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.