  We give a new proof of Cartan's fixed point theorem using topological fixed
point theory. For an odd dimensional, simply connected and complete manifold
having non-positive curvature, we further prove that every isometry with finite
order not only fixes a point, but also maps a pencil of geodesics to itself. We
apply similar techniques to the automorphism group of a smoothly bounded domain
in the complex plane having connectivity three or more. We show that any prime
dividing the order of this finite group must divide a certain integer depending
only on the connectivity of the domain.
