Billiards in Finsler and Minkowski geometries
This paper introduces the subject of Finsler billiards. The paper starts by deducing the law of reflection from the principles of Fermat and Huygens, and extending the basic properties of billiards on Euclidean and Riemannian spaces to Minkowski and Finsler spaces. In the case of Minkowski billiards, it extends Mather's theorem (i.e., If the boundary of billiard table is sufficiently smooth and has a point of zero curvature, the billiard map has no invariant curves ) and uncovers an orbit-to-orbit duality. This duality is the discrete version of that found by Álvarez in the proof of Schaeffer's girth conjecture.
The Hilbert metric and Gromov Hyperbolicity
Karlsson and Noskov give some sufficient conditions for a Hilbert metric in a convex domain to be Gromov-hyperbolic. In particular, the authors prove that if the boundary of the domain is twice differentiable and quadratically convex, then the Hilbert metric is Gromov-hyperbolic.
Space of geodesics of Zoll three-spheres
If the space of geodesics of a Riemannian or Finsler manifold is itself a manifold, then it carries a natural symplectic structure. When the manifold is an n-dimensional sphere and the metric is such that all geodesics are closed with the same length, it has been known for some time that the space of geodesics is a manifold whose cohomology coincides with that of a complex hyperquadric in CPn. The present paper shows that the space of manifolds of a Riemannian metric on the 3-sphere all of whose geodesics are closed with the same length is actually symplectomorphic to a complex hyperquadric on CP3. The proof -- an application of Seiberg-Witten theory and symplectic cutting -- also works for Finsler metrics.