[Paper Review] Spaces of pseudo-Riemannian geodesics and pseudo-Euclidean billiards

Boris Khesin, Serge Tabachnikov|arXiv (Cornell University)|Aug 24, 2006

Mathematical Dynamics and Fractals34 references3 citations

TL;DR

This paper establishes symplectic and contact structures on spaces of pseudo-Riemannian geodesics—space-like and time-like geodesics admit symplectic forms, while light-like geodesics carry a contact structure—extending classical Riemannian results. It introduces pseudo-Euclidean billiards and proves analogs of the Jacobi-Chasles theorems, demonstrating integrability of both geodesic flow on pseudo-Euclidean ellipsoids and billiards within them.

ABSTRACT

Many classical facts in Riemannian geometry have their pseudo-Riemannian analogs. For instance, the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. We discuss the geometry of these structures in detail, as well as introduce and study pseudo-Euclidean billiards. In particular, we prove pseudo-Euclidean analogs of the Jacobi-Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space.

Motivation & Objective

• To extend classical Riemannian geometric results on geodesic spaces to the pseudo-Riemannian setting.
• To investigate the natural geometric structures—symplectic and contact—on spaces of space-like, time-like, and light-like geodesics.
• To introduce and analyze the dynamics of pseudo-Euclidean billiards.
• To establish pseudo-Euclidean analogs of the Jacobi-Chasles theorems.
• To prove the integrability of geodesic flow on pseudo-Euclidean ellipsoids and billiards within such ellipsoids.

Proposed method

• Analyzing the natural symplectic structures on spaces of space-like and time-like geodesics in pseudo-Riemannian manifolds.
• Identifying the contact structure on the space of light-like geodesics as a key geometric invariant.
• Formulating the dynamics of pseudo-Euclidean billiards using reflection laws in indefinite quadratic forms.
• Applying symplectic and contact geometry techniques to extend classical integrability results to indefinite signatures.
• Using the Jacobi-Chasles correspondence to relate billiard dynamics to geodesic flows on quadrics.
• Establishing integrability via the existence of conserved quantities and invariant foliations in the phase space.

Experimental results

Research questions

• RQ1 What geometric structures naturally arise on spaces of pseudo-Riemannian geodesics, and how do they generalize the Riemannian case?
• RQ2 How can the theory of billiards be extended to pseudo-Euclidean spaces with indefinite metrics?
• RQ3 What are the pseudo-Euclidean analogs of the classical Jacobi-Chasles theorems in the context of geodesic and billiard dynamics?
• RQ4 Under what conditions is the geodesic flow on a pseudo-Euclidean ellipsoid integrable?
• RQ5 Can the billiard flow within a pseudo-Euclidean ellipsoid be shown to be integrable using symplectic and contact geometry?

Key findings

• The space of space-like and time-like geodesics on a pseudo-Riemannian manifold carries a natural symplectic structure, generalizing the Riemannian case.
• The space of light-like geodesics admits a natural contact structure, providing a new geometric invariant in indefinite signature.
• Pseudo-Euclidean billiards are defined via reflection laws in indefinite quadratic forms, extending classical Euclidean billiard theory.
• The paper proves pseudo-Euclidean analogs of the Jacobi-Chasles theorems, linking billiard dynamics to geodesic flows on quadrics.
• The geodesic flow on a pseudo-Euclidean ellipsoid is shown to be integrable, with conserved quantities arising from the underlying quadratic forms.
• The billiard flow within a pseudo-Euclidean ellipsoid is also proven integrable, establishing a direct extension of classical integrability results to indefinite geometries.