Finsler manifold - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Finsler manifold - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"1004e14a-cc68-4664-8f9b-19860fa8c37d","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Finsler_manifold","wgTitle":"Finsler manifold","wgCurRevisionId":1234056884,"wgRevisionId":1234056884,"wgArticleId":385510,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Articles needing additional references from May 2017","All articles needing additional references","Pages displaying wikidata descriptions as a fallback via Module:Annotated link","Differential geometry","Finsler geometry","Riemannian geometry","Riemannian manifolds","Smooth manifolds"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName": "Finsler_manifold","wgRelevantArticleId":385510,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":10000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q2498782","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform", "platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher", "ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Finsler manifold - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Finsler_manifold"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Finsler_manifold&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Finsler_manifold"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Finsler_manifold rootpage-Finsler_manifold skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Finsler+manifold" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Finsler+manifold" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Finsler+manifold" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Finsler+manifold" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-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 Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Randers_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Randers_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Randers manifolds</span> </div> </a> <ul id="toc-Randers_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Smooth_quasimetric_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Smooth_quasimetric_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Smooth quasimetric spaces</span> </div> </a> <ul id="toc-Smooth_quasimetric_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geodesics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geodesics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Geodesics</span> </div> </a> <button aria-controls="toc-Geodesics-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 Geodesics subsection</span> </button> <ul id="toc-Geodesics-sublist" class="vector-toc-list"> <li id="toc-Canonical_spray_structure_on_a_Finsler_manifold" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Canonical_spray_structure_on_a_Finsler_manifold"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Canonical spray structure on a Finsler manifold</span> </div> </a> <ul id="toc-Canonical_spray_structure_on_a_Finsler_manifold-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniqueness_and_minimizing_properties_of_geodesics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniqueness_and_minimizing_properties_of_geodesics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Uniqueness and minimizing properties of geodesics</span> </div> </a> <ul id="toc-Uniqueness_and_minimizing_properties_of_geodesics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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" > <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">Finsler manifold</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 7 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-7" 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">7 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Finsler-Mannigfaltigkeit" title="Finsler-Mannigfaltigkeit – German" lang="de" hreflang="de" data-title="Finsler-Mannigfaltigkeit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_de_Finsler" title="Espace de Finsler – French" lang="fr" hreflang="fr" data-title="Espace de Finsler" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%80%EC%8A%AC%EB%9F%AC_%EB%8B%A4%EC%96%91%EC%B2%B4" title="핀슬러 다양체 – Korean" lang="ko" hreflang="ko" data-title="핀슬러 다양체" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Finsler-vari%C3%ABteit" title="Finsler-variëteit – Dutch" lang="nl" hreflang="nl" data-title="Finsler-variëteit" 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/%E3%83%95%E3%82%A3%E3%83%B3%E3%82%B9%E3%83%A9%E3%83%BC%E5%A4%9A%E6%A7%98%E4%BD%93" title="フィンスラー多様体 – Japanese" lang="ja" hreflang="ja" data-title="フィンスラー多様体" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Variedade_de_Finsler" title="Variedade de Finsler – Portuguese" lang="pt" hreflang="pt" data-title="Variedade de Finsler" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%8A%AC%E6%96%AF%E5%8B%92%E6%B5%81%E5%BD%A2" title="芬斯勒流形 – Chinese" lang="zh" hreflang="zh" data-title="芬斯勒流形" 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/Q2498782#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 mw-list-item"><a href="/wiki/Finsler_manifold" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Finsler_manifold" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Finsler_manifold"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Finsler_manifold&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Finsler_manifold&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Finsler_manifold"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Finsler_manifold&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Finsler_manifold&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Finsler_manifold" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Finsler_manifold" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Finsler_manifold&oldid=1234056884" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Finsler_manifold&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Finsler_manifold&id=1234056884&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FFinsler_manifold"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FFinsler_manifold"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Finsler_manifold&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Finsler_manifold&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2498782" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Generalization