Oh right, this is supposed to be a math blog, isn’t it?
So, I was trying to understand the proof in do Carmo (which is a great book, incidentally) of Synge-Weinstein, and I tried to write an expository paper on it. Unfortunately I can’t get the formatting perfect in WordPress since some of the equations go out of the margins, but most if it should look ok. I thus made a PDF of this too. I dressed it up in formal article formatting just for the heck of it.
We give an exposition of the proof of a few results in global Riemannian geometry due to Synge and Weinstein using variations of the energy integral.
1. Introduction
One of the big refrains of modern Riemannian geometry is that curvature determines topology. Recall, for instance, the basic Cartan-Hadamard theorem that a complete, simply connected Riemannian manifold of nonpositive curvature is diffeomorphic to under the exponential map. We proved this basically by showing that
is nonsingular under the hypothesis of nonnegative curvature (using Jacobi fields) and that it was thus a covering map (the latter part was relatively easy). More difficult, and relevant to the present topic, was the Bonnet-Myers theorem, which asserted the compactness of a complete Riemannian manifold with bounded-below, positive Ricci curvature. The proof there showed that a long enough geodesic could not minimize energy (by using the second variation formula—recall that the second variation formula is intimately connected with curvature), and therefore could not minimize length. Since the distance between two points in a complete Riemanninan manifold is the length of the shortest geodesic between them (Hopf-Rinow!), this implied a bound on the diameter.
Today, however, we’re going to assume at the outset that the manifold in question is already compact. One of the theorems will be that a compact, even-dimensional orientable manifold of positive curvature is simply connected. In particular, there is no metric of everywhere positive sectional curvature on the torus .
How will we do this? Well, first consider the universal cover . The covering transformations of
are all smooth, and we can endow
with a metric in a natural way such that these are isometries, and
has positive curvature—hence, by comppleteness (a covering manifold of a complete manifold is also complete, easy exercise) and the Bonnet-Myers theorem,
is compact. It is also orientable since we can pull back the
-orientation. If
is not simply connected, then we can find a nontrivial covering transformation
.
But, we will show, using the that an isometry of a compact, oriented, even-dimensional manifold admits a fixed point. In particular, does, which means that it is the identity, contradiction.
2. The statement
We will now begin work on the more general fixed-point theorem.
So, we’re going to start with a compact oriented -dimensional Riemannian manifold
of positive sectional curvature and an isometry
.
Theorem 1 (Weinstein) Suppose
is as above and
preserves orientation if
is even and reverses orientation if
is odd. Then
has a fixed point.
The hypothesis about the dimension seems a little odd, but it comes from linear algebra used in the proof.
3. The strategy
Here is the strategy. Let be the metric. By compactness, there is
such that
is minimal. Assuimng this minimum is nonzero, we consider the minimal geodesic
from
to
and construct a variation
of it joining points
. By its construction and the second variation formula, we will show that
for
small, which contradicts minimality.
So, how are we going to whisk this variation out of thin air? We will construct a parallel vector field on
, perpendicular to
, and let
In order that connects
to
, we need
(assuming
is parametrized by
).
4. Construction of the vector field
Proposition 2 There exists a parallel vector field
on
, perpendicular to
, such that
.
The first step, paradoxically enough, will be to prove that itself satisfies these conditions (except orthogonality), in other words that:
Lemma 3
.
Proof: Now is a geodesic starting at
, and if we show that the piecewise smooth broken geodesic
(concatenation) is actually smooth, we will have established the first step.
Pick some point in the middle of
. Then
. But there is a path
from
to
of the same length
, namely
traversed starting at
to
. For instance, we could take
and then traverse the curve
from
to
, for a total distance of
. This means that
is smooth, hence so is
; the only point in doubt was at
. In particular the left and right-hand derivatives match, so
.
