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First Integrals of Geodesic Flows on Cones

Andrey E. Mironov, Siyao Yin

TL;DR

This work studies geodesic flows on the cone over a closed $C^3$-smooth manifold and shows they admit a rich set of first integrals that almost uniquely determine non-radial geodesics. Building on a cone-sphere correspondence, the authors construct $2N+2$ integrals that are continuous on the full tangent bundle and $C^1$ on the non-radial region, with radial (generatrix) geodesics mapped to the origin. Non-radial geodesics are shown to be determined by fixed-length geodesic segments on the cross-section $\\oldmath{\\Sigma}=K\\cap \\mathbb{S}^N$, via a correspondence with the sphere tangent condition. Consequently, the geodesic flow on the cone is integrable in a Liouville-type sense, in that almost all trajectories are uniquely defined by the integral data, extending billiard-type integrals to continuous Riemannian cone settings.

Abstract

In this paper we study the behavior of geodesics on cones over arbitrary $C^3$-smooth closed Riemannian manifolds. We show that the geodesic flow on such cones admits first integrals whose values uniquely determine almost all geodesics except for cone generatrices. This investigation is inspired by our results on billiards inside cones over manifolds where similar results hold true.

First Integrals of Geodesic Flows on Cones

TL;DR

This work studies geodesic flows on the cone over a closed -smooth manifold and shows they admit a rich set of first integrals that almost uniquely determine non-radial geodesics. Building on a cone-sphere correspondence, the authors construct integrals that are continuous on the full tangent bundle and on the non-radial region, with radial (generatrix) geodesics mapped to the origin. Non-radial geodesics are shown to be determined by fixed-length geodesic segments on the cross-section , via a correspondence with the sphere tangent condition. Consequently, the geodesic flow on the cone is integrable in a Liouville-type sense, in that almost all trajectories are uniquely defined by the integral data, extending billiard-type integrals to continuous Riemannian cone settings.

Abstract

In this paper we study the behavior of geodesics on cones over arbitrary -smooth closed Riemannian manifolds. We show that the geodesic flow on such cones admits first integrals whose values uniquely determine almost all geodesics except for cone generatrices. This investigation is inspired by our results on billiards inside cones over manifolds where similar results hold true.

Paper Structure

This paper contains 3 sections, 6 theorems, 66 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Geodesics on Cones
  3. Construction of first integrals, proof of Theorem \ref{['lem:integral']}

Key Result

Lemma 1

Let $\gamma(s)$ be a geodesic on a cone $K \subset \mathbb{R}^{N+1}$. Then remains constant along $\gamma$. Geometrically, $I$ represents the squared distance from the vertex $O$ to the tangent line of $\gamma$.

Figures (3)

  • Figure 1: The sphere as a caustic of the billiard inside a cone.
  • Figure 2: All tangent lines of a given non-generatrix geodesic are tangent to a common sphere.
  • Figure 3: The geodesic $\gamma_{x,v}(s)$ touches $\mathbb{S}^N(\sqrt{I})$ at the unique point $\gamma_{x,v}(s_0)$.

Theorems & Definitions (13)

  • Lemma 1
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • proof : Proof of Lemma \ref{['lem:distance']}
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['lem:cone-sphere-correspondence']}
  • Lemma 3
  • ...and 3 more