Today we will prove the fixed point theorem, which I restate here for convenience:

Theorem 1 (Elie Cartan) Let {K} be a compact Lie group acting by isometries on a simply connected, complete Riemannian manifold {M} of negative curvature. Then there is a common fixed point of all {k \in K}.

 

There is a Haar measure on {K}. In fact, we could even construct this by picking a nonzero alternating {n}-tensor (where {n=\dim K}) at {T_e(K)}, and choosing the corresponding {K}-invariant {n}-form on {K}. This yields a functional {C(K) \rightarrow \mathbb{R}}, which we can assume positive by choosing the orientation appropriately. This yields the Haar measure {d \mu} by the Riesz representation theorem.

Now define {J(q) := \int_K d^2(q,kp) d \mu(k).} This is a continuous function {M \rightarrow \mathbb{R}} which has a minimum, because {J(q)>J(p)} for {q} outside some compact set containing {p}. Let the minimum occur at {q_0}. I claim that the minimum is unique, which will imply that it is a fixed point of {K}.

It can be checked that {J} is continuously differentiable; indeed, let {q_t} be a curve. Then {d^2(q_t,kp)} can be computed as in yesterday when {kp \neq q_t}; when they are equal, it is still differentiable with zero derivative because of the {d^2}. (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 {q_t} to be a geodesic joining the minimal point {q_0} to some other point {q_1}. Now

\displaystyle \frac{d}{dt} J(q_t)|_{t=0} = \int_K \frac{d}{dt} d^2(q_t, kp) |_{t=0} d\mu(k) = 0.

 Then we get

\displaystyle \int_K d(q_0,kp) \cos \alpha d\mu(k) = 0

 where {\alpha} is an appropriate angle as in yesterday’s post. When {q_0=kp}, this {\alpha} is not well-defined, but {d(q_0,kp)=0}, so it is ok. Now

\displaystyle \int_K d^2(q_1, k.p) d\mu \geq \int_K \left( d^2(q_0, kp) + d^2(q_0,q_1) - 2 d(q_0, kp) \cos \alpha \right) d \mu(k) .

 This is because of the cosine inequality. But the cosine part vanishes, so this is strictly greater than {J(q_0)}. In particular, since {q_1} was arbitrary, {q_0} was a global minimum for {J}—and it is thus a fixed point.

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