https://diffgeom.subwiki.org/w/index.php?title=Complete_equals_geodesically_complete&feed=atom&action=history Complete equals geodesically complete - Revision history 2020-04-05T08:02:46Z Revision history for this page on the wiki MediaWiki 1.29.2 https://diffgeom.subwiki.org/w/index.php?title=Complete_equals_geodesically_complete&diff=1942&oldid=prev Vipul: New page: ==Statement== Let $(M,g)$ be a Riemannian manifold. Then, the following are equivalent: # $M$ is [[geodesically complete Riemannian manifold|geodesically comple... 2008-05-22T13:27:19Z <p>New page: ==Statement== Let &lt;math&gt;(M,g)&lt;/math&gt; be a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>. Then, the following are equivalent: # &lt;math&gt;M&lt;/math&gt; is [[geodesically complete Riemannian manifold|geodesically comple...</p> <p><b>New page</b></p><div>==Statement==<br /> <br /> Let &lt;math&gt;(M,g)&lt;/math&gt; be a [[Riemannian manifold]]. Then, the following are equivalent:<br /> <br /> # &lt;math&gt;M&lt;/math&gt; is [[geodesically complete Riemannian manifold|geodesically complete]]: in other words, geodesics can be extended indefinitely in both directions, or equivalently, the [[exponential map at a point]] is defined on the whole tangent space at the point<br /> # &lt;math&gt;M&lt;/math&gt; is geodesically complete at ''one point'': i.e. there exists &lt;math&gt;p \in M&lt;/math&gt; such that the exponential map &lt;math&gt;\exp_p&lt;/math&gt; is defined on the whole of &lt;math&gt;T_p(M)&lt;/math&gt;<br /> # &lt;math&gt;M&lt;/math&gt; is complete as a metric space, where the distance between two points is defined as the infimum of lengths of all curves between the two points.<br /> <br /> ==Facts used==<br /> <br /> * [[Uses::Hopf-Rinow theorem]]<br /> <br /> ==Proof==<br /> <br /> ===Complete implies geodesically complete===</div> Vipul