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Hopf-Rinow II and an application November 15, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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2 comments

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 {(M,g)} which is a metric space {d}, the existence of arbitrary geodesics from {p} implies that {M} is complete with respect to {d}. Actually, this is slightly stronger than what H-R states: geodesic completeness at one point {p} implies completeness.

The first thing to notice is that {\exp: T_p(M) \rightarrow M} is smooth by the global smoothness theorem and the assumption that arbitrary geodesics from {p} exist. Moreover, it is surjective by the second Hopf-Rinow theorem.

Now fix a {d}-Cauchy sequence {q_n \in M}. We will show that it converges. Draw minimal geodesics {\gamma_n} travelling at unit speed with

\displaystyle \gamma_n(0)=p, \quad \gamma_n( d(p,q_n)) = q_n.  (more…)

The Hopf-Rinow theorems and geodesic completeness November 14, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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10 comments

Ok, yesterday I covered the basic fact that given a Riemannian manifold {(M,g)}, the geodesics on {M} (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 {c: I \rightarrow M} is a piecewise {C^1}-path between {p,q} and has the smallest length among piecewise {C^1} paths, then {c} is, up to reparametrization, a geodesic (in particular smooth). The way to see this is to pick {a,b \in I} very close to each other, so that {c([a,b])} is contained in a neighborhood of {c\left( \frac{a+b}{2}\right)} satisfying the conditions of yesterday’s theorem; then {c|_{[a,b]}} must be length-minimizing, so it is a geodesic. We thus see that {c} is locally a geodesic, hence globally.

Say that {M} is geodesically complete if {\exp} can be defined on all of {TM}; in other words, a geodesic {\gamma} can be continued to {(-\infty,\infty)}. The name is justified by the following theorem:

Theorem 1 (Hopf-Rinow)

The following are equivalent:

  • {M} is geodesically complete.
  • In the metric {d} on {M} induced by {g} (see here), {M} is a complete metric space (more…)

Geodesics are locally length-minimizing November 13, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Fix a Riemannian manifold with metric {g} and Levi-Civita connection {\nabla}. Then we can talk about geodesics on {M} with respect to {\nabla}. We can also talk about the length of a piecewise smooth curve {c: I \rightarrow M} as

\displaystyle l(c) := \int g(c'(t),c'(t))^{1/2} dt .

 Our main goal today is:

Theorem 1 Given {p \in M}, there is a neighborhood {U} containing {p} such that geodesics from {p} to every point of {U} exist and also such that given a path {c} inside {U} from {p} to {q}, we have

 

\displaystyle l(\gamma_{pq}) \leq l(c)  

with equality holding if and only if {c} is a reparametrization of {\gamma_{pq}}.

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 \mathbb{R}^3, but it need not be the shortest path between two points. (more…)

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