Focal rigidity of hyperbolic surfaces

Abstract

In this note, we consider the rigidity of the focal decomposition of closed hyperbolic surfaces. We show that, generically, the focal decomposition of a closed hyperbolic surface does not allow for non-trivial topological deformations, without changing the hyperbolic structure of the surface. By classical rigidity theory this is also true in dimension $n\ge 3$. Our current result extends a previous result that flat tori in dimension $n\ge 2$ that are focally equivalent are isometric modulo rescaling.

Published in Annales Academiae Scientiarum Fennicae Mathematica, Volumen 37, 2012, 149-159, 11 pages