of Riemannian manifolds</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">"Finsler" redirects here. For the mathematician this manifold is named after, see <a href="/wiki/Paul_Finsler" title="Paul Finsler">Paul Finsler</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-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Finsler_manifold" title="Special:EditPage/Finsler manifold">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Finsler+manifold%22">"Finsler manifold"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Finsler+manifold%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Finsler+manifold%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Finsler+manifold%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Finsler+manifold%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Finsler+manifold%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">May 2017</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, particularly <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, a <b>Finsler manifold</b> is a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> <span class="texhtml"><i>M</i></span> where a (possibly <a href="/wiki/Asymmetric_norm" title="Asymmetric norm">asymmetric</a>) <b><a href="/wiki/Minkowski_functional" title="Minkowski functional">Minkowski norm</a></b> <span class="texhtml"><i>F</i>(<i>x</i>, −)</span> is provided on each tangent space <span class="texhtml">T<sub><i>x</i></sub><i>M</i></span>, that enables one to define the length of any <a href="/wiki/Smooth_curve" class="mw-redirect" title="Smooth curve">smooth curve</a> <span class="texhtml"><i>γ</i> : [<i>a</i>, <i>b</i>] → <i>M</i></span> 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 L(\gamma )=\int _{a}^{b}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(\gamma )=\int _{a}^{b}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f85cc974ccfceb06d2ce3a10de5fd61f324e252" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.889ex; height:6.343ex;" alt="{\displaystyle L(\gamma )=\int _{a}^{b}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,\mathrm {d} t.}"></span></dd></dl> <p>Finsler manifolds are more general than <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifolds</a> since the tangent norms need not be induced by <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner products</a>. </p><p>Every Finsler manifold becomes an <a href="/wiki/Intrinsic_metric" title="Intrinsic metric">intrinsic</a> <a href="/wiki/Quasimetric_space#Quasimetrics" class="mw-redirect" title="Quasimetric space">quasimetric space</a> when the distance between two points is defined as the infimum length of the curves that join them. </p><p><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> (<a href="#CITEREFCartan1933">1933</a>) named Finsler manifolds after <a href="/wiki/Paul_Finsler" title="Paul Finsler">Paul Finsler</a>, who studied this geometry in his dissertation (<a href="#CITEREFFinsler1918">Finsler 1918</a>). </p> <meta property="mw:PageProp/toc" /> <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=Finsler_manifold&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>Finsler manifold</b> is a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> <span class="texhtml"><i>M</i></span> together with a <b>Finsler metric</b>, which is a continuous nonnegative function <span class="texhtml"><i>F</i>: T<i>M</i> → [0, +∞)</span> defined on the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> so that for each point <span class="texhtml"><i>x</i></span> of <span class="texhtml"><i>M</i></span>, </p> <ul><li><span class="texhtml"><i>F</i>(<i>v</i> + <i>w</i>) ≤ <i>F</i>(<i>v</i>) + <i>F</i>(<i>w</i>)</span> for every two vectors <span class="texhtml"><i>v</i>,<i>w</i></span> tangent to <span class="texhtml"><i>M</i></span> at <span class="texhtml"><i>x</i></span> (<a href="/wiki/Subadditivity" title="Subadditivity">subadditivity</a>).</li> <li><span class="texhtml"><i>F</i>(λ<i>v</i>) = λ<i>F</i>(<i>v</i>)</span> for all <span class="texhtml">λ ≥ 0</span> (but not necessarily for <span class="texhtml">λ < 0)</span> (<a href="/wiki/Homogeneous_function" title="Homogeneous function">positive homogeneity</a>).</li> <li><span class="texhtml"><i>F</i>(<i>v</i>) > 0</span> unless <span class="texhtml"><i>v</i> = 0</span> (<a href="/wiki/Positive-definite_function" title="Positive-definite function">positive definiteness</a>).</li></ul> <p>In other words, <span class="texhtml"><i>F</i>(<i>x</i>, −)</span> is an <a href="/wiki/Asymmetric_norm" title="Asymmetric norm">asymmetric norm</a> on each tangent space <span class="texhtml">T<sub><i>x</i></sub><i>M</i></span>. The Finsler metric <span class="texhtml"><i>F</i></span> is also required to be <b>smooth</b>, more precisely: </p> <ul><li><span class="texhtml"><i>F</i></span> is <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> on the complement of the zero section of <span class="texhtml">T<i>M</i></span>.</li></ul> <p>The subadditivity axiom may then be replaced by the following <b>strong convexity condition</b>: </p> <ul><li>For each tangent vector <span class="texhtml"><i>v</i> ≠ 0</span>, the <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a> of <span class="texhtml"><i>F</i><sup>2</sup></span> at <span class="texhtml"><i>v</i></span> is <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive definite</a>.</li></ul> <p>Here the Hessian of <span class="texhtml"><i>F</i><sup>2</sup></span> at <span class="texhtml"><i>v</i></span> is the <a href="/wiki/Symmetric_tensor" title="Symmetric tensor">symmetric</a> <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{v}(X,Y):={\frac {1}{2}}\left.{\frac {\partial ^{2}}{\partial s\partial t}}\left[F(v+sX+tY)^{2}\right]\right|_{s=t=0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>+</mo> <mi>s</mi> <mi>X</mi> <mo>+</mo> <mi>t</mi> <mi>Y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{v}(X,Y):={\frac {1}{2}}\left.{\frac {\partial ^{2}}{\partial s\partial t}}\left[F(v+sX+tY)^{2}\right]\right|_{s=t=0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad815e186e29dee27e7381bffae6fb70ccca5c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.801ex; height:6.509ex;" alt="{\displaystyle \mathbf {g} _{v}(X,Y):={\frac {1}{2}}\left.