Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Superintegrable systems on conformal surfaces

Jonathan Kress, Konrad Schöbel, Andreas Vollmer

Abstract

We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of dimension three and higher. The governing equations of non-degenerate second order superintegrability in dimension two are structurally significantly different from those valid in higher dimensions. Specifically, we find conformally covariant structural equations, allowing one to classify the (conformal classes of) non-degenerate second order superintegrable systems on conformal surfaces geometrically. We then specialise to second order properly superintegrable systems on surfaces with a (pseudo-)Riemannian metric and obtain structural equations in accordance with the known equations for Euclidean space. We finally give a single explicit set of purely algebraic equations defining the variety parametrising such systems on all constant curvature surfaces.

Superintegrable systems on conformal surfaces

Abstract

We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of dimension three and higher. The governing equations of non-degenerate second order superintegrability in dimension two are structurally significantly different from those valid in higher dimensions. Specifically, we find conformally covariant structural equations, allowing one to classify the (conformal classes of) non-degenerate second order superintegrable systems on conformal surfaces geometrically. We then specialise to second order properly superintegrable systems on surfaces with a (pseudo-)Riemannian metric and obtain structural equations in accordance with the known equations for Euclidean space. We finally give a single explicit set of purely algebraic equations defining the variety parametrising such systems on all constant curvature surfaces.
Paper Structure (22 sections, 14 theorems, 145 equations, 1 table)

This paper contains 22 sections, 14 theorems, 145 equations, 1 table.

Table of Contents

  1. Introduction
  2. Novelty of the present paper
  3. Structure of the paper
  4. Notation
  5. Preliminaries
  6. Riemann surfaces as conformal surfaces
  7. Conformally superintegrable systems
  8. Bertrand-Darboux condition
  9. Non-degenerate systems
  10. Local description
  11. Conformal transformations
  12. Conformal superintegrability
  13. Useful identities
  14. Standard gauge
  15. Flat gauge
  16. ...and 7 more sections

Key Result

Lemma 2.7

A non-degenerate conformally superintegrable potential in dimension $2$, satisfying eq:conformal.Wilczynski, forms the closed system where are derived from and where $R$ denotes the Gauß curvature.

Theorems & Definitions (42)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Remark 2.9
  • ...and 32 more