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Integrable geodesic flow in 3D and webs of maximal rank

Sergey I. Agafonov

Abstract

We characterize geodesic flows, admitting two commuting quadratic integrals with common principal directions, in terms of the geodesic 4-webs such that the tangents to the web leaves are common zero directions of the integrals. We prove that, under some natural geometric hypothesis, the metric is of Stäckel type.

Integrable geodesic flow in 3D and webs of maximal rank

Abstract

We characterize geodesic flows, admitting two commuting quadratic integrals with common principal directions, in terms of the geodesic 4-webs such that the tangents to the web leaves are common zero directions of the integrals. We prove that, under some natural geometric hypothesis, the metric is of Stäckel type.
Paper Structure (15 sections, 12 theorems, 70 equations)

This paper contains 15 sections, 12 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Stäckel metrics and geodesic webs
  3. Maximal rank webs
  4. Stäckel metrics from webs of maximal rank
  5. On quadratic integrals with not normal principal congruences
  6. Integrability by two linear integrals or by one linear and one quadratic integrals
  7. Examples
  8. Linear and quadratic integral
  9. Translation $\partial _z$
  10. Rotation $y\partial _x-x\partial_y$
  11. Screw symmetry $\alpha \partial_z+y\partial _x-x\partial_y$
  12. Two quadratic integrals
  13. Concluding remarks
  14. Geodesically equivalent metrics
  15. Stäckel metrics and integrable 3D billiards

Key Result

Theorem 1

Suppose that a three-dimensional Riemannian manifold $M$ carries a geodesic 4-web $w_g$ such that at each point $m\in M$ there are 3 mutually orthogonal planes $\pi_i$, $i=1,2,3,$ such that Then the metric is of Stäckel type and $\pi_i$ are tangent to the Stäckel coordinate surfaces.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Definition 4
  • Proposition 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Definition 9
  • Corollary 10
  • ...and 4 more