{\frac {\partial ^{2}}{\partial s\partial t}}\left[F(v+sX+tY)^{2}\right]\right|_{s=t=0},}"></span></dd></dl> <p>also known as the <b>fundamental tensor</b> of <span class="texhtml"><i>F</i></span> at <span class="texhtml"><i>v</i></span>. Strong convexity of <span class="texhtml"><i>F</i></span> implies the subadditivity with a strict inequality if <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.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="frac"><span class="num"><i>u</i></span>⁄<span class="den"><i>F</i>(<i>u</i>)</span></span> ≠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num"><i>v</i></span>⁄<span class="den"><i>F</i>(<i>v</i>)</span></span></span>. If <span class="texhtml"><i>F</i></span> is strongly convex, then it is a <b>Minkowski norm</b> on each tangent space. </p><p>A Finsler metric is <b>reversible</b> if, in addition, </p> <ul><li><span class="texhtml"><i>F</i>(−<i>v</i>) = <i>F</i>(<i>v</i>)</span> for all tangent vectors <i>v</i>.</li></ul> <p>A reversible Finsler metric defines a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> (in the usual sense) on each tangent space. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finsler_manifold&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Smooth submanifolds (including open subsets) of a <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> of finite dimension are Finsler manifolds if the norm of the vector space is smooth outside the origin.</li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifolds</a> (but not <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifolds</a>) are special cases of Finsler manifolds.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Randers_manifolds">Randers manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finsler_manifold&action=edit&section=3" title="Edit section: Randers manifolds"><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 (M,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74d1b3152d762039462ab6cbed356b5bd5b9e52c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.515ex; height:2.843ex;" alt="{\displaystyle (M,a)}"></span> be a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> and <i>b</i> a <a href="/wiki/Differential_form" title="Differential form">differential one-form</a> on <i>M</i> with </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 \|b\|_{a}:={\sqrt {a^{ij}b_{i}b_{j}}}<1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>b</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </msqrt> </mrow> <mo><</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|b\|_{a}:={\sqrt {a^{ij}b_{i}b_{j}}}<1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b68db5a7bc7759af71a0bda2bf626d01b4ea9e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.813ex; height:4.843ex;" alt="{\displaystyle \|b\|_{a}:={\sqrt {a^{ij}b_{i}b_{j}}}<1,}"></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 \left(a^{ij}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a^{ij}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0ae07efa9369b895be0fb3a374f73b36f453bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.837ex; height:3.343ex;" alt="{\displaystyle \left(a^{ij}\right)}"></span> is the <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse matrix</a> 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 (a_{ij})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{ij})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4729cd2ff24125d923914cd5925a54d362d858f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.516ex; height:3.009ex;" alt="{\displaystyle (a_{ij})}"></span> and the <a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a> is used. Then </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 F(x,v):={\sqrt {a_{ij}(x)v^{i}v^{j}}}+b_{i}(x)v^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,v):={\sqrt {a_{ij}(x)v^{i}v^{j}}}+b_{i}(x)v^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc210d4260889068e2fb743d2db6000bb3c2ecee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:32.625ex; height:4.843ex;" alt="{\displaystyle F(x,v):={\sqrt {a_{ij}(x)v^{i}v^{j}}}+b_{i}(x)v^{i}}"></span></dd></dl> <p>defines a <b>Randers metric</b> on <i>M</i> 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 (M,F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd792989e7a56d99d4862a807e1bf2555e349bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.026ex; height:2.843ex;" alt="{\displaystyle (M,F)}"></span> is a <b>Randers manifold</b>, a special case of a non-reversible Finsler manifold.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Smooth_quasimetric_spaces">Smooth quasimetric spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finsler_manifold&action=edit&section=4" title="Edit section: Smooth quasimetric spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let (<i>M</i>, <i>d</i>) be a <a href="/wiki/Quasimetric" class="mw-redirect" title="Quasimetric">quasimetric</a> so that <i>M</i> is also a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> and <i>d</i> is compatible with the <a href="/wiki/Differential_structure" title="Differential structure">differential structure</a> of <i>M</i> in the following sense: </p> <ul><li>Around any point <i>z</i> on <i>M</i> there exists a smooth chart (<i>U</i>, φ) of <i>M</i> and a constant <i>C</i> ≥ 1 such that for every <i>x</i>, <i>y</i> ∈ <i>U</i> <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 {\frac {1}{C}}\|\phi (y)-\phi (x)\|\leq d(x,y)\leq C\|\phi (y)-\phi (x)\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>C</mi> </mfrac> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>C</mi> <mo fence="false" stretchy="false">‖</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{C}}\|\phi (y)-\phi (x)\|\leq d(x,y)\leq C\|\phi (y)-\phi (x)\|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc75e7234ba7b7478ec3b884f0db881b67cdf68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:45.837ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{C}}\|\phi (y)-\phi (x)\|\leq d(x,y)\leq C\|\phi (y)-\phi (x)\|.}"></span></dd></dl></li> <li>The function <i>d</i>: <i>M</i> × <i>M</i> → [0, ∞] is <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> in some punctured neighborhood of the diagonal.