex; height:2.509ex;" alt="{\displaystyle P\to P'}" /></span> among all points in the plane (or in space), that is, a rule for associating pairs of points, with the understanding that each pair has a first member <span class="texhtml mvar" style="font-style:italic;">P</span> and a second member <span class="texhtml mvar" style="font-style:italic;">P'</span> and that every point occurs as the first member of just one pair and also as the second member of just one pair... <div class="paragraphbreak" style="margin-top:0.5em"></div> In particular, an <i>isometry</i> (or "congruent transformation," or "congruence") is a transformation which preserves length ..." — Coxeter (1969) p. 29<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"> <p><b>3.11</b> <i>Any two congruent triangles are related by a unique isometry.</i>— Coxeter (1969) p. 39<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></p></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"> <br />Let <span class="texhtml mvar" style="font-style:italic;">T</span> be a transformation (possibly many-valued) of <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 E^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa22ce4506e8504dfccbc3266b236d0a64e394d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.012ex; height:2.343ex;" alt="{\displaystyle E^{n}}" /></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 2\leq n<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>≤</mo> <mi>n</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\leq n<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a35351f11f62583fd2b9da423cb58162c30d14b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.078ex; height:2.343ex;" alt="{\displaystyle 2\leq n<\infty }" /></span>) into itself.<br />Let <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 d(p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7f759b2db30abcd9b048d406e829f2024b8f18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:2.843ex;" alt="{\displaystyle d(p,q)}" /></span> be the distance between points <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">q</span> of <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 E^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa22ce4506e8504dfccbc3266b236d0a64e394d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.012ex; height:2.343ex;" alt="{\displaystyle E^{n}}" /></span>, and let <span class="texhtml mvar" style="font-style:italic;">Tp</span>, <span class="texhtml mvar" style="font-style:italic;">Tq</span> be any images of <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">q</span>, respectively.<br />If there is a length <span class="texhtml mvar" style="font-style:italic;">a</span> > 0 such that <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 d(Tp,Tq)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mi>p</mi> <mo>,</mo> <mi>T</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(Tp,Tq)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf6b3dd0a2739897e606ca2498771dd82506a965" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.899ex; height:2.843ex;" alt="{\displaystyle d(Tp,Tq)=a}" /></span> whenever <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 d(p,q)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(p,q)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4070d12f690091179cba3073ee6ec221bcb83d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.626ex; height:2.843ex;" alt="{\displaystyle d(p,q)=a}" /></span>, then <span class="texhtml mvar" style="font-style:italic;">T</span> is a Euclidean transformation of <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 E^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa22ce4506e8504dfccbc3266b236d0a64e394d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.012ex; height:2.343ex;" alt="{\displaystyle E^{n}}" /></span> onto itself.<sup id="cite_ref-Beckman-Quarles-1953_6-1" class="reference"><a href="#cite_note-Beckman-Quarles-1953-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1969">Coxeter 1969</a>, p. 46 <p><b>3.51</b> <i>Any direct isometry is either a translation or a rotation. Any opposite isometry is either a reflection or a glide reflection.</i></p></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1969">Coxeter 1969</a>, p. 29</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1969">Coxeter 1969</a>, p. 39</span> </li> <li id="cite_note-Beckman-Quarles-1953-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Beckman-Quarles-1953_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Beckman-Quarles-1953_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBeckmanQuarles1953" class="citation journal cs1">Beckman, F.S.; Quarles, D.A. Jr. (1953). