Today we will prove the fixed point theorem, which I restate here for convenience:
Theorem 1 (Elie Cartan) Let
be a compact Lie group acting by isometries on a simply connected, complete Riemannian manifold
of negative curvature. Then there is a common fixed point of all
.
There is a Haar measure on . In fact, we could even construct this by picking a nonzero alternating
-tensor (where
) at
, and choosing the corresponding
-invariant
-form on
. This yields a functional
, which we can assume positive by choosing the orientation appropriately. This yields the Haar measure
by the Riesz representation theorem.
Now define This is a continuous function
which has a minimum, because
for
outside some compact set containing
. Let the minimum occur at
. I claim that the minimum is unique, which will imply that it is a fixed point of
.
It can be checked that is continuously differentiable; indeed, let
be a curve. Then
can be computed as in yesterday when
; when they are equal, it is still differentiable with zero derivative because of the
. (I am sketching things here because I don’t currently want to dive into the technical details; see Helgason’s book for them.)
So now take to be a geodesic joining the minimal point
to some other point
. Now
Then we get
where is an appropriate angle as in yesterday’s post. When
, this
is not well-defined, but
, so it is ok. Now
This is because of the cosine inequality. But the cosine part vanishes, so this is strictly greater than . In particular, since
was arbitrary,
was a global minimum for
—and it is thus a fixed point.
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