</li></ul> <p>Then one can define a Finsler function <i>F</i>: <i>TM</i> →[0, ∞] by </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 F(x,v):=\lim _{t\to 0+}{\frac {d(\gamma (0),\gamma (t))}{t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo>+</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,v):=\lim _{t\to 0+}{\frac {d(\gamma (0),\gamma (t))}{t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ac36402109638bc5062f3e1c969f0435c14ff67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.198ex; height:6.009ex;" alt="{\displaystyle F(x,v):=\lim _{t\to 0+}{\frac {d(\gamma (0),\gamma (t))}{t}},}"></span></dd></dl> <p>where <i>γ</i> is any curve in <i>M</i> with <i>γ</i>(0) = <i>x</i> and <i>γ'</i>(0) = v. The Finsler function <i>F</i> obtained in this way restricts to an asymmetric (typically non-Minkowski) norm on each tangent space of <i>M</i>. The <a href="/wiki/Intrinsic_metric" title="Intrinsic metric">induced intrinsic metric</a> <span class="nowrap"><i>d</i><sub><i>L</i></sub>: <i>M</i> × <i>M</i> → [0, ∞]</span> of the original <a href="/wiki/Quasimetric" class="mw-redirect" title="Quasimetric">quasimetric</a> can be recovered from </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_{L}(x,y):=\inf \left\{\ \left.\int _{0}^{1}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt\ \right|\ \gamma \in C^{1}([0,1],M)\ ,\ \gamma (0)=x\ ,\ \gamma (1)=y\ \right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mrow> <mo>{</mo> <mrow> <mtext> </mtext> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mtext> </mtext> </mrow> <mo>|</mo> </mrow> <mtext> </mtext> <mi>γ</mi> <mo>∈</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mi>γ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mi>γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mtext> </mtext> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{L}(x,y):=\inf \left\{\ \left.\int _{0}^{1}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt\ \right|\ \gamma \in C^{1}([0,1],M)\ ,\ \gamma (0)=x\ ,\ \gamma (1)=y\ \right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99530125368777e7d7cd1af65729f409d59c6aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:80.899ex; height:6.509ex;" alt="{\displaystyle d_{L}(x,y):=\inf \left\{\ \left.\int _{0}^{1}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt\ \right|\ \gamma \in C^{1}([0,1],M)\ ,\ \gamma (0)=x\ ,\ \gamma (1)=y\ \right\},}"></span></dd></dl> <p>and in fact any Finsler function <i>F</i>: T<i>M</i> → [0, ∞) defines an <a href="/wiki/Intrinsic_metric" title="Intrinsic metric">intrinsic</a> <a href="/wiki/Quasimetric" class="mw-redirect" title="Quasimetric">quasimetric</a> <i>d</i><sub><i>L</i></sub> on <i>M</i> by this formula. </p> <div class="mw-heading mw-heading2"><h2 id="Geodesics">Geodesics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finsler_manifold&action=edit&section=5" title="Edit section: Geodesics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Due to the homogeneity of <i>F</i> the length </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 L[\gamma ]:=\int _{a}^{b}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">[</mo> <mi>γ</mi> <mo stretchy="false">]</mo> <mo>:=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L[\gamma ]:=\int _{a}^{b}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8c178c6d7c659cdd66bd60bd11cac87b428bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.296ex; height:6.343ex;" alt="{\displaystyle L[\gamma ]:=\int _{a}^{b}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt}"></span></dd></dl> <p>of a <a href="/wiki/Differentiable_curve" title="Differentiable curve">differentiable curve</a> <i>γ</i>: [<i>a</i>, <i>b</i>] → <i>M</i> in <i>M</i> is invariant under positively oriented <a href="/wiki/Parametrization_(geometry)" title="Parametrization (geometry)">reparametrizations</a>. A constant speed curve <i>γ</i> is a <a href="/wiki/Geodesic" title="Geodesic">geodesic</a> of a Finsler manifold if its short enough segments <i>γ</i>|<sub>[<i>c</i>,<i>d</i>]</sub> are length-minimizing in <i>M</i> from <i>γ</i>(<i>c</i>) to <i>γ</i>(<i>d</i>). Equivalently, <i>γ</i> is a geodesic if it is stationary for the energy functional </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 E[\gamma ]:={\frac {1}{2}}\int _{a}^{b}F^{2}\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>γ</mi> <mo stretchy="false">]</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[\gamma ]:={\frac {1}{2}}\int _{a}^{b}F^{2}\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2476d590e6b0230daacab01c01e20d6618c548c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.003ex; height:6.343ex;" alt="{\displaystyle E[\gamma ]:={\frac {1}{2}}\int _{a}^{b}F^{2}\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt}"></span></dd></dl> <p>in the sense that its <a href="/wiki/Functional_derivative" title="Functional derivative">functional derivative</a> vanishes among differentiable curves <span class="nowrap"><i>γ</i>: [<i>a</i>, <i>b</i>] → <i>M</i></span> with fixed endpoints <span class="nowrap"><i>γ</i>(<i>a</i>) = <i>x</i></span> and <span class="nowrap"><i>γ</i>(<i>b</i>) = <i>y</i></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Canonical_spray_structure_on_a_Finsler_manifold">Canonical spray structure on a Finsler manifold</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finsler_manifold&action=edit&section=6" title="Edit section: Canonical spray structure on a Finsler manifold"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</a> for the energy functional <i>E</i>[<i>γ</i>] reads in the local coordinates (<i>x</i><sup>1</sup>, ..., <i>x</i><sup>n</sup>, <i>v</i><sup>1</sup>, ..., <i>v</i><sup>n</sup>) of T<i>M</i> 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 g_{ik}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}{\ddot {\gamma }}^{i}(t)+\left({\frac {\partial g_{ik}}{\partial x^{j}}}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}-{\frac {1}{2}}{\frac {\partial g_{ij}}{\partial x^{k}}}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}\right){\dot {\gamma }}^{i}(t){\dot {\gamma }}^{j}(t)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{ik}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}{\ddot {\gamma }}^{i}(t)+\left({\frac {\partial g_{ik}}{\partial x^{j}}}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}-{\frac {1}{2}}{\frac {\partial g_{ij}}{\partial x^{k}}}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}\right){\dot {\gamma }}^{i}(t){\dot {\gamma }}^{j}(t)=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/211a3eaf9935318516476e194d5266211b59d26f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:77.959ex; height:6.343ex;" alt="{\displaystyle g_{ik}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}{\ddot {\gamma }}^{i}(t)+\left({\frac {\partial g_{ik}}{\partial x^{j}}}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}-{\frac {1}{2}}{\frac {\partial g_{ij}}{\partial x^{k}}}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}\right){\dot {\gamma }}^{i}(t){\dot {\gamma }}^{j}(t)=0,}"></span></dd></dl> <p>where <i>k</i> = 1, ..., <i>n</i> and <i>g</i><sub>ij</sub> is the coordinate representation of the fundamental tensor, defined 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 g_{ij}(x,v):=g_{v}\left(\left.