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. 

Then we can write

\displaystyle q_n = \gamma_n(d(p,q_n)) = \exp_p( d(p,q_n) v_n)  

where the {v_n} are unit vectors in {T_p(M)}. Taking a subsequence, we may assume that both the {v_n} and the {d(p,q_n)} (which are bounded) tend to a limit. (Recall that a Cauchy sequence converges iff a subsequence does.) By continuity of {\exp_p}, we find that the {q_n} tend to a limit.

From the metric-completeness criterion, we find:

Corollary 1 A compact Riemannian manifold is geodesically complete, and any two points are joined by a geodesic of minimal length.

I now want to discuss an application of some of these ideas.

Theorem 2

Let {X} be a topological space and {M} a compact Riemannian manifold. Then there exists {\epsilon>0} such that if {f,h: X \rightarrow M} satisfy\displaystyle d(f(x),g(x))<\epsilon  

for all {x \in X}, then they are homotopic.

 

The proof of this doesn’t actually require all the machinery of the H-R theorems. The idea is that we can find a covering {U_i} of {M} such that any two points in any {U_i} can be joined by a unique geodesic lying in {U_i}. This “normal neighborhood theorem” apparently due to Whitney is actually true without the compactness assumption and is a refinement of the usual geodesic theorem. It’s in Helgason’s book.

In the compact case, choose {r>0} so small that the map {\exp_p: B_r(0) \subset T_p(M) \rightarrow M} is an open imbedding for all {p}. ({r} can be chosen locally, so this is ok.) Now I claim that any two points {p,q \in M} with {d(p,q)<r} are joined by a unique geodesic in {D_r(p) \cap D_r(q) \subset M}. Uniqueness is clear from the choice of {r}. For existence, by H-R there is a unit-speed geodesic from {p} to {q} of length {d(p,q)}. This is given by the image of some straight line through the origin in {T_p(M)} that must end at a point in {B_r(0)}, which line is thus contained in {B_r(0)}. Thus the geodesic is contained in {D_r(p)}.

So take this {r} as the {\epsilon} in the theorem. Given {f,h}, define {H(x,t)} by considering the unique shortest geodesic parametrized by {[0,1]} between {f(x),h(x)} and taking the value at {t}. This is a homotopy; it is even smooth if {f,h} are.

In particular, if {X} is a manifold, a continuous function {f: X \rightarrow M} can be approximated by a smooth function {h: X \rightarrow M}. In particular, {f} is homotopic to a smooth map {h: X \rightarrow M}. This is a useful fact.

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