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/proc/1953-004-05/S0002-9939-1953-0058193-5/S0002-9939-1953-0058193-5.pdf">"On isometries of Euclidean spaces"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society">Proceedings of the American Mathematical Society</a></i>. <b>4</b> (5): <span class="nowrap">810–</span>815. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2032415">10.2307/2032415</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2032415">2032415</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0058193">0058193</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=On+isometries+of+Euclidean+spaces&rft.volume=4&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E810-%3C%2Fspan%3E815&rft.date=1953&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0058193%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2032415%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2032415&rft.aulast=Beckman&rft.aufirst=F.S.&rft.au=Quarles%2C+D.A.+Jr.&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fproc%2F1953-004-05%2FS0002-9939-1953-0058193-5%2FS0002-9939-1953-0058193-5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLe_Donne2013" class="citation journal cs1">Le Donne, Enrico (2013-10-01). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/s10711-012-9785-2">"Lipschitz and path isometric embeddings of metric spaces"</a>. <i>Geometriae Dedicata</i>. <b>166</b> (1): <span class="nowrap">47–</span>66. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10711-012-9785-2">10.1007/s10711-012-9785-2</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1572-9168">1572-9168</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geometriae+Dedicata&rft.atitle=Lipschitz+and+path+isometric+embeddings+of+metric+spaces&rft.volume=166&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E47-%3C%2Fspan%3E66&rft.date=2013-10-01&rft_id=info%3Adoi%2F10.1007%2Fs10711-012-9785-2&rft.issn=1572-9168&rft.aulast=Le+Donne&rft.aufirst=Enrico&rft_id=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs10711-012-9785-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBuragoBuragoIvanov2001" class="citation book cs1">Burago, Dmitri; Burago, Yurii; Ivanov, Sergeï (2001). 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Providence, RI: American Mathematical Society (AMS). pp. <span class="nowrap">86–</span>87. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-2129-6" title="Special:BookSources/0-8218-2129-6"><bdi>0-8218-2129-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3+Constructions%2C+%C2%A73.5+Arcwise+isometries&rft.btitle=A+course+in+metric+geometry&rft.series=Graduate+Studies+in+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E86-%3C%2Fspan%3E87&rft.pub=Providence%2C+RI%3A+American+Mathematical+Society+%28AMS%29&rft.date=2001&rft.isbn=0-8218-2129-6&rft.aulast=Burago&rft.aufirst=Dmitri&rft.au=Burago%2C+Yurii&rft.au=Ivanov%2C+Serge%C3%AF&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTENariciBeckenstein2011275–339-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein2011275–339_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011275–339_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 275–339.</span> </li> <li id="cite_note-FOOTNOTEWilansky201321–26-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWilansky201321–26_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWilansky2013">Wilansky 2013</a>, pp. 21–26.</span> </li> <li id="cite_note-Thomsen_2017_p125-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Thomsen_2017_p125_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFThomsen2017" class="citation book cs1 cs1-prop-foreign-lang-source">Thomsen, Jesper Funch (2017). <i>Lineær algebra</i> [<i>Linear Algebra</i>]. 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"Nonlinear dimensionality reduction by locally linear embedding". <i><a href="/wiki/Science_(journal)" title="Science (journal)">Science</a></i>. <b>290</b> (5500): <span class="nowrap">2323–</span>2326. