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Rational First Integrals and Relative Killing Tensors

Boris Kruglikov

TL;DR

The paper analyzes rational first integrals of geodesic flows and their connection to relative Killing tensors, introducing spaces $\mathcal{R}_{m,r,s}(g)$ and $\mathcal{K}^L_{m,d}(g)$ and establishing finite-dimensionality with explicit bounds. In spaceforms, it proves that rational integrals are generated by Killing tensors with a sharp dimension formula $\dim\mathcal{R}_{m,r,s}(g_0)=\Lambda_{m,r}+\Lambda_{m,s}-1$, and it clarifies the 2D case where fractional-linear integrals have dimension $5$ for constant curvature and $3$ otherwise. The work develops a robust algebraic and geometric framework (gauge, projective invariance, and finite-type PDEs) and provides concrete 2D and higher-dimensional examples that illustrate how rational integrals arise as ratios of relative Killing objects, including instances of reducibility. Overall, the results give a structural, finiteness-based understanding of rational integrals and their construction from Killing data, with implications for integrability of geodesic flows on varied geometries.

Abstract

We relate rational integrals of the geodesic flow of a (pseudo-)Riemannian metric to relative Killig tensors, describe the spaces they span and discuss upper bounds on their dimensions.

Rational First Integrals and Relative Killing Tensors

TL;DR

The paper analyzes rational first integrals of geodesic flows and their connection to relative Killing tensors, introducing spaces and and establishing finite-dimensionality with explicit bounds. In spaceforms, it proves that rational integrals are generated by Killing tensors with a sharp dimension formula , and it clarifies the 2D case where fractional-linear integrals have dimension for constant curvature and otherwise. The work develops a robust algebraic and geometric framework (gauge, projective invariance, and finite-type PDEs) and provides concrete 2D and higher-dimensional examples that illustrate how rational integrals arise as ratios of relative Killing objects, including instances of reducibility. Overall, the results give a structural, finiteness-based understanding of rational integrals and their construction from Killing data, with implications for integrability of geodesic flows on varied geometries.

Abstract

We relate rational integrals of the geodesic flow of a (pseudo-)Riemannian metric to relative Killig tensors, describe the spaces they span and discuss upper bounds on their dimensions.

Paper Structure

This paper contains 8 sections, 14 theorems, 49 equations.

Table of Contents

  1. Rational integrals and relative Killing tensors
  2. Relative Killing tensors
  3. Rational integrals with analytic coefficients
  4. Finite dimensionality of the space of rational integrals
  5. Finite dimensionality of the space of relative Killing tensors
  6. Computation for spaceforms
  7. The case of a Riemann surface
  8. Examples

Key Result

Lemma 1

Let $F=\frac{P}{Q}$ be an integral of $H$, where $P=\sum_{i=r_0}^{r_1}P_i$, $Q=\sum_{i=s_0}^{s_1}Q_i$, with $P_i,Q_i$ being homogeneous polynomials in $p$ of degree $i$. Then both $F_0=\frac{P_{r_0}}{Q_{s_0}}$ and $F_1=\frac{P_{r_1}}{Q_{s_1}}$ are rational integrals.

Theorems & Definitions (34)

  • Lemma 1
  • proof
  • Definition
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3
  • Remark 4
  • Lemma 2
  • ...and 24 more