Línea geodésica - Wikipedia, la enciclopedia libre

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class="vector-toc-link" href="#Introducción"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Introducción</span> </div> </a> <ul id="toc-Introducción-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Las_geodésicas_como_curvas_de_aceleración_nula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Las_geodésicas_como_curvas_de_aceleración_nula"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Las geodésicas como curvas de aceleración nula</span> </div> </a> <button aria-controls="toc-Las_geodésicas_como_curvas_de_aceleración_nula-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>Alternar subsección Las geodésicas como curvas de aceleración nula</span> </button> <ul id="toc-Las_geodésicas_como_curvas_de_aceleración_nula-sublist" class="vector-toc-list"> <li id="toc-Conexión_afín" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conexión_afín"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Conexión afín</span> </div> </a> <ul id="toc-Conexión_afín-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivada_covariante_y_transporte_paralelo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivada_covariante_y_transporte_paralelo"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Derivada covariante y transporte paralelo</span> </div> </a> <ul id="toc-Derivada_covariante_y_transporte_paralelo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geodésicas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geodésicas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Geodésicas</span> </div> </a> <ul id="toc-Geodésicas-sublist" class="vector-toc-list"> <li id="toc-Ejemplos" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ejemplos"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Ejemplos</span> </div> </a> <ul id="toc-Ejemplos-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Curvas_integrales_en_el_fibrado_tangente" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Curvas_integrales_en_el_fibrado_tangente"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Curvas integrales en el fibrado tangente</span> </div> </a> <ul id="toc-Curvas_integrales_en_el_fibrado_tangente-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Véase_también" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véase_también"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Véase también</span> </div> </a> <ul id="toc-Véase_también-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencias" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencias"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Referencias</span> </div> </a> <ul id="toc-Referencias-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografía" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografía"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bibliografía</span> </div> </a> <ul id="toc-Bibliografía-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="Contenidos" 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="Cambiar a la tabla de contenidos" > <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">Cambiar a la tabla de contenidos</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">Línea geodésica</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="Ir a un artículo en otro idioma. Disponible en 42 idiomas" > <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-42" 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">42 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D9%8A%D9%88%D8%AF%D9%8A%D8%B3%D9%8A" title="جيوديسي (árabe)" lang="ar" hreflang="ar" data-title="جيوديسي" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AD%E0%A7%82%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A6%95%E0%A7%8D%E0%A6%B0%E0%A6%B0%E0%A7%87%E0%A6%96%E0%A6%BE" title="ভূমিতিক বক্ররেখা (bengalí)" lang="bn" hreflang="bn" data-title="ভূমিতিক বক্ররেখা" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Geod%C3%A8sica" title="Geodèsica (catalán)" lang="ca" hreflang="ca" data-title="Geodèsica" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Geodetika" title="Geodetika (checo)" lang="cs" hreflang="cs" data-title="Geodetika" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D0%BB%D0%BB%D0%B5_%D0%B9%C4%95%D1%80" title="Геодезилле йĕр (chuvasio)" lang="cv" hreflang="cv" data-title="Геодезилле йĕр" data-language-autonym="Чӑвашла" data-language-local-name="chuvasio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Geod%C3%A4te" title="Geodäte (alemán)" lang="de" hreflang="de" data-title="Geodäte" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CF%89%CE%B4%CE%B1%CE%B9%CF%83%CE%B9%CE%B1%CE%BA%CE%AE" title="Γεωδαισιακή (griego)" lang="el" hreflang="el" data-title="Γεωδαισιακή" data-language-autonym="Ελληνικά" data-language-local-name="griego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Geodesic" title="Geodesic (inglés)" lang="en" hreflang="en" data-title="Geodesic" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Geodezia_linio" title="Geodezia linio (esperanto)" lang="eo" hreflang="eo" data-title="Geodezia linio" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Geodeetiline_joon" title="Geodeetiline joon (estonio)" lang="et" hreflang="et" data-title="Geodeetiline joon" data-language-autonym="Eesti" data-language-local-name="estonio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%98%D8%A6%D9%88%D8%AF%D8%B2%DB%8C%DA%A9" title="ژئودزیک (persa)" lang="fa" hreflang="fa" data-title="ژئودزیک" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Geodeesi" title="Geodeesi (finés)" lang="fi" hreflang="fi" data-title="Geodeesi" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/G%C3%A9od%C3%A9sique" title="Géodésique (francés)" lang="fr" hreflang="fr" data-title="Géodésique" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Geodasach" title="Geodasach (irlandés)" lang="ga" hreflang="ga" data-title="Geodasach" data-language-autonym="Gaeilge" data-language-local-name="irlandés" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Xeod%C3%A9sica" title="Xeodésica (gallego)" lang="gl" hreflang="gl" data-title="Xeodésica" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%99%D7%9C%D7%94_%D7%92%D7%90%D7%95%D7%93%D7%96%D7%99%D7%AA" title="מסילה גאודזית (hebreo)" lang="he" hreflang="he" data-title="מסילה גאודזית" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Geodetikus_vonalak" title="Geodetikus vonalak (húngaro)" lang="hu" hreflang="hu" data-title="Geodetikus vonalak" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D5%A5%D5%B8%D5%A4%D5%A5%D5%A6%D5%AB%D5%AF_%D5%A3%D5%AE%D5%A5%D6%80" title="Գեոդեզիկ գծեր (armenio)" lang="hy" hreflang="hy" data-title="Գեոդեզիկ գծեր" data-language-autonym="Հայերեն" data-language-local-name="armenio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Geodesik" title="Geodesik (indonesio)" lang="id" hreflang="id" data-title="Geodesik" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Geodetica" title="Geodetica (italiano)" lang="it" hreflang="it" data-title="Geodetica" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%B8%AC%E5%9C%B0%E7%B7%9A" title="測地線 (japonés)" lang="ja" hreflang="ja" data-title="測地線" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D1%81%D1%8B%D0%B7%D1%8B%D2%9B%D1%82%D0%B0%D1%80" title="Геодезиялық сызықтар (kazajo)" lang="kk" hreflang="kk" data-title="Геодезиялық сызықтар" data-language-autonym="Қазақша" data-language-local-name="kazajo" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B8%A1%EC%A7%80%EC%84%A0" title="측지선 (coreano)" lang="ko" hreflang="ko" data-title="측지선" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Geodetesch_Linn" title="Geodetesch Linn (luxemburgués)" lang="lb" hreflang="lb" data-title="Geodetesch Linn" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxemburgués" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Geodezin%C4%97_kreiv%C4%97" title="Geodezinė