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000Sci...290.2323R">2000Sci...290.2323R</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.111.3313">10.1.1.111.3313</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.290.5500.2323">10.1126/science.290.5500.2323</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11125150">11125150</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Nonlinear+dimensionality+reduction+by+locally+linear+embedding&rft.volume=290&rft.issue=5500&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2323-%3C%2Fspan%3E2326&rft.date=2000&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.111.3313%23id-name%3DCiteSeerX&rft_id=info%3Apmid%2F11125150&rft_id=info%3Adoi%2F10.1126%2Fscience.290.5500.2323&rft_id=info%3Abibcode%2F2000Sci...290.2323R&rft.aulast=Roweis&rft.aufirst=S.T.&rft.au=Saul%2C+L.K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSaulRoweis2003" class="citation journal cs1">Saul, Lawrence K.; Roweis, Sam T. (June 2003). "Think globally, fit locally: Unsupervised learning of nonlinear manifolds". <i><a href="/wiki/Journal_of_Machine_Learning_Research" title="Journal of Machine Learning Research">Journal of Machine Learning Research</a></i>. <b>4</b> (June): <span class="nowrap">119–</span>155. <q>Quadratic optimisation of <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 {M} =(I-W)^{\top }(I-W)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>−</mo> <mi>W</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo>−</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} =(I-W)^{\top }(I-W)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa105cbf8687ceb78cfafc793c423743172dee06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.66ex; height:3.176ex;" alt="{\displaystyle \mathbf {M} =(I-W)^{\top }(I-W)}" /></span> (page 135) such that <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 {M} \equiv YY^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>≡</mo> <mi>Y</mi> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} \equiv YY^{\top }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e28b5bbc7e467ac3ae0734f6b96d7ef4363aa210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.82ex; height:2.676ex;" alt="{\displaystyle \mathbf {M} \equiv YY^{\top }}" /></span></q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Machine+Learning+Research&rft.atitle=Think+globally%2C+fit+locally%3A+Unsupervised+learning+of+nonlinear+manifolds&rft.volume=4&rft.issue=June&rft.pages=%3Cspan+class%3D%22nowrap%22%3E119-%3C%2Fspan%3E155&rft.date=2003-06&rft.aulast=Saul&rft.aufirst=Lawrence+K.&rft.au=Roweis%2C+Sam+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></span> </li> <li id="cite_note-Zhang-Zha-2004-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Zhang-Zha-2004_15-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZhangZha2004" class="citation journal cs1">Zhang, Zhenyue; Zha, Hongyuan (2004). "Principal manifolds and nonlinear dimension reduction via local tangent space alignment". <i>SIAM Journal on Scientific Computing</i>. <b>26</b> (1): <span class="nowrap">313–</span>338. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.211.9957">10.1.1.211.9957</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2Fs1064827502419154">10.1137/s1064827502419154</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Scientific+Computing&rft.atitle=Principal+manifolds+and+nonlinear+dimension+reduction+via+local+tangent+space+alignment&rft.volume=26&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E313-%3C%2Fspan%3E338&rft.date=2004&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.211.9957%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1137%2Fs1064827502419154&rft.aulast=Zhang&rft.aufirst=Zhenyue&rft.au=Zha%2C+Hongyuan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></span> </li> <li id="cite_note-Zhang-Wang-2006-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Zhang-Wang-2006_16-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZhangWang2006" class="citation conference cs1">Zhang, Zhenyue; Wang, Jing (2006). <a rel="nofollow" class="external text" href="https://papers.nips.cc/paper/3132-mlle-modified-locally-linear-embedding-using-multiple-weights">"MLLE: Modified locally linear embedding using multiple weights"</a>. In Schölkopf, B.; Platt, J.; Hoffman, T. (eds.). <i><a href="/wiki/Advances_in_Neural_Information_Processing_Systems" class="mw-redirect" title="Advances in Neural Information Processing Systems">Advances in Neural Information Processing Systems</a></i>. NIPS 2006. NeurIPS Proceedings. Vol. 19. pp. <span class="nowrap">1593–</span>1600. