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
Assume the second item: let be complete in the appropriate metric. Then if
for
an open interval
is a geodesic, consider a sequence
. Then
, since geodesics move at constant speed (cf. the remark after lemma 2 in the link). Thus the
form a Cauchy sequence, converging to some
. It is easy to check (by splicing two sequences together) that the limit does not depend on the choice of
. In local coordinates we can write
, when the geodesic property implies
for suitable Christoffel symbols . The first derivatives
are bounded (indeed, constant), so the second derivatives are too. Thus
extends to the interval
with a right-handed derivative
at
, and moreover the right-hand derivative at
exists and is uniformly continuous in a neighborhood of
(by the mean value theorem). Now there is locally a geodesic
at
with
. The function
satisfies the geodesic equation everywhere and is defined on . So we can extend these geodesics to the right, and similarly it is done to the left.
To go the other way, we first prove another Hopf-Rinow theorem, interesting in its own right:
Theorem 2 (Hopf-Rinow) Suppose
is defined all of
. Then for any
, there is a geodesic
from
to
that minimizes length.
The proof is really a nice bit of geometry. This is a global result, unlike yesterday’s theorem.
Consider a small sphere around
with respect to the metric
such that
satisfies the conclusion of yesterday’s theorem. Take the point
in
(which is compact if
is not too big at least) with
minimized. Then
because of the definition of via lengths of curves.
There is a geodesic travelling at unit speed with
,
. In particular,
Let be the set of all
with
. The boxed statement means that
, and
is evidently closed. If
, then we’ll be done—we’ll have a geodesic from
to
that minimizes length.
Since is closed, pick its largest element
, and let
. Choose a small neighborhood
satisfying the conditions of yesterday’s theorem. Now if we pick the point
in
closest to
, we have evidently
I claim that . First,
. The path
from
to
catenated with the geodesic from
to
forms a path from
to
of minimizing length
, so it is smooth and a geodesic.
In particular, , so we get a contradiction and hence the second theorem.
Since this post has already reached a certain length, I’ll defer the proof of the second implication in the first Hopf-Rinow theorem for tomorrow. Also, I should add that I’ve followed Milnor’s Morse Theory, chapter 2, in the proof of the second H-R theorem.
November 15, 2009 at 8:34 pm
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