Hopf-Rinow II and an application November 15, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: geodesic completeness, geodesics, homotopy, Hopf-Rinow theorem, Riemannian manifolds
Now, let’s finish the proof of the Hopf-Rinow theorem (the first one) started yesterday. We need to show that given a Riemannian manifold which is a metric space , the existence of arbitrary geodesics from implies that is complete with respect to . Actually, this is slightly stronger than what H-R states: geodesic completeness at one point implies completeness.
The first thing to notice is that is smooth by the global smoothness theorem and the assumption that arbitrary geodesics from exist. Moreover, it is surjective by the second Hopf-Rinow theorem.
Now fix a -Cauchy sequence . We will show that it converges. Draw minimal geodesics travelling at unit speed with
The Hopf-Rinow theorems and geodesic completeness November 14, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: completeness, geodesic completeness, geodesics, Hopf-Rinow theorem, Riemannian manifolds
Ok, yesterday I covered the basic fact that given a Riemannian manifold , the geodesics on (with respect to the Levi-Civita connection) locally minimize length. Today I will talk about the phenomenon of “geodesic completeness.”
Henceforth, all manifolds are assumed connected.
The first basic remark to make is the following. If is a piecewise -path between and has the smallest length among piecewise paths, then is, up to reparametrization, a geodesic (in particular smooth). The way to see this is to pick very close to each other, so that is contained in a neighborhood of satisfying the conditions of yesterday’s theorem; then must be length-minimizing, so it is a geodesic. We thus see that is locally a geodesic, hence globally.
Say that is geodesically complete if can be defined on all of ; in other words, a geodesic can be continued to . The name is justified by the following theorem:
Theorem 1 (Hopf-Rinow)The following are equivalent:
- is geodesically complete.
- In the metric on induced by (see here), is a complete metric space (more…)
Geodesics are locally length-minimizing November 13, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: Gauss lemma, geodesics, Riemannian manifolds
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Our main goal today is:
Theorem 1 Given , there is a neighborhood containing such that geodesics from to every point of exist and also such that given a path inside from to , we have
with equality holding if and only if is a reparametrization of .
In other words, geodesics are locally path-minimizing. Not necessarily globally–a great circle is a geodesic on a sphere with the Riemannian metric coming from the embedding in , but it need not be the shortest path between two points. (more…)
Geodesics and the exponential map November 4, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, exponential map, geodesics, ordinary differential equations
Ok, we know what connections and covariant derivatives are. Now we can use them to get a map from the tangent space at one point to the manifold which is a local isomorphism. This is interesting because it gives a way of saying, “start at point and go five units in the direction of the tangent vector ,” in a rigorous sense, and will be useful in proofs of things like the tubular neighborhood theorem—which I’ll get to shortly.
Anyway, first I need to talk about geodesics. A geodesic is a curve such that the vector field along created by the derivative is parallel. In local coordinates , here’s what this means. Let the Christoffel symbols be . Then using the local formula for covariant differentiation along a curve, we get
so being a geodesic is equivalent to the system of differential equations