kreivė (lituano)" lang="lt" hreflang="lt" data-title="Geodezinė kreivė" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geodeet_(wiskunde)" title="Geodeet (wiskunde) (neerlandés)" lang="nl" hreflang="nl" data-title="Geodeet (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Geodetisk_kurve" title="Geodetisk kurve (noruego nynorsk)" lang="nn" hreflang="nn" data-title="Geodetisk kurve" data-language-autonym="Norsk nynorsk" data-language-local-name="noruego nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Geodetisk_kurve" title="Geodetisk kurve (noruego bokmal)" lang="nb" hreflang="nb" data-title="Geodetisk kurve" data-language-autonym="Norsk bokmål" data-language-local-name="noruego bokmal" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Linia_geodezyjna" title="Linia geodezyjna (polaco)" lang="pl" hreflang="pl" data-title="Linia geodezyjna" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Geod%C3%A9sica" title="Geodésica (portugués)" lang="pt" hreflang="pt" data-title="Geodésica" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Geodezic%C4%83" title="Geodezică (rumano)" lang="ro" hreflang="ro" data-title="Geodezică" data-language-autonym="Română" data-language-local-name="rumano" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F" title="Геодезическая (ruso)" lang="ru" hreflang="ru" data-title="Геодезическая" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Geodezijska_linija" title="Geodezijska linija (serbocroata)" lang="sh" hreflang="sh" data-title="Geodezijska linija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Geodesic" title="Geodesic (Simple English)" lang="en-simple" hreflang="en-simple" data-title="Geodesic" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Geodetka" title="Geodetka (esloveno)" lang="sl" hreflang="sl" data-title="Geodetka" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D1%98%D1%81%D0%BA%D0%B0_%D0%BB%D0%B8%D0%BD%D0%B8%D1%98%D0%B0" title="Геодезијска линија (serbio)" lang="sr" hreflang="sr" data-title="Геодезијска линија" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Jeodezik" title="Jeodezik (turco)" lang="tr" hreflang="tr" data-title="Jeodezik" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D1%87%D0%BD%D0%B0_%D0%BB%D1%96%D0%BD%D1%96%D1%8F" title="Геодезична лінія (ucraniano)" lang="uk" hreflang="uk" data-title="Геодезична лінія" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Geodezik_chiziq" title="Geodezik chiziq (uzbeko)" lang="uz" hreflang="uz" data-title="Geodezik chiziq" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeko" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba" title="Đường trắc địa (vietnamita)" lang="vi" hreflang="vi" data-title="Đường trắc địa" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B5%8B%E5%9C%B0%E7%BA%BF" title="测地线 (chino)" lang="zh" 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</div> <div id="siteSub" class="noprint">De Wikipedia, la enciclopedia libre</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirigido desde «<a href="/w/index.php?title=Geod%C3%A9sica&redirect=no" class="mw-redirect" title="Geodésica">Geodésica</a>»)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="es" dir="ltr"><div class="rellink noprint hatnote">Este artículo trata sobre el término línea geodésica en <a href="/wiki/Geometr%C3%ADa" title="Geometría">geometría</a>. Para el término en <a href="/wiki/Teor%C3%ADa_de_grafos" title="Teoría de grafos">teoría de grafos</a>, véase <a href="/wiki/Problema_del_camino_m%C3%A1s_corto" title="Problema del camino más corto">problema del camino más corto</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Archivo:Geodesiques.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Geodesiques.png/300px-Geodesiques.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/1f/Geodesiques.png 1.5x" data-file-width="400" data-file-height="300" /></a><figcaption>Dos líneas geodésicas, en rojo, sobre una superficie curva, esas geodésicas coinciden con las trayectorias de dos partículas en el <a href="/wiki/M%C3%A9trica_de_Schwarzschild" title="Métrica de Schwarzschild">campo gravitatorio esférico</a> de una masa central de acuerdo con la <a href="/wiki/Relatividad_general" title="Relatividad general">teoría general de la relatividad</a>.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/Archivo:Spherical_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Spherical_triangle.svg/230px-Spherical_triangle.svg.png" decoding="async" width="230" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Spherical_triangle.svg/345px-Spherical_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Spherical_triangle.svg/460px-Spherical_triangle.svg.png 2x" data-file-width="356" data-file-height="358" /></a><figcaption>Triángulo geodésico sobre una esfera. La línea geodésica sería cualquiera de los arcos que forman los triángulos.</figcaption></figure> <p>En <a href="/wiki/Geometr%C3%ADa" title="Geometría">geometría</a>, la <b>línea</b> <b>geodésica</b> se define como la línea de mínima <a href="/wiki/Longitud" title="Longitud">longitud</a> que une dos puntos en una superficie dada, y está contenida en esta superficie. El <a href="/wiki/Plano_osculador" class="mw-redirect" title="Plano osculador">plano osculador</a> de la geodésica es perpendicular en cualquier punto al plano tangente a la superficie. Las geodésicas de una superficie son las líneas "más rectas" posibles (con menor curvatura) fijado un punto y una dirección dada sobre dicha superficie. </p><p>Más generalmente, se puede hablar de geodésicas en "espacios curvados" de dimensión superior llamados <a href="/wiki/Variedad_de_Riemann" title="Variedad de Riemann">variedades riemannianas</a> en donde, si el espacio contiene una métrica natural, entonces las geodésicas son (<a href="/wiki/Localmente" title="Localmente">localmente</a>) la <a href="/wiki/Distancia" title="Distancia">distancia</a> más corta entre dos puntos en el espacio. Un ejemplo físico, de variedad semirriemanniana es el que aparece en la <a href="/wiki/Teor%C3%ADa_general_de_la_relatividad" class="mw-redirect" title="Teoría general de la relatividad">teoría de la relatividad general</a>, que establece que las partículas materiales se mueven a lo largo de geodésicas temporales del <a href="/wiki/Espacio-tiempo" title="Espacio-tiempo">espacio-tiempo</a> curvo. </p><p>El término "geodésico" proviene de la palabra <i><a href="/wiki/Geodesia" title="Geodesia">geodesia</a></i>, la ciencia de medir el tamaño y forma del planeta <a href="/wiki/Tierra" title="Tierra">Tierra</a>; en el sentido original, fue la ruta más corta entre dos puntos sobre la superficie de la <a href="/wiki/Tierra" title="Tierra">Tierra</a>, específicamente, el <a href="/wiki/Segmento" title="Segmento">segmento</a> de un <a href="/wiki/Gran_c%C3%ADrculo" title="Gran círculo">círculo máximo</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introducción"><span id="Introducci.C3.B3n"></span>Introducción</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%ADnea_geod%C3%A9sica&action=edit&section=1" title="Editar sección: Introducción"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una <a href="/wiki/Variedad_de_Riemann" title="Variedad de Riemann">variedad Riemanianna</a> (M, g) es una <a href="/wiki/Variedad_(matem%C3%A1ticas)" title="Variedad (matemáticas)">variedad</a> diferenciable dotada de una estructura adicional de <a href="/wiki/Espacio_m%C3%A9trico" title="Espacio métrico">espacio métrico</a> que permite generalizar conceptos de la geometría euclídea a espacios más generales. En concreto, el espacio tangente a cada punto se dota de un <a href="/wiki/Producto_escalar" title="Producto escalar">producto escalar</a>, que determina la métrica en la variedad, dada por: </p><p><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}{\big |}_{p}={\bigg \langle }{\frac {\partial }{\partial x_{i}}}{\bigg |}_{p},{\frac {\partial }{\partial x_{j}}}{\bigg |}_{p}{\bigg \rangle }}"> <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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">⟨</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">⟩</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{ij}{\big |}_{p}={\bigg \langle }{\frac {\partial }{\partial x_{i}}}{\bigg |}_{p},{\frac {\partial }{\partial x_{j}}}{\bigg |}_{p}{\bigg \rangle }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7cd38d51a7aeab8d2321844feaf5cc1f200003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24ex; height:6.343ex;" alt="{\displaystyle g_{ij}{\big |}_{p}={\bigg \langle }{\frac {\partial }{\partial x_{i}}}{\bigg |}_{p},{\frac {\partial }{\partial x_{j}}}{\bigg |}_{p}{\bigg \rangle }}"></span> </p><p>donde <,> es el producto escalar anteriormente definido y p es cualquier punto de la variedad M. </p><p>En función de esta métrica, la longitud <i>L<sub>C</sub></i> a lo largo de una curva contenida en ella se evalúa gracias a las componentes <i>g<sub>ij</sub></i> del <a href="/wiki/Tensor_m%C3%A9trico" title="Tensor métrico">tensor métrico</a> <i>g</i> del siguiente modo: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><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_{C}=\int _{C}{\sqrt {\sum _{i,j}g_{ij}x'_{i}(t)x'_{j}(t)}}\ dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{C}=\int _{C}{\sqrt {\sum _{i,j}g_{ij}x'_{i}(t)x'_{j}(t)}}\ dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a0ff00783836295262508064c0f8a339eeb15f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:30.278ex; height:7.509ex;" alt="{\displaystyle L_{C}=\int _{C}{\sqrt {\sum _{i,j}g_{ij}x'_{i}(t)x'_{j}(t)}}\ dt}"></span> </p> </blockquote> <p>Donde <i>x<sub>i</sub></i>(<i>t</i>) es la expresión paramétrica de los puntos de la <a href="/wiki/Curva" title="Curva">curva parametrizada</a> mediante el parámetro <i>t</i>. Todas las líneas geodésicas son extremales de la integral anterior. Una de las formas de obtener las ecuaciones de las geodésicas es minimizar el funcional anterior. En ese caso las <a href="/wiki/Ecuaciones_de_Euler-Lagrange" title="Ecuaciones de Euler-Lagrange">ecuaciones de Euler-Lagrange</a> proporcionan las curvas geodésicas. En el siguiente apartado seguiremos un enfoque diferente. </p> <div class="mw-heading mw-heading2"><h2 id="Las_geodésicas_como_curvas_de_aceleración_nula"><span id="Las_geod.C3.A9sicas_como_curvas_de_aceleraci.C3.B3n_nula"></span>Las geodésicas como curvas de aceleración nula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%ADnea_geod%C3%A9sica&action=edit&section=2" title="Editar sección: Las geodésicas como curvas de aceleración nula"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>posee una estructura afín que permite 'conectar' espacios tangentes de distintos puntos de una forma natural. Dado un <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d5b6202943d58e511d6616d84cd13ca0bc7547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.996ex; height:2.676ex;" alt="{\displaystyle p\in \mathbb {R} ^{n}}"></span>, podemos identificar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}\mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{p}\mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ff0cee923f9ef0490325b09a0450dc77b3e8d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.313ex; height:3.009ex;" alt="{\displaystyle T_{p}\mathbb {R} ^{n}}"></span> con el propio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>, lo cual permite diferenciar campos vectoriales de manera sencilla e intuitiva. Este no es el caso para variedades Riemannianas más generales, en las que los espacios tangentes a cada punto son espacios vectoriales abstractos, de modo que no tiene sentido realizar operaciones directamente entre ellos. Para ilustrar esto, sea <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\in \chi (M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∈</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\in \chi (M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e35820a940e0628b39c011f2d914ae3cd5fe7ff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.527ex; height:2.843ex;" alt="{\displaystyle X\in \chi (M)}"></span> un <a href="/wiki/Campo_vectorial" title="Campo vectorial">campo vectorial</a> en M y <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 \gamma (t)\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (t)\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c68ac1f9ba5afbbaece10a1f53885fafe115b93d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.194ex; height:2.843ex;" alt="{\displaystyle \gamma (t)\in M}"></span> una curva <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 \gamma :(-\varepsilon ,\varepsilon )\rightarrow M;\quad t\mapsto \gamma (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ε</mi> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo>;</mo> <mspace width="1em" /> <mi>t</mi> <mo stretchy="false">↦</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma :(-\varepsilon ,\varepsilon )\rightarrow M;\quad t\mapsto \gamma (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78afc5b36b97f4b9e1b2bba4c247476b50ab7e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.796ex; height:2.843ex;" alt="{\displaystyle \gamma :(-\varepsilon ,\varepsilon )\rightarrow M;\quad t\mapsto \gamma (t)}"></span>. De forma intuitiva, aplicando las técnicas del cálculo en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>, si quisiéramos calcular la derivada del campo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> en un <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}\in (-\varepsilon ,\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ε</mi> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}\in (-\varepsilon ,\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc76cdd3a95d02570ec16439c5a403b82b84635b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.553ex; height:2.843ex;" alt="{\displaystyle t_{0}\in (-\varepsilon ,\varepsilon )}"></span> a lo largo de la curva <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>, escribiríamos: </p><p><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'|_{t_{0}}=\lim _{h\to 0}{\frac {X|_{\gamma (t_{0}+h)}-X_{\gamma (t_{0})}}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mo>′</mo> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>X</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X'|_{t_{0}}=\lim _{h\to 0}{\frac {X|_{\gamma (t_{0}+h)}-X_{\gamma (t_{0})}}{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51300975bd02827b10da0b13cc33039e891a7ab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.995ex; height:6.343ex;" alt="{\displaystyle X'|_{t_{0}}=\lim _{h\to 0}{\frac {X|_{\gamma (t_{0}+h)}-X_{\gamma (t_{0})}}{h}}}"></span> </p><p><br /> Sin embargo, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X|_{\gamma }(t_{0}+h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X|_{\gamma }(t_{0}+h)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9427cbe9c313fb5a1524993fed9e8e17b64da9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.634ex; height:3.343ex;" alt="{\displaystyle X|_{\gamma }(t_{0}+h)}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X|_{\gamma }(t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X|_{\gamma }(t_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e15db8ea0b1d0a93c5ca96e254368173042f73a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.455ex; height:3.343ex;" alt="{\displaystyle X|_{\gamma }(t_{0})}"></span> pertenecen a espacios vectoriales distintos y la operación anterior no tiene sentido. Para ello es necesario 'conectar' los espacios vectoriales de los distintos puntos de la variedad de forma que la operación anterior tenga sentido. Para ello surge el concepto de <a href="/wiki/Conexi%C3%B3n_(matem%C3%A1tica)" title="Conexión (matemática)">conexión afín</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Conexión_afín"><span id="Conexi.C3.B3n_af.C3.ADn"></span>Conexión afín</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%ADnea_geod%C3%A9sica&action=edit&section=3" title="Editar sección: Conexión afín"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una conexión afín es una aplicación <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 \nabla :\chi (M)\times \chi (M)\rightarrow \chi (M),\quad (X,Y)\mapsto \nabla _{Y}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>:</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla :\chi (M)\times \chi (M)\rightarrow \chi (M),\quad (X,Y)\mapsto \nabla _{Y}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac70a1ef7bd092fa117e3253e9b4e9737d12acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.417ex; height:2.843ex;" alt="{\displaystyle \nabla :\chi (M)\times \chi (M)\rightarrow \chi (M),\quad (X,Y)\mapsto \nabla _{Y}X}"></span> que cumple: <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 i)\;\;\quad \nabla _{f\cdot Y_{1}+g\cdot Y_{2}}X=f\nabla _{Y_{1}}X+g\nabla _{Y_{2}}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="1em" /> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>⋅</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>g</mi> <mo>⋅</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mi>X</mi> <mo>=</mo> <mi>f</mi> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mi>X</mi> <mo>+</mo> <mi>g</mi> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i)\;\;\quad \nabla _{f\cdot Y_{1}+g\cdot Y_{2}}X=f\nabla _{Y_{1}}X+g\nabla _{Y_{2}}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e467414b01cca1403082b303d5fd9a18b498d85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.131ex; height:3.009ex;" alt="{\displaystyle i)\;\;\quad \nabla _{f\cdot Y_{1}+g\cdot Y_{2}}X=f\nabla _{Y_{1}}X+g\nabla _{Y_{2}}X}"></span> </p><p><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 ii)\;\quad \nabla _{Y}(X_{1}+X_{2})=\nabla _{Y}X_{1}+\nabla _{Y}X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ii)\;\quad \nabla _{Y}(X_{1}+X_{2})=\nabla _{Y}X_{1}+\nabla _{Y}X_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e36522600ed40c6112318e6ecb0a2274e22aed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.246ex; height:2.843ex;" alt="{\displaystyle ii)\;\quad \nabla _{Y}(X_{1}+X_{2})=\nabla _{Y}X_{1}+\nabla _{Y}X_{2}}"></span> </p><p><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 iii)\quad \nabla _{Y}(f\cdot X)=f\cdot \nabla _{y}X+X\cdot Y(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>i</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋅</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo>⋅</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>X</mi> <mo>+</mo> <mi>X</mi> <mo>⋅</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle iii)\quad \nabla _{Y}(f\cdot X)=f\cdot \nabla _{y}X+X\cdot Y(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1cb33f4a733ab01ef729deae03989dca9258c1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.186ex; height:3.009ex;" alt="{\displaystyle iii)\quad \nabla _{Y}(f\cdot X)=f\cdot \nabla _{y}X+X\cdot Y(f)}"></span> </p><p>para funciones <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in C^{\infty }(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in C^{\infty }(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0c3fbb7f70cd0f1da1e09cc81f21ec9e1606e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.194ex; height:2.843ex;" alt="{\displaystyle f,g\in C^{\infty }(M)}"></span> y campos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},Y_{1},Y_{2}\in \chi (M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},Y_{1},Y_{2}\in \chi (M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999b5a112ca4fdc9987bcdd3c57611ae4e8b716c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.416ex; height:2.843ex;" alt="{\displaystyle X_{1},X_{2},Y_{1},Y_{2}\in \chi (M)}"></span>. </p><p>Si elegimos una <a href="/wiki/Carta_(matem%C3%A1tica)" title="Carta (matemática)">carta</a> alrededor de un punto p de la variedad: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U\ni p,\varphi =(x_{1},\dots ,x_{n}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>U</mi> <mo>∋</mo> <mi>p</mi> <mo>,</mo> <mi>φ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (U\ni p,\varphi =(x_{1},\dots ,x_{n}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126504f8dec7a3fdfdcc25b03966365af8a2eca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.174ex; height:2.843ex;" alt="{\displaystyle (U\ni p,\varphi =(x_{1},\dots ,x_{n}))}"></span>, entonces podemos expresar la conexión en función de las coordenadas locales. En efecto, sean: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\sum _{i=1}^{n}f_{i}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\sum _{i=1}^{n}f_{i}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95832caffb1ce638f5153d923131a90aaddc96c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.963ex; height:6.843ex;" alt="{\displaystyle X=\sum _{i=1}^{n}f_{i}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{i}}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\sum _{j=1}^{n}g_{j}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\sum _{j=1}^{n}g_{j}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{j}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/403ff955af2c3748009aee6d5daa99a6f8a86bbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.946ex; height:7.176ex;" alt="{\displaystyle Y=\sum _{j=1}^{n}g_{j}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{j}}}}"></span>. Expresemos la conexión <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 \nabla _{Y}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{Y}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb93fdc1285b766125fbac96f26d788c9fdd698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.402ex; height:2.509ex;" alt="{\displaystyle \nabla _{Y}X}"></span> en función de las coordenadas locales: </p><p><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 \nabla _{Y}X=\nabla _{\sum _{j=1}^{n}g_{j}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{j}}}}{\bigg (}\sum _{i=1}^{n}f_{i}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{i}}}{\bigg )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mi>X</mi> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{Y}X=\nabla _{\sum _{j=1}^{n}g_{j}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{j}}}}{\bigg (}\sum _{i=1}^{n}f_{i}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{i}}}{\bigg )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37368d6d064ce93ef8e8975c9606ec53de5df5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.237ex; height:6.843ex;" alt="{\displaystyle \nabla _{Y}X=\nabla _{\sum _{j=1}^{n}g_{j}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{j}}}}{\bigg (}\sum _{i=1}^{n}f_{i}(x_{1},\dots ,x_{n}){\frac {\partial }{\partial x_{i}}}{\bigg )}}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sum _{i=1}^{n}f_{i}\sum _{j=1}^{n}{\bigg (}{\frac {\partial g_{j}}{\partial x_{j}}}{\frac {\partial }{\partial x_{j}}}+g_{j}\nabla _{\frac {\partial }{\partial x_{i}}}{\frac {\partial }{\partial x_{j}}}{\bigg )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{i=1}^{n}f_{i}\sum _{j=1}^{n}{\bigg (}{\frac {\partial g_{j}}{\partial x_{j}}}{\frac {\partial }{\partial x_{j}}}+g_{j}\nabla _{\frac {\partial }{\partial x_{i}}}{\frac {\partial }{\partial x_{j}}}{\bigg )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d58df87eea18b7b0c473042e16f95998fa0805c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.843ex; height:7.176ex;" alt="{\displaystyle =\sum _{i=1}^{n}f_{i}\sum _{j=1}^{n}{\bigg (}{\frac {\partial g_{j}}{\partial x_{j}}}{\frac {\partial }{\partial x_{j}}}+g_{j}\nabla _{\frac {\partial }{\partial x_{i}}}{\frac {\partial }{\partial