Two Applications of Topological Fixed Point Theory

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.