There are a whole bunch of theorems in Riemannian geometry to the effect that “if the Riemannian manifold has property A of the curvature, then it has the topological property B.” Over the rest of MaBloWriMo and in the following weeks, I aim to talk about a few such results. The first one characterizes manifolds of negative curvature.
Negative curvature
Let be a Riemannian manifold with Riemannian metric
Say that
has negative curvature if for all
,
(Later I will interpret this in terms of the sectional curvature, which I have not yet defined.)
Statements
Theorem 1 (Cartan-Hadamard) Let be a complete Riemannian manifold of negative curvature. Then for
, the map
is a covering map. In particular, if
is simply connected, then it is diffeomorphic to
.
Of course, the diffeomorphism doesn’t have to preserve the Riemannian metric.
The strategy of the proof is as follows. First, we will show that the map is an immersion (though in general not injective), using the discussion yesterday about how Jacobi fields determine the differential of the exponential map. Then we will invoke
Proposition 2 Let
be a complete Riemannian manifold. Suppose
and the map
is an immersion. Then
is a covering map.
The condition of the result is often stated to the effect that “ has no conjugate points to
.”
To see this, we will appeal to yet another result:
Theorem 3 (Ambrose) Let
be a surjective morphisms of Riemannian manifolds with
complete. Suppose
preserves the metric on the tangent spaces. Then
is a covering map.
I will work backwards to prove these three results.
Proof of Ambrose’s theorem
The idea is to use exponential coordinates. Firstly takes geodesics to geodesics, at least locally. Indeed,
if
is a curve in
, geodesics locally minimize length, and
is locally an isomorphism of manifolds. By splitting a geodesic in
into small pieces, it follows that
maps geodesics in
to a broken geodesic in
, but since
is smooth,
just maps
-geodesics to
-geodesics.
So let . We will find a small neighborhood
of
such that
has components diffeomorphic to
. Indeed, take
an open star-shaped set containing
mapped diffeomorphically onto some open set, say
. If
is taken very small and
is chosen as
, we can take
to be
if
is the metric on
induced by the Riemannian metric.
Let . Because
is a local diffeomorphism,
is discrete. I claim that
Indeed, if , then
because
preserves curve lengths. This implies that the second set is contained in the first.
For the other inclusion, let us choose and draw the commutative diagram
The image in is
, while by Hopf-Rinow II the image in
is
. From the diagram it is even clear that
is a diffeomorphism.
If we can choose a geodesic between
and
which we can locally, piece-by-piece, lift it to a piecewise-smooth curve
starting at
and ending at some point of
. Now
is proportional to
so
is actually a geodesic (hence smooth) by these arguments. In particular, the length of
is
, and
. This proves the other inclusion.
So we have . Each
is diffeomorphic to
. All we need to show is that
are disjoint for
. Otherwise we could choose
and choose geodesics from
to
in
, and
to
in
. The images give two different geodesics from
to
, contradiction. This proves Ambrose’s theorem.
Proof of the proposition
Since the map is surjective (H-R II) and an immersion by assumption, we can consider the pulled back metric
on
and apply Ambrose’s theorem to it. The only thing we have to check is that with the Riemannian metric
on
, it is complete. But geodesics through the origin exist of arbitrary length—they are just the straight lines through the origin, since these lines are mapped onto geodesics in
. By H-R, this “completeness at a point” implies the completeness of the whole manifold.
I will actually prove the Cartan-Hadamard theorem in the next post.
November 22, 2009 at 9:33 am
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