On the role of the surface geometry in convex billiards

TL;DR

This paper develops a framework for convex billiards on Riemannian surfaces, establishing fundamental properties and analyzing conditions for invariant curves, extending classical planar billiard results to curved geometries.

Contribution

It introduces a new framework for convex billiards on Riemannian manifolds and proves key properties like the twist condition and criteria for invariant curves.

Findings

Established basic properties of convex billiards on Riemannian surfaces.

Proved the twist property in this new setting.

Analyzed conditions for the existence of rotational invariant curves.

Abstract

This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some basic properties that have long been known for billiards on the plane are established. We prove the twist property and investigate conditions on the billiard for the existence and non existence of rotational invariant curves.