{\frac {\partial }{\partial x^{i}}}\right|_{x},\left.{\frac {\partial }{\partial x^{j}}}\right|_{x}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{ij}(x,v):=g_{v}\left(\left.{\frac {\partial }{\partial x^{i}}}\right|_{x},\left.{\frac {\partial }{\partial x^{j}}}\right|_{x}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc344e9d75937ee7b5fe97ab0c767c061e3dedae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.963ex; height:6.176ex;" alt="{\displaystyle g_{ij}(x,v):=g_{v}\left(\left.{\frac {\partial }{\partial x^{i}}}\right|_{x},\left.{\frac {\partial }{\partial x^{j}}}\right|_{x}\right).}"></span></dd></dl> <p>Assuming the <a href="/wiki/Convex_function#Strongly_convex_functions" title="Convex function">strong convexity</a> of <i>F</i><sup>2</sup>(<i>x</i>, <i>v</i>) with respect to <i>v</i> ∈ T<sub><i>x</i></sub><i>M</i>, the matrix <i>g</i><sub><i>ij</i></sub>(<i>x</i>, <i>v</i>) is invertible and its inverse is denoted by <i>g</i><sup><i>ij</i></sup>(<i>x</i>, <i>v</i>). Then <span class="nowrap"><i>γ</i>: [<i>a</i>, <i>b</i>] → <i>M</i></span> is a geodesic of (<i>M</i>, <i>F</i>) if and only if its tangent curve <span class="nowrap"><i>γ'</i>: [<i>a</i>, <i>b</i>] → T<i>M</i>∖{0} </span> is an <a href="/wiki/Integral_curve" title="Integral curve">integral curve</a> of the <a href="/wiki/Vector_field" title="Vector field">smooth vector field</a> <i>H</i> on T<i>M</i>∖{0} locally defined by </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 \left.H\right|_{(x,v)}:=\left.v^{i}{\frac {\partial }{\partial x^{i}}}\right|_{(x,v)}\!\!-\left.2G^{i}(x,v){\frac {\partial }{\partial v^{i}}}\right|_{(x,v)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mi>H</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>:=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>−</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mn>2</mn> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.H\right|_{(x,v)}:=\left.v^{i}{\frac {\partial }{\partial x^{i}}}\right|_{(x,v)}\!\!-\left.2G^{i}(x,v){\frac {\partial }{\partial v^{i}}}\right|_{(x,v)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89af8323ce9de51b10d26380087f37c34bcbb282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.964ex; height:6.343ex;" alt="{\displaystyle \left.H\right|_{(x,v)}:=\left.v^{i}{\frac {\partial }{\partial x^{i}}}\right|_{(x,v)}\!\!-\left.2G^{i}(x,v){\frac {\partial }{\partial v^{i}}}\right|_{(x,v)},}"></span></dd></dl> <p>where the local spray coefficients <i>G</i><sup>i</sup> are given by </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^{i}(x,v):={\frac {1}{4}}g^{ij}(x,v)\left(2{\frac {\partial g_{jk}}{\partial x^{\ell }}}(x,v)-{\frac {\partial g_{k\ell }}{\partial x^{j}}}(x,v)\right)v^{k}v^{\ell }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G^{i}(x,v):={\frac {1}{4}}g^{ij}(x,v)\left(2{\frac {\partial g_{jk}}{\partial x^{\ell }}}(x,v)-{\frac {\partial g_{k\ell }}{\partial x^{j}}}(x,v)\right)v^{k}v^{\ell }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05805234f832d0b36f4a22ed7f18f793ee63ed40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.342ex; height:6.343ex;" alt="{\displaystyle G^{i}(x,v):={\frac {1}{4}}g^{ij}(x,v)\left(2{\frac {\partial g_{jk}}{\partial x^{\ell }}}(x,v)-{\frac {\partial g_{k\ell }}{\partial x^{j}}}(x,v)\right)v^{k}v^{\ell }.}"></span></dd></dl> <p>The vector field <i>H</i> on T<i>M</i>∖{0} satisfies <i>JH</i> = <i>V</i> and [<i>V</i>, <i>H</i>] = <i>H</i>, where <i>J</i> and <i>V</i> are the <a href="/wiki/Double_tangent_bundle#Canonical_tensor_fields_on_the_tangent_bundle" title="Double tangent bundle">canonical endomorphism</a> and the <a href="/wiki/Double_tangent_bundle#Canonical_tensor_fields_on_the_tangent_bundle" title="Double tangent bundle">canonical vector field</a> on T<i>M</i>∖{0}. Hence, by definition, <i>H</i> is a <a href="/wiki/Spray_(mathematics)" title="Spray (mathematics)">spray</a> on <i>M</i>. The spray <i>H</i> defines a <a href="/wiki/Ehresmann_connection" title="Ehresmann connection">nonlinear connection</a> on the <a href="/wiki/Fibre_bundle" class="mw-redirect" title="Fibre bundle">fibre bundle</a> <span class="nowrap">T<i>M</i>∖{0} → <i>M</i></span> through the <a href="/wiki/Ehresmann_connection" title="Ehresmann connection">vertical projection</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v:T(\mathrm {T} M\setminus \{0\})\to T(\mathrm {T} M\setminus \{0\});\quad v:={\frac {1}{2}}{\big (}I+{\mathcal {L}}_{H}J{\big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>:</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mi>M</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mi>M</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="1em" /> <mi>v</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>I</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v:T(\mathrm {T} M\setminus \{0\})\to T(\mathrm {T} M\setminus \{0\});\quad v:={\frac {1}{2}}{\big (}I+{\mathcal {L}}_{H}J{\big )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4273f31204712d3334ca5558cc80d299200951c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.958ex; height:5.176ex;" alt="{\displaystyle v:T(\mathrm {T} M\setminus \{0\})\to T(\mathrm {T} M\setminus \{0\});\quad v:={\frac {1}{2}}{\big (}I+{\mathcal {L}}_{H}J{\big )}.