x_{j}}}{\bigg )}}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sum _{k=1}^{n}{\bigg (}\sum _{i,j}f_{i}g_{j}\Gamma _{i,j}^{k}+Y(f_{k}){\bigg )}{\frac {\partial }{\partial x_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>+</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{k=1}^{n}{\bigg (}\sum _{i,j}f_{i}g_{j}\Gamma _{i,j}^{k}+Y(f_{k}){\bigg )}{\frac {\partial }{\partial x_{k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2932846412e2a44b2686814cfb1f179e6157917b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:34.314ex; height:7.176ex;" alt="{\displaystyle =\sum _{k=1}^{n}{\bigg (}\sum _{i,j}f_{i}g_{j}\Gamma _{i,j}^{k}+Y(f_{k}){\bigg )}{\frac {\partial }{\partial x_{k}}}}"></span>, donde <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 \Gamma _{i,j}^{k}(x_{1},\dots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{i,j}^{k}(x_{1},\dots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33718167611b544df74ce08bd372a6da931808b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.307ex; height:3.509ex;" alt="{\displaystyle \Gamma _{i,j}^{k}(x_{1},\dots ,x_{n})}"></span> son los <a href="/wiki/S%C3%ADmbolos_de_Christoffel" title="Símbolos de Christoffel">símbolos de Christoffel</a> de la conexión y vienen dados por la siguiente expresión: </p><p><br /> <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 \nabla _{\frac {\partial }{\partial x_{i}}}{\frac {\partial }{\partial x_{j}}}=\sum _{k=1}^{n}\Gamma _{i,j}^{k}{\frac {\partial }{\partial x_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\frac {\partial }{\partial x_{i}}}{\frac {\partial }{\partial x_{j}}}=\sum _{k=1}^{n}\Gamma _{i,j}^{k}{\frac {\partial }{\partial x_{k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac296a6ee37d0a9c330c5995728a778d1ab905f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.312ex; height:6.843ex;" alt="{\displaystyle \nabla _{\frac {\partial }{\partial x_{i}}}{\frac {\partial }{\partial x_{j}}}=\sum _{k=1}^{n}\Gamma _{i,j}^{k}{\frac {\partial }{\partial x_{k}}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Derivada_covariante_y_transporte_paralelo">Derivada covariante y transporte paralelo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%ADnea_geod%C3%A9sica&action=edit&section=4" title="Editar sección: Derivada covariante y transporte paralelo"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ahora sea <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\in \chi (M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∈</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\in \chi (M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a201281a6272244c06e376a64e28e44c7a2ab9c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.321ex; height:2.843ex;" alt="{\displaystyle Y\in \chi (M)}"></span> un campo en M y <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 c:(-\varepsilon ,\varepsilon )\rightarrow M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ε</mi> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c:(-\varepsilon ,\varepsilon )\rightarrow M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e33b30f6a24f34503876772b3c3c5e9fb170e4f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.819ex; height:2.843ex;" alt="{\displaystyle c:(-\varepsilon ,\varepsilon )\rightarrow M}"></span> una <a href="/wiki/Curva_integral_de_un_campo_vectorial" title="Curva integral de un campo vectorial">curva integral</a> de dicho campo, de modo que <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 c'(t)=Y|_{c(t)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c'(t)=Y|_{c(t)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b658b3ce469878c2b3eaae663d5f5690e0b762b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.676ex; height:3.509ex;" alt="{\displaystyle c'(t)=Y|_{c(t)}}"></span> y sea <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\in \chi (M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∈</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\in \chi (M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e35820a940e0628b39c011f2d914ae3cd5fe7ff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.527ex; height:2.843ex;" alt="{\displaystyle X\in \chi (M)}"></span> otro campo. Sea <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 {\overline {c}}(t)=(\varphi \circ c)(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>∘</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {c}}(t)=(\varphi \circ c)(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deacff84fe4b66c6a2a2253155448ecb39a9e58c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.049ex; height:2.843ex;" alt="{\displaystyle {\overline {c}}(t)=(\varphi \circ c)(t)}"></span> la expresión local de la curva. En esta situación, el campo <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 \nabla _{Y}X|_{c(t)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mi>X</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{Y}X|_{c(t)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010413e9cb2b48f716fa77020c4fdc45df55660c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.866ex; height:3.343ex;" alt="{\displaystyle \nabla _{Y}X|_{c(t)}}"></span> únicamente depende de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X|_{c(t)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X|_{c(t)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd97bb80061a8cda78cc9f746bd5d489df80aab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.444ex; height:3.343ex;" alt="{\displaystyle X|_{c(t)}}"></span> y de <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 c(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77d04462d26c4fdbfe8f988b182babd15059df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.656ex; height:2.843ex;" alt="{\displaystyle c(t)}"></span>. Esto motiva la siguiente notación: <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 \nabla _{c'(t)}v(t):=\nabla _{Y}X|_{c(t)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mi>X</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{c'(t)}v(t):=\nabla _{Y}X|_{c(t)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b913b6173d5175a11be1f61242f740a1aa0fca36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.673ex; height:3.343ex;" alt="{\displaystyle \nabla _{c'(t)}v(t):=\nabla _{Y}X|_{c(t)}}"></span>, donde <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)=X|_{c(t)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)=X|_{c(t)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7814be8fef436b2ae64614528afb6de107de9f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.319ex; height:3.343ex;" alt="{\displaystyle v(t)=X|_{c(t)}}"></span> es la restricción del campo X a la curva c. La expresión anterior particulariza el concepto de conexión afín a las restricciones de campos sobre una curva integral. Definimos <a href="/wiki/Derivada_covariante" title="Derivada covariante">derivada covariante</a> a lo largo de una curva como: </p><p><br /> <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 {Dv}{dt}}=\nabla _{c'(t)}v(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {Dv}{dt}}=\nabla _{c'(t)}v(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b908b6040d4e6d31a1488b441ed52efeb42bb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.048ex; height:5.343ex;" alt="{\displaystyle {\frac {Dv}{dt}}=\nabla _{c'(t)}v(t)}"></span>. </p><p><br /> La derivada covariante es una generalización de la aceleración en variedades Riemannianas. </p><p>Un campo de vectores <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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243a0bf98a12f48552ba6a70302122d81b237b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.777ex; height:2.843ex;" alt="{\displaystyle v(t)}"></span> se dice paralelo a lo largo de una curva <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 