CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> geodesic convexity in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> geodesic convexity </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussions/?CategoryID=0" title="Discuss this page on the nForum. It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="riemannian_geometry">Riemannian geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></strong></p> <h2 id="basic_definitions">Basic definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/metric">metric</a>, <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/isometry+group">isometry group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space+of+Riemannian+metrics">moduli space of Riemannian metrics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo-Riemannian manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian manifold</a>, <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic">geodesic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic+convexity">geodesic convexity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic+flow">geodesic flow</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Levi-Civita+connection">Levi-Civita connection</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+curvature">Riemann curvature</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+metric+connection">torsion of a metric connection</a></li> </ul> </li> </ul> <h2 id="further_concepts">Further concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+inner+product">Hodge inner product</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gradient">gradient</a>, <a class="existingWikiWord" href="/nlab/show/gradient+flow">gradient flow</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a>-theorem</li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a>, <a class="existingWikiWord" href="/nlab/show/Einstein+equations">Einstein equations</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/Riemannian+geometry+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#in_a_riemannian_manifold'>In a Riemannian manifold</a></li> <li><a href='#in_a_metric_space'>In a metric space</a></li> </ul> <li><a href='#properties'>Properties</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The definition of <em>geodesic convexity</em> is like that of <em><a class="existingWikiWord" href="/nlab/show/convexity">convexity</a></em>, but with straight lines in an <a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a> generalized to <a class="existingWikiWord" href="/nlab/show/geodesics">geodesics</a> in a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> or <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>.</p> <h2 id="definition">Definition</h2> <h3 id="in_a_riemannian_manifold">In a Riemannian manifold</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C \subset X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/subset">subset</a>. We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is</p> <ul> <li> <p><strong>weakly geodesically convex</strong> if for any two points from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> there exists exactly one minimizing <a class="existingWikiWord" href="/nlab/show/geodesic">geodesic</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> connecting them;</p> </li> <li> <p><strong>geodesically convex</strong> if for any two points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> there exists exactly one minimizing geodesic in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> connecting them, and that geodesic arc lies completely in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></p> </li> <li> <p><strong>strongly geodesically convex</strong> if for any two points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{C}</annotation></semantics></math> there exists exactly one minimizing geodesic in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> connecting them, and that geodesic arc lies completely in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, except possibly the endpoints; and furthermore there exists no nonminimizing geodesic inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> connecting the two points.</p> </li> </ul> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>The <strong>convexity radius</strong> at a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p \in X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/supremum">supremum</a> (which may be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">+ \infty</annotation></semantics></math>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">r \in \mathbb{R}</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo><</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">\eta \lt r</annotation></semantics></math> the geodesic ball <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_p(r)</annotation></semantics></math> is strongly geodesically convex.</p> <p>The convexity radius of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/infimum">infimum</a> over the points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p \in X</annotation></semantics></math> of the convexity radii at these points.</p> </div> <h3 id="in_a_metric_space">In a metric space</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, a <em>distance-preserving path</em> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \colon [a,b]\to X</annotation></semantics></math> which is an <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>. This is a metric-space analogue of an “arc-length-parametrized minimizing geodesic” on a Riemannian manifold. In particular, the existence of such a path implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mo>−</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">d(x(a),x(b)) = b-a</annotation></semantics></math>. We then say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <strong>geodesic</strong> (or <em>geodesically convex</em>) if any two points can be connected by a distance-preserving path.</p> <p>We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <strong>length space</strong> if for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, the distance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(x,y)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/infimum">infimum</a> of the lengths of all continuous paths from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>. The <span class="newWikiWord">Hopf-Rinow theorem<a href="/nlab/new/Hopf-Rinow+theorem">?</a></span> says that for a metric space in which every <a class="existingWikiWord" href="/nlab/show/closed+subset">closed</a> <a class="existingWikiWord" href="/nlab/show/bounded+subset">bounded</a> subset is <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a>, being a length space is equivalent to being geodesic.</p> <h2 id="properties">Properties</h2> <div class="un_proposition"> <h6 id="proposition">Proposition</h6> <p>At any point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p \in X</annotation></semantics></math> of a Riemannian manifold, the convexity radius is positive.</p> </div> <p>This is due to (<a href="#Whitehead">Whitehead</a>).</p> <div class="un_proposition"> <h6 id="proposition_2">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a> then the convexity radius of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> is positive.</p> </div> <p>This is reproduced for instance as proposition 95 in (<a href="#Berger">Berger</a>)</p> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/paracompact+manifold">paracompact manifold</a> admits a complete <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> with bounded absolute sectional curvature and positive injectivity radius.</p> </div> <p>This is shown in (<a href="#Greene">Greene</a>).</p> <h2 id="references">References</h2> <p>Original literature includes</p> <ul id="Whitehead"> <li>J. H. C. Whitehead, <em>Convex regions in the geometry of paths</em> Quart. J. Math. 3, 33–42 (1932).</li> </ul> <ul id="Greene"> <li>R. Greene, <em>Complete metrics of bounded curvature on noncompact manifolds</em> Archiv der Mathematik Volume 31, Number 1 (1978)</li> </ul> <p>A review of geodesic convexity in Riemannian manifolds is in</p> <ul id="Chavel"> <li>Isaac Chavel, <em>Riemannian geometry – A modern introduction</em> Cambridge University Press (1993)</li> </ul> <ul id="Berger"> <li>Marcel Berger, <em>A panoramic view of Riemannian geometry</em></li> </ul> <p>A categorical perspective on geodesic convexity for metric spaces can be found in</p> <ul> <li><em>Taking categories seriously</em>, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. (<a href="http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf">pdf</a>)</li> </ul> <p>Much of the issues on geodesic convexity becomes more complicated when trying to generalize from Riemannian to <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifolds">Lorentzian manifolds</a>, as discussed at length at</p> <ul> <li>John K. Beem, Paul E. Ehrlich, Kevin L. Easley, <em>Global Lorentzian geometry</em>, Marcel Dekker, 1996, 635 pages</li> <li>John K. Beem, <em>Lorentzian geometry in the large</em>, Math. of gravitation I, Lorentzian geometry and Einstein equations, Banach Center Publications <strong>41</strong>, Inst. of Math. Polish Acad. of Sci. Warszawa 1997 <a href="http://matwbn.icm.edu.pl/ksiazki/bcp/bcp41/bcp4111.pdf">pdf</a></li> </ul> <p>In particular, the conclusions of the Hopf-Rinow Theorem fail to hold for complete Lorentzian manifolds.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on January 21, 2014 at 15:01:05. See the <a href="/nlab/history/geodesic+convexity" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/geodesic+convexity" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/geodesic+convexity/6" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/geodesic+convexity" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/geodesic+convexity" accesskey="S" class="navlink" id="history" rel="nofollow">History (6 revisions)</a> <a href="/nlab/show/geodesic+convexity/cite" style="color: black">Cite</a> <a href="/nlab/print/geodesic+convexity" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/geodesic+convexity" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>