Proof:
There was, in fact, method to this madness. We are now going to use this fact and linear algebra to construct the vector field . So, the goal is to find some vector
such that the transformation
obtained by first applying
(and sending to
) and then parallel translating back along
has an eigenvector perpendicular to
—which we just proved is a fixed point. Then the parallel field extending
can be taken as our
, which proves the lemma. Now consider the subspace
. Now
is an isometry so fixes
, and
is of dimension one smaller. Also
(and hence
) preserves (resp. reverses) orientation if
is odd (resp. even). By now invoking the following result from linear algebra, such a vector falls into our lap.
Lemma 4 (Linear algebra) Let
be an orthogonal linear transformation of a real vector space
. Suppose
fixes orientation if
is odd and reverses it if
is even. Then
has a nontrivial fixed point.
This will be proved later (in the appendix). Anyway, we now can use Proposition 1.
5. The second variation formula
5.1. The approach
Recall that we have defined the variation ; by what has been discussed,
for all
. In particular, we have paths between
and
. Recall also the energy
of a piecewise-smooth path
; we shall use this in the sequel because it is easier to work with than the length (which has annoying square roots). Now
because has a minimum at
. Indeed,
—since
moves at constant speed, being a geodesic—and
by Schwarz’s inequality. When we prove
it will follow that there is some small with
but
contradiction.
5.2. Proof of the variation formula
First, let us recall a more general version of the second variation formula and a sketch of the proof. Let be a geodesic,
a smooth variation of
(not necessarily fixing endpoints) with variation vector field
. Then
This becomes (where, by abuse of notation denotes differentiation w.r.t.
)
i.e.
Differentiate with respect to again:
We shall now analyze each term separately. The first two terms become
The last term becomes
Since is a geodesic, evaluation at
of this yields
which in total yields
This is the version of the second variation formula that we shall use.
6. Computation of the variation
Now let’s apply the formula to the constructed in the proof of Weinstein’s theorem. Fortunately, most of the mess clears up. By parallelism,
, so all that we are left with is
by hypothesis on the sectional curvature and since are orthogonal. It now follows, as discussed previously, that
is not minimal, which gives a contradiction.
7. Consequences
Theorem 5 (Synge)
Letbe a compact
-dimensional Riemannian manifold of positive curvature.
- If
is even and
is oriented, then
is simply connected.
- If
is odd, then
is orientable.
Proof: We have already discussed case a) in the introduction. In case b), if is not orientable, then there is an orientable double cover
. The manifold
is compact, has an induced Riemannian metric of positive curvature, and has an orientation-reversing covering transformation
when considered as a convering space of
. This transformation
must thus have no fixed points, which contradicts Weinstein’s theorem.
8. Appendix: Proof of the linear algebra lemma
For convenience, we restate the lemma:
Let be an orthogonal linear transformation of
. Suppose
fixes orientation if
is odd and reverses it if
is even. Then
has a nontrivial fixed point.
Proof: First, in either case, the nonreal eigenvalues of occur in conjugate pairs, so the product of nonreal eigenvalues is positive. All the real eigenvalues are
since
is orthogonal.
is odd. Then
and
has an odd number of real eigenvalues; they thus cannot all be
.
is even. Then
and
has an even number of real eigenvalues; they thus cannot all be
.
October 27, 2010 at 9:50 pm
Which Do Carmo by the way? Do Carmo’s diff. geometry book or Do Carmo’s Riemannian geometry book? There’re two!
October 28, 2010 at 3:35 pm
The one on Riemannian geometry.
November 18, 2011 at 4:35 pm
I think there is a typo in the intro. Cartan-Hadamard theorem applies to manifolds of non-positive rather than non-negative curvature.
November 18, 2011 at 8:08 pm
Fixed, thanks.
January 19, 2012 at 6:02 pm
Hi, I want make a remark and a question. First, the compactness in Synge can be replaced by completness by the Soul Theorem of Perelman. And the question: can you do the same in Weinstein Thm?
Thanks.
January 19, 2012 at 6:26 pm
I haven’t thought about this in years and don’t remember this at all. Sorry! You might try mathoverflow.