c(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77d04462d26c4fdbfe8f988b182babd15059df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.656ex; height:2.843ex;" alt="{\displaystyle c(t)}"></span> si su derivada covariante se anula a lo largo de esta. Dada una curva <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 c(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77d04462d26c4fdbfe8f988b182babd15059df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.656ex; height:2.843ex;" alt="{\displaystyle c(t)}"></span> y un vector <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_{0}\in TpM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>T</mi> <mi>p</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}\in TpM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/820053baeb69da70217d3e1fbc5fd2f3f25fb84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.27ex; height:2.509ex;" alt="{\displaystyle v_{0}\in TpM}"></span> para algún <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ad2c18a15749505c928763cd4fdb56f4982816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.542ex; height:2.509ex;" alt="{\displaystyle p\in M}"></span>, la existencia de dicho campo paralelo está garantizada por el teorema de existencia y unicidad de ecuaciones diferenciales ordinarias. En efecto, si tomamos la expresión de la derivada covariante e igualamos a 0 obtenemos un sistema lineal de ecuaciones diferenciales ordinarias: </p><p><br /> <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 {Dv(t)}{dt}}=\sum _{k=1}^{n}{\bigg (}v'_{k}(t)+\sum _{i,j}v_{i}(t){\overline {c}}_{j}(t)\Gamma _{j,i}^{k}({\overline {c}}_{1},\dots ,{\overline {c}}_{n}){\bigg )}{\frac {\partial }{\partial x_{k}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {Dv(t)}{dt}}=\sum _{k=1}^{n}{\bigg (}v'_{k}(t)+\sum _{i,j}v_{i}(t){\overline {c}}_{j}(t)\Gamma _{j,i}^{k}({\overline {c}}_{1},\dots ,{\overline {c}}_{n}){\bigg )}{\frac {\partial }{\partial x_{k}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47fef473f0b2d5b7582ce1dae0e0d1183fa1a1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:61.228ex; height:7.176ex;" alt="{\displaystyle {\frac {Dv(t)}{dt}}=\sum _{k=1}^{n}{\bigg (}v'_{k}(t)+\sum _{i,j}v_{i}(t){\overline {c}}_{j}(t)\Gamma _{j,i}^{k}({\overline {c}}_{1},\dots ,{\overline {c}}_{n}){\bigg )}{\frac {\partial }{\partial x_{k}}}=0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(0)=v_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(0)=v_{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc8a8ba2a3b6c25a09df316d49892ce60b6c4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.027ex; height:2.843ex;" alt="{\displaystyle v(0)=v_{0}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Geodésicas"><span id="Geod.C3.A9sicas"></span>Geodésicas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%ADnea_geod%C3%A9sica&action=edit&section=5" title="Editar sección: Geodésicas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Continuando con el concepto de que la derivada covariante representa de alguna manera la aceleración, es natural concebir las geodésicas como curvas de derivada covariante nula. Como la aceleración no es una magnitud que posean las curvas sino los vectores, entonces parece natural que si <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 \gamma (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54fa4a5d64e164410e4a18106677bebefe1a1f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.911ex; height:2.843ex;" alt="{\displaystyle \gamma (t)}"></span> es una geodésica cumpla: </p><p><br /> <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 \nabla _{\gamma '}\gamma '=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>γ</mi> <mo>′</mo> </msup> </mrow> </msub> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\gamma '}\gamma '=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d8fe25f5d614e5e5e7133d3286b061e0cd92d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.477ex; height:3.176ex;" alt="{\displaystyle \nabla _{\gamma '}\gamma '=0.}"></span> </p><p><br /> Es decir, las geodésicas son curvas cuyo vector tangente es un campo paralelo a lo largo de sí misma. Aquí, como se ve, <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)=\gamma '(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)=\gamma '(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb26a896b96ca39d6bfd05615df68c6db62e85eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.488ex; height:3.009ex;" alt="{\displaystyle v(t)=\gamma '(t)}"></span>. Expandiendo la ecuación anterior obtenemos el sistema de ecuaciones de segundo orden: </p><p><br /> <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 \gamma ''_{k}(t)+\sum _{i,j}\Gamma _{i,j}^{k}\gamma '_{i}\gamma '_{j}=0,\quad k=1,2,...,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>″</mo> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ''_{k}(t)+\sum _{i,j}\Gamma _{i,j}^{k}\gamma '_{i}\gamma '_{j}=0,\quad k=1,2,...,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ea8cb14ffdf1bae65e958aa249a6f97974f637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:41.004ex; height:5.843ex;" alt="{\displaystyle \gamma ''_{k}(t)+\sum _{i,j}\Gamma _{i,j}^{k}\gamma '_{i}\gamma '_{j}=0,\quad k=1,2,...,n}"></span>, </p><p><br /> que se resuelve con las condiciones iniciales <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 \gamma (0)=p;\quad \gamma '(0)=v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo>;</mo> <mspace width="1em" /> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (0)=p;\quad \gamma '(0)=v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f1a6a5132ea46c365abe82700588d01cf63fbb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.021ex; height:3.009ex;" alt="{\displaystyle \gamma (0)=p;\quad \gamma '(0)=v}"></span>. Nótese que si <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> es una geodésica, entonces <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 |\gamma '(t)|=cte}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>c</mi> <mi>t</mi> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\gamma '(t)|=cte}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de72f21b26c3ae35445b75eee0f21367665faecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.936ex; height:3.009ex;" alt="{\displaystyle |\gamma '(t)|=cte}"></span>. Esto implica, en efecto, que las geodésicas tienen aceleración intrínseca nula. Para entender esto mejor, imaginemos que nuestra variedad es la esfera unidad <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=\mathbb {S} ^{2}\subset \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\mathbb {S} ^{2}\subset \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3cca2a8e245ae4e00419da4026c14b21f2341e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.718ex; height:2.676ex;" alt="{\displaystyle M=\mathbb {S} ^{2}\subset \mathbb {R} ^{3}}"></span>. Entonces, las geodésicas son curvas que se recorren con velocidad constante en la esfera, lo que implica que tienen, en términos físicos, aceleración tangencial nula, pero vistas como curvas de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>todavía tendrían aceleración normal distinta de 0. Ese vector aceleración normal, no obstante, no pertenece al espacio tangente a un punto de la esfera, de manera que, intrínsecamente, para un observador que viviera en la esfera, efectivamente esa curva se recorrería sin aceleración. </p> <div class="mw-heading mw-heading4"><h4 id="Ejemplos">Ejemplos</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%ADnea_geod%C3%A9sica&action=edit&section=6" title="Editar sección: Ejemplos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Las geodésicas en el <a href="/wiki/Espacio_eucl%C3%ADdeo" title="Espacio euclídeo">espacio euclídeo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8fa8586d428ff5706c6d0a00a7939950fad89b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {E} ^{n}}"></span> son las líneas rectas.