Hopf-Rinow theorem: Difference between revisions

Latest revision as of 13:17, 22 May 2008

Statement

Let $(M,g)$ be a Riemannian manifold (viz $M$ is a differential manifold equipped with a Riemannian metric $g$ ). Then the Hopf-Rinow theorem states the following equivalent things:

If $M$ is geodesically complete, then given any two points $p,q\in M$ there is a minimizing geodesic joining $p$ and $q$ .
For any point $p\in M$ , the exponential map at $p$ is surjective from $T_{p}(M)$ to $M$ , and any length-minimizing curve from $p$ to $q$ for some $q\in M$ , occurs as the image of a straight line from the origin, under the exponential map.

Proof

Given $p\in M$ , there exists $\epsilon >0$ , such that the exponential map takes the unit sphere of radius $\epsilon$ about $p$ to the set of points on the manifold at distance $\epsilon$ from $p$ , with the minimal geodesics from $p$ being exponentials of the radial paths.

The upshot of this is: given $p\in M$ , there is an $\epsilon$ such that the set of points at distance $\epsilon$ from $p$ is a compact set, and such that there is a unique minimizing geodesic from $p$ to each of them

Any geodesic can be extended indefinitely (and uniquely) in both directions (this follows from the assumption that our manifold is geodesically complete)

The proof beyond these two facts is purely geometric (that is, it only uses standard properties of metrics and continuous maps).

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 ==Statement==
-Let <math>(M,g)</math> be a [[Riemannian manifold]] (viz <math>M</math> is a [[differential manifold]] equipped with a [[Riemannian metric]] <math>g</math>). Then, the Hopf-Rinow theorem states that the following are equivalent:
+Let <math>(M,g)</math> be a [[Riemannian manifold]] (viz <math>M</math> is a [[differential manifold]] equipped with a [[Riemannian metric]] <math>g</math>). Then the Hopf-Rinow theorem states the following equivalent things:
-* <math>M</math> is [[geodesically complete Riemannian manifold|geodesically complete]], viz any geodesic can be extended indefinitely in both directions
+# If <math>M</math> is geodesically complete, then given any two points <math>p,q \in M</math> there is a minimizing geodesic joining <math>p</math> and <math>q</math>.
-* <math>M</math> is complete as a metric space (with metric induced by the Riemannian distance function)
+# For any point <math>p \in M</math>, the [[exponential map at a point|exponential map]] at <math>p</math> is surjective from <math>T_p(M)</math> to <math>M</math>, and any length-minimizing curve from <math>p</math> to <math>q</math> for some <math>q \in M</math>, occurs as the image of a straight line from the origin, under the exponential map.
-In some texts, the Hopf-Rinow theorem is expressed in another form (which is more geometric):
-If <math>M</math> is geodesically complete, then given any two points <math>p,q \in M</math> there is a minimizing geodesic joining <math>p</math> and <math>q</math>.
 ==Proof==
-The geometric version of Hopf-Rinow theorem (from which the other version easily falls out) can be proved by applying triangle inequalities once we have established the following basic facts:
 * Given <math>p \in M</math>, there exists <math>\epsilon > 0</math>, such that the exponential map takes the unit sphere of radius <math>\epsilon</math> about <math>p</math> to the set of points on the manifold at distance <math>\epsilon</math> from <math>p</math>, with the minimal geodesics from <math>p</math> being exponentials of the radial paths.