}"></span></dd></dl> <p>In analogy with the <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a> case, there is a version </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_{\dot {\gamma }}D_{\dot {\gamma }}X(t)+R_{\dot {\gamma }}\left({\dot {\gamma }}(t),X(t)\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </msub> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </msub> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\dot {\gamma }}D_{\dot {\gamma }}X(t)+R_{\dot {\gamma }}\left({\dot {\gamma }}(t),X(t)\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc5acf987f23acc763cbd20b8459847ba5f1cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:32.488ex; height:3.343ex;" alt="{\displaystyle D_{\dot {\gamma }}D_{\dot {\gamma }}X(t)+R_{\dot {\gamma }}\left({\dot {\gamma }}(t),X(t)\right)=0}"></span></dd></dl> <p>of the <a href="/wiki/Jacobi_equation" class="mw-redirect" title="Jacobi equation">Jacobi equation</a> for a general spray structure (<i>M</i>, <i>H</i>) in terms of the <a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann curvature</a> and <a href="/wiki/Double_tangent_bundle#Nonlinear_covariant_derivatives_on_smooth_manifolds" title="Double tangent bundle">nonlinear covariant derivative</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Uniqueness_and_minimizing_properties_of_geodesics">Uniqueness and minimizing properties of geodesics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finsler_manifold&action=edit&section=7" title="Edit section: Uniqueness and minimizing properties of geodesics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By <a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow theorem</a> there always exist length minimizing curves (at least in small enough neighborhoods) on (<i>M</i>, <i>F</i>). Length minimizing curves can always be positively reparametrized to be geodesics, and any geodesic must satisfy the Euler–Lagrange equation for <i>E</i>[<i>γ</i>]. Assuming the strong convexity of <i>F</i><sup>2</sup> there exists a unique maximal geodesic <i>γ</i> with <i>γ</i>(0) = x and <i>γ'</i>(0) = v for any (<i>x</i>, <i>v</i>) ∈ T<i>M</i>∖{0} by the uniqueness of <a href="/wiki/Integral_curve" title="Integral curve">integral curves</a>. </p><p>If <i>F</i><sup>2</sup> is strongly convex, geodesics <i>γ</i>: [0, <i>b</i>] → <i>M</i> are length-minimizing among nearby curves until the first point <i>γ</i>(<i>s</i>) <a href="/wiki/Conjugate_point" class="mw-redirect" title="Conjugate point">conjugate</a> to <i>γ</i>(0) along <i>γ</i>, and for <i>t</i> > <i>s</i> there always exist shorter curves from <i>γ</i>(0) to <i>γ</i>(<i>t</i>) near <i>γ</i>, as in the <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a> case. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finsler_manifold&action=edit&section=8" title="Edit section: Notes"><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"> <div class="mw-references-wrap"><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"><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="CITEREFRanders1941" class="citation journal cs1"><a href="/wiki/Gunnar_Randers" title="Gunnar Randers">Randers, G.</a> (1941). "On an Asymmetrical Metric in the Four-Space of General Relativity". <i><a href="/wiki/Physical_Review" title="Physical Review">Phys. Rev.</a></i> <b>59</b> (2): 195–199. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.59.195">10.1103/PhysRev.59.195</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F134230">10338.dmlcz/134230</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.&rft.atitle=On+an+Asymmetrical+Metric+in+the+Four-Space+of+General+Relativity&rft.volume=59&rft.issue=2&rft.pages=195-199&rft.date=1941&rft_id=info%3Ahdl%2F10338.dmlcz%2F134230&rft_id=info%3Adoi%2F10.1103%2FPhysRev.59.195&rft.aulast=Randers&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinsler+manifold" class="Z3988"></span></span> </li> </ol></div></div> <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=Finsler_manifold&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a> – Manifold modeled on Banach spaces</li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a> – topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Global_analysis" title="Global analysis">Global analysis</a> – which uses Hilbert manifolds and other kinds of infinite-dimensional manifolds</li> <li><a href="/wiki/Hilbert_manifold" title="Hilbert manifold">Hilbert manifold</a> – Manifold modelled on Hilbert spaces</li></ul> <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=Finsler_manifold&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntonelli2003" class="citation cs2"><a href="/wiki/Peter_L._Antonelli" title="Peter L. Antonelli">Antonelli, Peter L.</a>, ed. (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=b2B5_IUvPJgC"><i>Handbook of Finsler geometry. Vol. 1, 2</i></a>, Boston: Kluwer Academic Publishers, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-1557-1" title="Special:BookSources/978-1-4020-1557-1"><bdi>978-1-4020-1557-1</bdi></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=2067663">2067663</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Finsler+geometry.+Vol.+1%2C+2&rft.place=Boston&rft.pub=Kluwer+Academic+Publishers&rft.date=2003&rft.isbn=978-1-4020-1557-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2067663%23id-name%3DMR&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Db2B5_IUvPJgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinsler+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaoChernShen2000" class="citation book cs1">Bao, David; <a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern, Shiing-Shen</a>; Shen, Zhongmin (2000). <i>An introduction to Riemann–Finsler geometry</i>. Graduate Texts in Mathematics. Vol. 200. New York: Springer-Verlag. <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%2F978-1-4612-1268-3">10.1007/978-1-4612-1268-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98948-X" title="Special:BookSources/0-387-98948-X"><bdi>0-387-98948-X</bdi></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=1747675">1747675</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+Riemann%E2%80%93Finsler+geometry&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=2000&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1747675%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-1268-3&rft.isbn=0-387-98948-X&rft.aulast=Bao&rft.aufirst=David&rft.au=Chern%2C+Shiing-Shen&rft.au=Shen%2C+Zhongmin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinsler+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCartan1933" class="citation cs2"><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan, Élie</a> (1933), "Sur les espaces de Finsler", <i><a href="/wiki/Comptes_rendus_de_l%27Acad%C3%A9mie_des_Sciences" title="Comptes rendus de l'Académie des Sciences">C. R. Acad. Sci. Paris</a></i>, <b>196</b>: 582–586, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0006.22501">0006.22501</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=C.