Proper homothetic maps and fixed points

Abstract

Let f be a proper homothetic map of the pseudo-Riemannian manifold M and assume f has a fixed point p. If all of the eigenvalues of either f*p or f-1*p have absolute values less than unity, then M is topologically Rn and M has a flat metric. This yields three characterizations of Minkowski spacetime. In general, a homothetic map of a complete pseudo-Riemannian manifold need not have fixed points. Furthermore, an example shows the existence of a proper homothetic map with a fixed point does not imply M is flat. The scalar curvature vanishes at a fixed point, but some of the sectional curvatures may be nonzero.