</li> <li>Las geodésicas en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd949bba96edbb143fd05b712e274aa07ab1c75f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle \mathbb {H} ^{2}}"></span>, el <a href="/wiki/Espacio_hiperb%C3%B3lico" title="Espacio hiperbólico">espacio hiperbólico</a>, son arcos de circunferencia.</li> <li>Las geodésicas en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/518268b00344a37811c08a236412bffaa68f75d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.347ex; height:2.676ex;" alt="{\displaystyle \mathbb {S} ^{2}}"></span> son arcos de circunferencias máximas. Si se considera la esfera como encajada en el espacio euclídeo tridimensional entonces los círculos máximos se obtienen como intersección de la esfera con un plano que pase por su centro. En particular, los <a href="/wiki/Meridiano" title="Meridiano">meridianos</a> de una esfera y el <a href="/wiki/Ecuador_terrestre" title="Ecuador terrestre">ecuador</a> son líneas geodésicas. Usando coordenadas <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 \scriptstyle {(r,\theta ,\phi )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {(r,\theta ,\phi )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f03548651aa24feeefc063283286717a534e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.686ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {(r,\theta ,\phi )}}"></span>esféricas para una esfera de radio <i>R</i>, las ecuaciones de las geodésicas son simplemente:</li></ul> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span style="float: right; width: 10%; text-align: right;">(<cite id="Equation_*" style="font-style: normal;"><a href="#Eqnref_*">*</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\theta }}-\sin(\theta )\cos(\theta )=0,\qquad {\ddot {\phi }}+{\frac {2\cos(\theta )}{\sin(\theta )}}{\dot {\theta }}{\dot {\phi }}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\theta }}-\sin(\theta )\cos(\theta )=0,\qquad {\ddot {\phi }}+{\frac {2\cos(\theta )}{\sin(\theta )}}{\dot {\theta }}{\dot {\phi }}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae0151eab45b17edcb215cffcfe6b354076b1a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.078ex; height:6.509ex;" alt="{\displaystyle {\ddot {\theta }}-\sin(\theta )\cos(\theta )=0,\qquad {\ddot {\phi }}+{\frac {2\cos(\theta )}{\sin(\theta )}}{\dot {\theta }}{\dot {\phi }}=0}"></span> </p> </blockquote> <p>En particular un meridiano que atraviese los polos norte y sur, responde a las ecuaciones paramétricas: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span style="float: right; width: 10%; text-align: right;">(<cite id="Equation_**" style="font-style: normal;"><a href="#Eqnref_**">**</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (s)={\frac {s}{R}},\qquad \phi (s)=\phi _{0}={\mbox{cte.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>R</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>cte.</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta (s)={\frac {s}{R}},\qquad \phi (s)=\phi _{0}={\mbox{cte.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80aa9db55e78b7405100b7626c11f113f3dc28e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.906ex; height:4.843ex;" alt="{\displaystyle \theta (s)={\frac {s}{R}},\qquad \phi (s)=\phi _{0}={\mbox{cte.}}}"></span> </p> </blockquote> <p>Que satisface las ecuaciones (<span id="Eqnref_*" class="plainlinks neverexpand"><a class="external text" href="https://es.wikipedia.org/wiki/L%C3%ADnea_geod%C3%A9sica#Equation_*">*</a></span>) trivialmente. </p> <div class="mw-heading mw-heading3"><h3 id="Curvas_integrales_en_el_fibrado_tangente">Curvas integrales en el fibrado tangente</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%ADnea_geod%C3%A9sica&action=edit&section=7" title="Editar sección: Curvas integrales en el fibrado tangente"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El sistema de ecuaciones de las geodésicas obtenido antes no depende de <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> explícitamente, solo de <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 \gamma ''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8bb7728f3ec6d2715921dbd21f68b5be7e22d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.417ex; height:3.009ex;" alt="{\displaystyle \gamma ''}"></span> y <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 \gamma '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e2b9e9e12e56bd62af445be6803ec4843e919a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.965ex; height:3.009ex;" alt="{\displaystyle \gamma '}"></span>. Esto implica que dicho sistema de ecuaciones de segundo orden y tamaño n, puede transformarse fácilmente en un sistema de ecuaciones de primer orden y tamaño 2n. En efecto, llamando <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 \delta _{k}(t)=\gamma '_{k}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{k}(t)=\gamma '_{k}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d1d8526e72319a77d1d2cc3868ade6169e4d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.81ex; height:3.009ex;" alt="{\displaystyle \delta _{k}(t)=\gamma '_{k}(t)}"></span> obtenemos el sistema: </p><p><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 \delta _{k}(t)=\gamma '_{k}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{k}(t)=\gamma '_{k}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d1d8526e72319a77d1d2cc3868ade6169e4d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.81ex; height:3.009ex;" alt="{\displaystyle \delta _{k}(t)=\gamma '_{k}(t)}"></span>; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta '_{k}=-\sum _{i,j}^{k}\delta _{i}\delta _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta '_{k}=-\sum _{i,j}^{k}\delta _{i}\delta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4077efa8143ce8aa58b78e705bb74dfc8ae65ff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.931ex; height:7.676ex;" alt="{\displaystyle \delta '_{k}=-\sum _{i,j}^{k}\delta _{i}\delta _{j}}"></span> </p><p>con condiciones iniciales <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 \gamma _{k}(0)=\gamma _{k}^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{k}(0)=\gamma _{k}^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ecc734ac7d750eb404b5e2719b7b9a7e86e4b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.697ex; height:3.176ex;" alt="{\displaystyle \gamma _{k}(0)=\gamma _{k}^{0}}"></span> y <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 \delta _{k}(0)=\gamma _{k}'^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{k}(0)=\gamma _{k}'^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c744d4b7f362b5be3fcf96ba906cac2839510d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.978ex; height:3.176ex;" alt="{\displaystyle \delta _{k}(0)=\gamma _{k}'^{0}}"></span>. Observando bien, este sistema puede interpretarse como un sistema de EDOs de primer orden sobre el <a href="/wiki/Fibrado_tangente" title="Fibrado tangente">fibrado tangente</a> de la variedad. En efecto, nuestras nuevas variables <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 (\gamma _{1}\dots ,\gamma _{n},\delta _{1}\dots ,\delta _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mo>,</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\gamma _{1}\dots ,\gamma _{n},\delta _{1}\dots ,\delta _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a2082c8375f75c22d71b05579b9716f383b8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.925ex; height:2.843ex;" alt="{\displaystyle (\gamma _{1}\dots ,\gamma _{n},\delta _{1}\dots ,\delta _{n})}"></span> son un conjunto de N coordenadas y N velocidades, tales que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{-1}:(\gamma _{1}\dots ,\gamma _{n},\delta _{1}\dots ,\delta _{n})\mapsto \sum _{i}\delta _{1}{\frac {\partial }{\partial x}}{\bigg |}_{(\gamma _{1},...,\gamma _{n})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mo>,</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ^{-1}:(\gamma _{1}\dots ,\gamma _{n},\delta _{1}\dots ,\delta _{n})\mapsto \sum _{i}\delta _{1}{\frac {\partial }{\partial x}}{\bigg |}_{(\gamma _{1},...,\gamma _{n})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1d5f4d86fcdb3dee90b7372c23b401dc53419c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.452ex; height:6.509ex;" alt="{\displaystyle \phi ^{-1}:(\gamma _{1}\dots ,\gamma _{n},\delta _{1}\dots ,\delta _{n})\mapsto \sum _{i}\delta _{1}{\frac {\partial }{\partial x}}{\bigg |}_{(\gamma _{1},...,\gamma _{n})}}"></span> es la inversa del homeomorfismo de una carta en el fibrado tangente. Así, las geodésicas pueden interpretarse como curvas integrales de un campo sobre el fibrado tangente, llamado campo geodésico. </p> <figure typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Geodesic_of_Enneper_surface.webm/300px--Geodesic_of_Enneper_surface.