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Integrable geodesic flows with simultaneously diagonalisable quadratic integrals

Sergey I. Agafonov, Vladimir S. Matveev

Abstract

We show that if $n$ functionally independent commutative quadratic in momenta integrals for the geodesic flow of a Riemannian or pseudo-Riemannian metric on an $n$-dimensional manifold are simultaneously diagonalisable at the tangent space to every point, then they come from the Stäckel construction, so the metric admits orthogonal separation of variables.

Integrable geodesic flows with simultaneously diagonalisable quadratic integrals

Abstract

We show that if functionally independent commutative quadratic in momenta integrals for the geodesic flow of a Riemannian or pseudo-Riemannian metric on an -dimensional manifold are simultaneously diagonalisable at the tangent space to every point, then they come from the Stäckel construction, so the metric admits orthogonal separation of variables.
Paper Structure (2 sections, 2 theorems, 6 equations)

This paper contains 2 sections, 2 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:1']}

Key Result

Theorem 1

Under the assumptions above, for almost every point $x$ the restrictions of the tensor fields $\overset{\alpha}K^{ij}$, $\alpha=1,\dots, n$, to $T_xM$ are linearly independent. In particular, for a generic linear combination $I= \sum_{\alpha=2}^n \lambda_\alpha \overset{\alpha}I$ of the integrals,

Theorems & Definitions (3)

  • Theorem 1
  • Corollary 1.1
  • Remark 1