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) 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.