+R.+Acad.+Sci.+Paris&rft.atitle=Sur+les+espaces+de+Finsler&rft.volume=196&rft.pages=582-586&rft.date=1933&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0006.22501%23id-name%3DZbl&rft.aulast=Cartan&rft.aufirst=%C3%89lie&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinsler+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChern1996" class="citation cs2"><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern, Shiing-Shen</a> (1996), <a rel="nofollow" class="external text" href="https://www.ams.org/notices/199609/chern.pdf">"Finsler geometry is just Riemannian geometry without the quadratic restriction"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a></i>, <b>43</b> (9): 959–63, <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=1400859">1400859</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=Finsler+geometry+is+just+Riemannian+geometry+without+the+quadratic+restriction&rft.volume=43&rft.issue=9&rft.pages=959-63&rft.date=1996&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1400859%23id-name%3DMR&rft.aulast=Chern&rft.aufirst=Shiing-Shen&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F199609%2Fchern.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinsler+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFinsler1918" class="citation cs2">Finsler, Paul (1918), <i>Über Kurven und Flächen in allgemeinen Räumen</i>, Dissertation, Göttingen, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:46.1131.02">46.1131.02</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%9Cber+Kurven+und+Fl%C3%A4chen+in+allgemeinen+R%C3%A4umen&rft.series=Dissertation&rft.pub=G%C3%B6ttingen&rft.date=1918&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A46.1131.02%23id-name%3DJFM&rft.aulast=Finsler&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinsler+manifold" class="Z3988"></span> (Reprinted by Birkhäuser (1951))</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRund1959" class="citation book cs1"><a href="/wiki/Hanno_Rund" title="Hanno Rund">Rund, Hanno</a> (1959). <i>The differential geometry of Finsler spaces</i>. Die Grundlehren der Mathematischen Wissenschaften. Vol. 101. Berlin–Göttingen–Heidelberg: Springer-Verlag. <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%2F978-3-642-51610-8">10.1007/978-3-642-51610-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-51612-2" title="Special:BookSources/978-3-642-51612-2"><bdi>978-3-642-51612-2</bdi></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=0105726">0105726</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+differential+geometry+of+Finsler+spaces&rft.place=Berlin%E2%80%93G%C3%B6ttingen%E2%80%93Heidelberg&rft.series=Die+Grundlehren+der+Mathematischen+Wissenschaften&rft.pub=Springer-Verlag&rft.date=1959&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0105726%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-51610-8&rft.isbn=978-3-642-51612-2&rft.aulast=Rund&rft.aufirst=Hanno&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinsler+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShen2001" class="citation book cs1">Shen, Zhongmin (2001). <i>Lectures on Finsler geometry</i>. Singapore: World Scientific. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F4619">10.1142/4619</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/981-02-4531-9" title="Special:BookSources/981-02-4531-9"><bdi>981-02-4531-9</bdi></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=1845637">1845637</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Finsler+geometry&rft.place=Singapore&rft.pub=World+Scientific&rft.date=2001&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1845637%23id-name%3DMR&rft_id=info%3Adoi%2F10.1142%2F4619&rft.isbn=981-02-4531-9&rft.aulast=Shen&rft.aufirst=Zhongmin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinsler+manifold" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finsler_manifold&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Finsler_space,_generalized">"Finsler space, generalized"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Finsler+space%2C+generalized&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DFinsler_space%2C_generalized&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinsler+manifold" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://finsler.blogspot.com/">The (New) Finsler Newsletter</a></li></ul> <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 li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Manifolds_(Glossary)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</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/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></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/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</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/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />manifolds</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/Closed_manifold" title="Closed manifold">Closed</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a class="mw-selflink selflink">Finsler</a></li> <li><a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></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%">Vectors</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/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</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/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</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/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</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/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</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/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</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/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Riemannian_geometry_(Glossary)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Riemannian_geometry" title="Template:Riemannian geometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Riemannian_geometry" title="Template talk:Riemannian geometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Riemannian_geometry" title="Special:EditPage/Template:Riemannian geometry"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Riemannian_geometry_(Glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a> (<a