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="300" height="300" data-durationhint="16" data-mwtitle="Geodesic_of_Enneper_surface.webm" data-mwprovider="wikimediacommons" resource="/wiki/Archivo:Geodesic_of_Enneper_surface.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/62/Geodesic_of_Enneper_surface.webm/Geodesic_of_Enneper_surface.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="480" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/62/Geodesic_of_Enneper_surface.webm/Geodesic_of_Enneper_surface.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="720" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/6/62/Geodesic_of_Enneper_surface.webm" type="video/webm; codecs="vp9"" data-width="1080" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/62/Geodesic_of_Enneper_surface.webm/Geodesic_of_Enneper_surface.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1080" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/62/Geodesic_of_Enneper_surface.webm/Geodesic_of_Enneper_surface.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="144" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/62/Geodesic_of_Enneper_surface.webm/Geodesic_of_Enneper_surface.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="240" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/62/Geodesic_of_Enneper_surface.webm/Geodesic_of_Enneper_surface.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="360" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/62/Geodesic_of_Enneper_surface.webm/Geodesic_of_Enneper_surface.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="360" data-height="360" /></video></span><figcaption>La trisectriz de Tschirnhaus-Catalan es geodésica en la <a href="/wiki/Superficie_de_Enneper" title="Superficie de Enneper">superficie de Enneper</a>.<sup id="cite_ref-1" class="reference separada"><a href="#cite_note-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup></figcaption></figure> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%ADnea_geod%C3%A9sica&action=edit&section=8" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/C%C3%BApula_geod%C3%A9sica" title="Cúpula geodésica">Cúpula geodésica</a></li> <li><a href="/wiki/Variedad_de_Riemann" title="Variedad de Riemann">Variedad de Riemann</a></li> <li><a href="/wiki/Curvatura" title="Curvatura">Curvatura</a></li> <li><a href="/wiki/Curvas_en_secci%C3%B3n_de_la_Tierra" title="Curvas en sección de la Tierra">Curvas en sección de la Tierra</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%ADnea_geod%C3%A9sica&action=edit&section=9" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span id="CITAREFFerréol2011" class="citation web">Ferréol, Robert (2011). <a rel="nofollow" class="external text" href="http://www.mathcurve.com/surfaces/enneper/enneper.shtml">«SURFACE D'ENNEPER»</a>. <i><a rel="nofollow" class="external free" href="http://www.mathcurve.com/">http://www.mathcurve.com/</a></i> <span style="color:var(--color-subtle, #555 );">(en francés)</span><span class="reference-accessdate">. Consultado el 11 de abril de 2020</span>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%ADnea+geod%C3%A9sica&rft.atitle=SURFACE+D%27ENNEPER&rft.au=Ferr%C3%A9ol%2C+Robert&rft.aufirst=Robert&rft.aulast=Ferr%C3%A9ol&rft.date=2011&rft.genre=article&rft.jtitle=http%3A%2F%2Fwww.mathcurve.com%2F&rft_id=http%3A%2F%2Fwww.mathcurve.com%2Fsurfaces%2Fenneper%2Fenneper.shtml&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%ADnea_geod%C3%A9sica&action=edit&section=10" title="Editar sección: Bibliografía"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="CITAREFdo_Carmo1993" class="citation libro">do Carmo, Manfredo Perdigao (1993). Birkhäuser, ed. <i>Riemannian Geometry</i>. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/0-8176-3490-8" title="Especial:FuentesDeLibros/0-8176-3490-8">0-8176-3490-8</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%ADnea+geod%C3%A9sica&rft.au=do+Carmo%2C+Manfredo+Perdigao&rft.aufirst=Manfredo+Perdigao&rft.aulast=do+Carmo&rft.btitle=Riemannian+Geometry&rft.date=1993&rft.genre=book&rft.isbn=0-8176-3490-8&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFEinstein1916" class="citation publicación"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein, Albert</a> (1916). <a rel="nofollow" class="external text" href="https://www.academia.edu/112192888/Einstein_1916_Los_Fundamentos_de_la_Teor%C3%ADa_General_de_la_Relatividad_Die_Grundlage_der_allgemeinen_Relativitatstheorie_">«Los Fundamentos de la Teoría General de la Relatividad (Die Grundlage der allgemeinen Relativitatstheorie)»</a>. <i>Annalen der Physik</i>: 769-822.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%ADnea+geod%C3%A9sica&rft.atitle=Los+Fundamentos+de+la+Teor%C3%ADa+General+de+la+Relatividad+%28Die+Grundlage+der+allgemeinen+Relativitatstheorie%29&rft.au=Einstein%2C+Albert&rft.aufirst=Albert&rft.aulast=Einstein&rft.date=1916&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.pages=769-822&rft_id=https%3A%2F%2Fwww.academia.edu%2F112192888%2FEinstein_1916_Los_Fundamentos_de_la_Teor%25C3%25ADa_General_de_la_Relatividad_Die_Grundlage_der_allgemeinen_Relativitatstheorie_&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span> (Texto en español)</li></ul> <ul><li><span id="CITAREFShoshichi_Kobayashi1996" class="citation libro">Katsumi Nomizu (1996). Wiley-Interscience, ed. <i>Foundations of Differential Geometry, Vol. 1</i>. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/0-471-15733-3" title="Especial:FuentesDeLibros/0-471-15733-3">0-471-15733-3</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%ADnea+geod%C3%A9sica&rft.au=Shoshichi+Kobayashi&rft.aulast=Shoshichi+Kobayashi&rft.btitle=Foundations+of+Differential+Geometry%2C+Vol.+1&rft.date=1996&rft.genre=book&rft.isbn=0-471-15733-3&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFO'Neill1983" class="citation libro">O'Neill, Barret (1983). Academic Press,London, ed. <i>Semi-Riemannian Geometry</i>. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/0-12-526740-1" title="Especial:FuentesDeLibros/0-12-526740-1">0-12-526740-1</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%ADnea+geod%C3%A9sica&rft.au=O%27Neill%2C+Barret&rft.aufirst=Barret&rft.aulast=O%27Neill&rft.btitle=Semi-Riemannian+Geometry&rft.date=1983&rft.genre=book&rft.isbn=0-12-526740-1&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span> Para el caso semiriemanniano.</li> <li><span id="CITAREFArellano1993" class="citation libro">Arellano, Arturo (1993). <i>Geometry Differential</i>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%ADnea+geod%C3%A9sica&rft.au=Arellano%2C+Arturo&rft.aufirst=Arturo&rft.aulast=Arellano&rft.btitle=Geometry+Differential&rft.date=1993&rft.genre=book&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul 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Línea geodésica - Wikipedia, la enciclopedia libre