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781622760381" title="Special:BookSources/9781622760381"><bdi>9781622760381</bdi></a>. <q>It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=MLLE%3A+Modified+locally+linear+embedding+using+multiple+weights&rft.btitle=Advances+in+Neural+Information+Processing+Systems&rft.series=NeurIPS+Proceedings&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1593-%3C%2Fspan%3E1600&rft.date=2006&rft.isbn=9781622760381&rft.aulast=Zhang&rft.aufirst=Zhenyue&rft.au=Wang%2C+Jing&rft_id=https%3A%2F%2Fpapers.nips.cc%2Fpaper%2F3132-mlle-modified-locally-linear-embedding-using-multiple-weights&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isometry&action=edit&section=12" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115" /><div class="div-col" style="column-width: 25em;"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRudin1991" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi"><i>Functional Analysis</i></a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/wiki/McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-054236-5" title="Special:BookSources/978-0-07-054236-5"><bdi>978-0-07-054236-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21163277">21163277</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis&rft.place=New+York%2C+NY&rft.series=International+Series+in+Pure+and+Applied+Mathematics&rft.edition=Second&rft.pub=McGraw-Hill+Science%2FEngineering%2FMath&rft.date=1991&rft_id=info%3Aoclcnum%2F21163277&rft.isbn=978-0-07-054236-5&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffunctionalanalys00rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNariciBeckenstein2011" class="citation book cs1">Narici, Lawrence; Beckenstein, Edward (2011). <i>Topological Vector Spaces</i>. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1584888666" title="Special:BookSources/978-1584888666"><bdi>978-1584888666</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/144216834">144216834</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces&rft.place=Boca+Raton%2C+FL&rft.series=Pure+and+applied+mathematics&rft.edition=Second&rft.pub=CRC+Press&rft.date=2011&rft_id=info%3Aoclcnum%2F144216834&rft.isbn=978-1584888666&rft.aulast=Narici&rft.aufirst=Lawrence&rft.au=Beckenstein%2C+Edward&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchaeferWolff1999" class="citation book cs1"><a href="/wiki/Helmut_H._Schaefer" title="Helmut H. Schaefer">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). <i>Topological Vector Spaces</i>. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-7155-0" title="Special:BookSources/978-1-4612-7155-0"><bdi>978-1-4612-7155-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840278135">840278135</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces&rft.place=New+York%2C+NY&rft.series=GTM&rft.edition=Second&rft.pub=Springer+New+York+Imprint+Springer&rft.date=1999&rft_id=info%3Aoclcnum%2F840278135&rft.isbn=978-1-4612-7155-0&rft.aulast=Schaefer&rft.aufirst=Helmut+H.&rft.au=Wolff%2C+Manfred+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTrèves2006" class="citation book cs1"><a href="/wiki/Fran%C3%A7ois_Tr%C3%A8ves" title="François Trèves">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. Mineola, N.Y.: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45352-1" title="Special:BookSources/978-0-486-45352-1"><bdi>978-0-486-45352-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/853623322">853623322</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces%2C+Distributions+and+Kernels&rft.place=Mineola%2C+N.Y.&rft.pub=Dover+Publications&rft.date=2006&rft_id=info%3Aoclcnum%2F853623322&rft.isbn=978-0-486-45352-1&rft.aulast=Tr%C3%A8ves&rft.aufirst=Fran%C3%A7ois&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWilansky2013" class="citation book cs1"><a href="/wiki/Albert_Wilansky" title="Albert Wilansky">Wilansky, Albert</a> (2013). <i>Modern Methods in Topological Vector Spaces</i>. Mineola, New York: Dover Publications, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49353-4" title="Special:BookSources/978-0-486-49353-4"><bdi>978-0-486-49353-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/849801114">849801114</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Methods+in+Topological+Vector+Spaces&rft.place=Mineola%2C+New+York&rft.pub=Dover+Publications%2C+Inc&rft.date=2013&rft_id=info%3Aoclcnum%2F849801114&rft.isbn=978-0-486-49353-4&rft.aulast=Wilansky&rft.aufirst=Albert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCoxeter1969" class="citation book cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a> (1969). <i>Introduction to Geometry, Second edition</i>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471504580" title="Special:BookSources/9780471504580"><bdi>9780471504580</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Geometry%2C+Second+edition&rft.pub=Wiley&rft.date=1969&rft.isbn=9780471504580&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLee2009" class="citation book cs1">Lee, Jeffrey M. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QqHdHy9WsEoC"><i>Manifolds and Differential Geometry</i></a>. Providence, RI: American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4815-9" title="Special:BookSources/978-0-8218-4815-9"><bdi>978-0-8218-4815-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Manifolds+and+Differential+Geometry&rft.place=Providence%2C+RI&rft.pub=American+Mathematical+Society&rft.date=2009&rft.isbn=978-0-8218-4815-9&rft.aulast=Lee&rft.aufirst=Jeffrey+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQqHdHy9WsEoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsometry" class="Z3988"></span></li></ul> </div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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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href="/wiki/Template:Metric_spaces" title="Template:Metric spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Metric_spaces&action=edit&redlink=1" class="new" title="Template talk:Metric spaces (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Metric_spaces" title="Special:EditPage/Template:Metric spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Metric_spaces_(Category)86" style="font-size:114%;margin:0 4em"><a href="/wiki/Metric_space" title="Metric space">Metric spaces</a> (<a href="/wiki/Category:Metric_spaces" title="Category:Metric spaces">Category</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Metric_space" title="Metric space">Metric space</a></li> <li><a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li> <li><a href="/wiki/Complete_metric_space" title="Complete metric space">Completeness</a></li> <li><a href="/wiki/Equivalence_of_metrics" title="Equivalence of metrics">Equivalent metrics</a></li> <li><a href="/wiki/Metrizable_space" title="Metrizable space">Metrizable space</a></li> <li><a href="/wiki/Triangle_inequality" title="Triangle inequality">Triangle inequality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Baire_category_theorem" title="Baire category theorem">Baire category theorem</a></li> <li><a href="/wiki/Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point</a></li> <li><a href="/wiki/Kuratowski_embedding" title="Kuratowski embedding">Kuratowski embedding</a></li> <li><a href="/wiki/Lebesgue%27s_number_lemma" title="Lebesgue's number lemma">Lebesgue's number lemma</a></li> <li><a href="/wiki/Metrization_theorem" class="mw-redirect" title="Metrization theorem">Metrization theorems</a>: <ul><li><a href="/wiki/Bing_metrization_theorem" title="Bing metrization theorem">Bing</a></li> <li><a href="/wiki/Nagata%E2%80%93Smirnov_metrization_theorem" title="Nagata–Smirnov metrization theorem">Nagata–Smirnov</a></li> <li><a href="/wiki/Urysohn%27s_metrization_theorem" class="mw-redirect" title="Urysohn's metrization theorem">Urysohn's</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contraction_mapping" title="Contraction mapping">Contraction</a> <ul><li><a href="/wiki/Metric_map" title="Metric map">Metric map</a></li></ul></li> <li><a href="/wiki/Dilation_(metric_space)" title="Dilation (metric space)">Dilation</a></li> <li><a href="/wiki/Equicontinuity" title="Equicontinuity">Equicontinuity</a></li> <li>(<a href="/wiki/Quasi-isometry" title="Quasi-isometry">Quasi-</a>) <a class="mw-selflink selflink">Isometry</a></li> <li><a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a></li> <li><a href="/wiki/Metric_derivative" title="Metric derivative">Metric derivative</a></li> <li><a href="/wiki/Metric_outer_measure" title="Metric outer measure">Metric outer measure</a></li> <li><a href="/wiki/Metric_projection" title="Metric projection">Metric projection</a></li> <li><a href="/wiki/Motion_(geometry)" title="Motion (geometry)">Motion</a></li> <li><a href="/wiki/Quasisymmetric_map" title="Quasisymmetric map">Quasisymmetric</a></li> <li><a href="/wiki/Stretch_factor" title="Stretch factor">Stretch factor</a></li> <li><a href="/wiki/Uniform_continuity" title="Uniform continuity">Uniform continuity</a> <ul><li><a href="/wiki/Uniform_isomorphism" title="Uniform isomorphism">Isomorphism</a></li></ul></li> <li><a href="/wiki/Uniform_convergence" title="Uniform