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Torsion-free connections of second-order maximally superintegrable systems

Andreas Vollmer

TL;DR

The work analyzes irreducible second-order maximally conformally superintegrable systems by introducing torsion-free connections built from primary structure tensors and linking the secondary/semi-degenerate tensors to curvature. For non-degenerate systems, the secondary tensor is derived from the induced connection’s curvature, and a form of curvature is obtained; when the secondary tensor vanishes, one recovers a special curvature structure. For semi-degenerate systems, the secondary tensor is proportional to the Ricci tensor and the induced connection is projectively flat, with proper cases yielding flat connections and Laplacian eigenfunction potentials. Proper systems exhibit curvature obstructions and Ricci-symmetric or Ricci-flat features, while examples (Smorodinsky–Winternitz and Escobar-Ruiz-Miller) illustrate the framework and confirm projective-flatness and BD-Wilczynski compatibility. Overall, the paper argues that projective differential geometry provides a natural geometric setting for understanding Wilczynski equations and Bertrand-Darboux compatibility in second-order superintegrable systems, bridging conformal and projective perspectives.

Abstract

Second-order (maximally) conformally superintegrable systems play an important role as models of mechanical systems, including systems such as the Kepler-Coulomb system and the isotropic harmonic oscillator. The present paper is dedicated to understanding non- and semi-degenerate systems. We obtain "projective flatness" results for two torsion-free connections naturally associated to such systems. This viewpoint sheds some light onto the interrelationship of properly and conformally (second-order maximally) superintegrable systems from a geometrical perspective. It is shown that the semi-degenerate secondary structure tensor can be viewed as the Ricci curvature of a natural torsion-free connection defined by the primary structure tensor (and similarly in the non-degenerate case). It is also shown that properly semi-degenerate systems are characterised, similar to the non-degenerate case, by the vanishing of the secondary structure tensor.

Torsion-free connections of second-order maximally superintegrable systems

TL;DR

The work analyzes irreducible second-order maximally conformally superintegrable systems by introducing torsion-free connections built from primary structure tensors and linking the secondary/semi-degenerate tensors to curvature. For non-degenerate systems, the secondary tensor is derived from the induced connection’s curvature, and a form of curvature is obtained; when the secondary tensor vanishes, one recovers a special curvature structure. For semi-degenerate systems, the secondary tensor is proportional to the Ricci tensor and the induced connection is projectively flat, with proper cases yielding flat connections and Laplacian eigenfunction potentials. Proper systems exhibit curvature obstructions and Ricci-symmetric or Ricci-flat features, while examples (Smorodinsky–Winternitz and Escobar-Ruiz-Miller) illustrate the framework and confirm projective-flatness and BD-Wilczynski compatibility. Overall, the paper argues that projective differential geometry provides a natural geometric setting for understanding Wilczynski equations and Bertrand-Darboux compatibility in second-order superintegrable systems, bridging conformal and projective perspectives.

Abstract

Second-order (maximally) conformally superintegrable systems play an important role as models of mechanical systems, including systems such as the Kepler-Coulomb system and the isotropic harmonic oscillator. The present paper is dedicated to understanding non- and semi-degenerate systems. We obtain "projective flatness" results for two torsion-free connections naturally associated to such systems. This viewpoint sheds some light onto the interrelationship of properly and conformally (second-order maximally) superintegrable systems from a geometrical perspective. It is shown that the semi-degenerate secondary structure tensor can be viewed as the Ricci curvature of a natural torsion-free connection defined by the primary structure tensor (and similarly in the non-degenerate case). It is also shown that properly semi-degenerate systems are characterised, similar to the non-degenerate case, by the vanishing of the secondary structure tensor.
Paper Structure (11 sections, 14 theorems, 56 equations)

This paper contains 11 sections, 14 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. The induced connection
  5. The non-degenerate case
  6. The semi-degenerate case
  7. Proper systems
  8. Examples
  9. Non-degenerate potentials
  10. Semi-degenerate potentials
  11. Discussion and conclusion

Key Result

Theorem 1.1

Let $(M^n,g)$ with $n\geq3$ and let $V$ be a non-degenerate potential with primary structure tensor $T$ and secondary structure tensor $\tau$. We introduce the torsion-free connection where $\nabla^g$ is the Levi-Civita connection. Then:

Theorems & Definitions (29)

  • Definition 1
  • Remark 1
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 19 more