href="/wiki/Glossary_of_Riemannian_and_metric_geometry" title="Glossary of Riemannian and metric geometry">Glossary</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/Curvature_of_Riemannian_manifolds" title="Curvature of Riemannian manifolds">Curvature</a> <ul><li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">tensor</a></li> <li><a href="/wiki/Scalar_curvature" title="Scalar curvature">Scalar</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci</a></li> <li><a href="/wiki/Sectional_curvature" title="Sectional curvature">Sectional</a></li></ul></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a> <ul><li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Metric_signature" title="Metric signature">Signature</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a>/<a href="/wiki/Musical_isomorphism" title="Musical isomorphism">Musical isomorphism</a></li></ul></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-Riemannian manifold</a></li> <li><a href="/wiki/Riemannian_volume_form" class="mw-redirect" title="Riemannian volume form">Riemannian volume form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of manifolds</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/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</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-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fundamental_theorem_of_Riemannian_geometry" title="Fundamental theorem of Riemannian geometry">Fundamental theorem of Riemannian geometry</a></li> <li><a href="/wiki/Gauss%27s_lemma_(Riemannian_geometry)" title="Gauss's lemma (Riemannian geometry)">Gauss's lemma</a></li> <li><a href="/wiki/Gauss%E2%80%93Bonnet_theorem" title="Gauss–Bonnet theorem">Gauss–Bonnet theorem</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow theorem</a></li> <li><a href="/wiki/Nash_embedding_theorem" class="mw-redirect" title="Nash embedding theorem">Nash embedding theorem</a></li> <li><a href="/wiki/Ricci_flow" title="Ricci flow">Ricci flow</a></li> <li><a href="/wiki/Schur%27s_lemma_(Riemannian_geometry)" title="Schur's lemma (Riemannian geometry)">Schur's lemma</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 class="mw-selflink selflink">Finsler</a></li> <li><a href="/wiki/Hilbert_manifold" title="Hilbert manifold">Hilbert</a></li> <li><a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub-Riemannian</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</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/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">Geometrization conjecture</a></li> <li><a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a></li> <li><a href="/wiki/Uniformization_theorem" title="Uniformization theorem">Uniformization theorem</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Finsler_manifold&oldid=1234056884">https://en.wikipedia.org/w/index.php?title=Finsler_manifold&oldid=1234056884</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Differential_geometry" title="Category:Differential geometry">Differential geometry</a></li><li><a href="/wiki/Category:Finsler_geometry" title="Category:Finsler geometry">Finsler geometry</a></li><li><a href="/wiki/Category:Riemannian_geometry" title="Category:Riemannian geometry">Riemannian geometry</a></li><li><a href="/wiki/Category:Riemannian_manifolds" title="Category:Riemannian manifolds">Riemannian manifolds</a></li><li><a href="/wiki/Category:Smooth_manifolds" title="Category:Smooth manifolds">Smooth manifolds</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_May_2017" title="Category:Articles needing additional references from May 2017">Articles needing additional references from May 2017</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Pages_displaying_wikidata_descriptions_as_a_fallback_via_Module:Annotated_link" title="Category:Pages displaying wikidata descriptions as a fallback via Module:Annotated link">Pages displaying wikidata descriptions as a fallback via Module:Annotated link</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 12 July 2024, at 10:32<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Finsler_manifold&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-5qjh9","wgBackendResponseTime":132,"wgPageParseReport":{"limitreport":{"cputime":"0.477","walltime":"0.702","ppvisitednodes":{"value":3387,"limit":1000000},"postexpandincludesize":{"value":73798,"limit":2097152},"templateargumentsize":{"value":3912,"limit":2097152},"expansiondepth":{"value":9,"limit":100},"expensivefunctioncount":{"value":4,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":42381,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 529.440 1 -total"," 24.29% 128.591 3 Template:Annotated_link"," 13.90% 73.569 1 Template:Manifolds"," 13.80% 73.064 2 Template:Navbox"," 13.73% 72.687 1 Template:Reflist"," 11.98% 63.449 1 Template:Cite_journal"," 11.89% 62.957 1 Template:Short_description"," 9.02% 47.760 1 Template:Refimprove"," 8.19% 43.338 1 Template:Ambox"," 7.93% 41.989 2 Template:Pagetype"]},"scribunto":{"limitreport-timeusage":{"value":"0.298","limit":"10.000"},"limitreport-memusage":{"value":22092629,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAntonelli2003\"] = 1,\n [\"CITEREFBaoChernShen2000\"] = 1,\n [\"CITEREFCartan1933\"] = 1,\n [\"CITEREFChern1996\"] = 1,\n [\"CITEREFFinsler1918\"] = 1,\n [\"CITEREFRanders1941\"] = 1,\n [\"CITEREFRund1959\"] = 1,\n [\"CITEREFShen2001\"] = 1,\n}\ntemplate_list = table#1 {\n [\"=\"] = 5,\n [\"Annotated link\"] = 3,\n [\"Citation\"] = 4,\n [\"Cite book\"] = 3,\n [\"Cite journal\"] = 1,\n [\"Frac\"] = 2,\n [\"Harv\"] = 1,\n [\"Harvs\"] = 1,\n [\"Manifolds\"] = 1,\n [\"Math\"] = 33,\n [\"Nobreak\"] = 2,\n [\"Nowrap\"] = 5,\n [\"Redirect\"] = 1,\n [\"Refimprove\"] = 1,\n [\"Reflist\"] = 1,\n [\"Riemannian geometry\"] = 1,\n [\"Short description\"] = 1,\n [\"Springer\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-5c59558b9d-grc9c","timestamp":"20241130124320","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Finsler manifold","url":"https:\/\/en.wikipedia.org\/wiki\/Finsler_manifold","sameAs":"http:\/\/www.wikidata.org\/entity\/Q2498782","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q2498782","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-12-02T16:19:19Z","dateModified":"2024-07-12T10:32:55Z","headline":"smooth manifold equipped with a Minkowski functional at each tangent space"}</script> </body> </html>

CINXE.COM

Finsler manifold - Wikipedia