convergence">Uniform convergence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />metric spaces</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Complete_metric_space" title="Complete metric space">Complete</a></li> <li><a href="/wiki/Convex_metric_space" title="Convex metric space">Convex</a></li> <li><a href="/wiki/Doubling_space" title="Doubling space">Doubling</a></li> <li><a href="/wiki/Hyperbolic_metric_space" title="Hyperbolic metric space">Hyperbolic</a></li> <li><a href="/wiki/Injective_metric_space" title="Injective metric space">Injective</a></li> <li><a href="/wiki/Length_metric_space" class="mw-redirect" title="Length metric space">Length metric space</a></li> <li><a href="/wiki/Metric_space_aimed_at_its_subspace" title="Metric space aimed at its subspace">Metric space aimed at its subspace</a></li> <li><a href="/wiki/Polish_space" title="Polish space">Polish</a></li> <li><a href="/wiki/Totally_bounded_space" title="Totally bounded space">Totally bounded</a></li> <li><a href="/wiki/Tree-graded_space" title="Tree-graded space">Tree-graded</a></li> <li><a href="/wiki/Ultrametric_space" title="Ultrametric space">Ultrametric space</a></li> <li><a href="/wiki/Uniformly_disconnected_space" title="Uniformly disconnected space">Uniformly disconnected</a></li> <li><a href="/wiki/Urysohn_universal_space" title="Urysohn universal space">Urysohn universal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">Balls</a></li> <li><a href="/wiki/Borel_set" title="Borel set">Borel</a></li> <li><a href="/wiki/Bounded_set" title="Bounded set">Bounded</a></li> <li><a href="/wiki/Delone_set" title="Delone set">Delone</a></li> <li><a href="/wiki/Diameter_of_a_set" title="Diameter of a set">Diameter</a></li> <li><a href="/wiki/Distance_set" title="Distance set">Distance set</a></li> <li><a href="/wiki/Gromov_product" title="Gromov product">Gromov product</a></li> <li><a href="/wiki/Gromov%E2%80%93Hausdorff_convergence" title="Gromov–Hausdorff convergence">Gromov–Hausdorff convergence</a></li> <li><a href="/wiki/Hausdorff_distance" title="Hausdorff distance">Hausdorff distance</a></li> <li><a href="/wiki/Kuratowski_convergence" title="Kuratowski convergence">Kuratowski convergence</a></li> <li><a href="/wiki/Meyer_set" title="Meyer set">Meyer</a></li> <li><a href="/wiki/Packing_dimension" title="Packing dimension">Packing dimension</a></li> <li><a href="/wiki/Porous_set" title="Porous set">Porous</a></li> <li><a href="/wiki/Positively_separated_sets" title="Positively separated sets">Positively separated sets</a></li> <li><a href="/wiki/Tight_span" title="Tight span">Tight span</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Manifold" title="Manifold">Manifolds</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a><br />and <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/L%C3%A9vy_metric" title="Lévy metric">Lévy metric</a></li> <li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable topological vector space</a></li> <li><a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">Normed space</a></li> <li><a href="/wiki/Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li> <li><a href="/wiki/Wasserstein_metric" title="Wasserstein metric">Wasserstein metric</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/General_topology" title="General topology">General topology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_space" title="Discrete space">Discrete space</a></li> <li><a href="/wiki/Intrinsic_metric" title="Intrinsic metric">Intrinsic metric</a></li> <li><a href="/wiki/Laakso_space" title="Laakso space">Laakso space</a></li> <li><a href="/wiki/Product_metric" title="Product metric">Product metric</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_of_metric_spaces" title="Category of metric spaces">Category of metric spaces</a></li> <li><a href="/wiki/Cantor_space" title="Cantor space">Cantor space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approach_space" title="Approach space">Approach space</a></li> <li><a href="/wiki/Cauchy_space" title="Cauchy space">Cauchy space</a></li> <li><a href="/wiki/Coarse_structure" title="Coarse structure">Coarse structure</a></li> <li><a href="/wiki/Cosmic_space" title="Cosmic space">Cosmic space</a></li> <li><a href="/wiki/Diversity_(mathematics)" title="Diversity (mathematics)">Diversity</a></li> <li><a href="/wiki/Generalised_metric" title="Generalised metric">Generalised metric</a></li> <li><a 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