Focal rigidity of hyperbolic surfaces

TL;DR

This paper investigates the rigidity of the focal decomposition in closed hyperbolic surfaces, demonstrating that generically, such structures are resistant to non-trivial topological deformations without altering their hyperbolic geometry.

Contribution

It extends previous results on flat tori to hyperbolic surfaces, showing focal decomposition rigidity in higher dimensions and general settings.

Findings

Focal decomposition of generic hyperbolic surfaces is topologically rigid.

Focal equivalence implies isometry modulo rescaling for flat tori.

Rigidity extends to higher dimensions based on classical theory.

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 $n \geq 3$. Our current result extends a previous result that flat tori in dimension $n \geq 2$ that are focally equivalent are isometric modulo rescaling.