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Symmetric Poisson geometry, totally geodesic foliations and Jacobi-Jordan algebras

Filip Moučka, Roberto Rubio

TL;DR

This work develops symmetric Poisson geometry as the natural analogue of Poisson geometry for degenerate metrics, formulating symmetric Poisson structures as pairs $(\vartheta,\nabla)$ with the key integrability $[\vartheta,\vartheta]_s=0$ and distinguishing strong from involutive cases. It builds a comprehensive toolkit—symmetric Cartan calculus, the symmetric Schouten bracket, and the Patterson–Walker construction—to interpret these structures geometrically via characteristic distributions, leaf metrics, and leaf connections, and dynamically via Patterson–Walker dynamics on $T^*M$. The authors show that involutive structures induce totally geodesic partitions with leafwise non-degenerate Poisson data, and that strong cases provide leaf Levi-Civita connections; they also establish a deep link between linear symmetric Poisson structures and Jacobi–Jordan algebras, offering a clear algebraic-geometric correspondence. Collectively, the paper lays foundational connections between symmetric bivectors, geodesic invariance, foliations, and algebraic structures, with potential implications for geometry, dynamics, and pseudo-Riemannian or sub-Riemannian contexts.

Abstract

We introduce symmetric Poisson structures as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying an integrability condition analogous to that in usual Poisson geometry. Equivalent conditions in Poisson geometry have inequivalent analogues in symmetric Poisson geometry and we distinguish between symmetric and strong symmetric Poisson structures. We prove that symmetric Poisson structures correspond to locally geodesically invariant distributions together with a characteristic metric, whereas strong symmetric Poisson structures correspond to totally geodesic foliations together with a leaf metric and a leaf connection. We introduce, using the Patterson-Walker metric, a dynamics on the cotangent bundle and show its connection to symmetric Poisson geometry, the parallel transport equation and the Newtonian equation for conservative systems. Finally, we prove that linear symmetric Poisson structures are in correspondence with Jacobi-Jordan algebras, whereas strong symmetric correspond to those that are moreover associative.

Symmetric Poisson geometry, totally geodesic foliations and Jacobi-Jordan algebras

TL;DR

This work develops symmetric Poisson geometry as the natural analogue of Poisson geometry for degenerate metrics, formulating symmetric Poisson structures as pairs with the key integrability and distinguishing strong from involutive cases. It builds a comprehensive toolkit—symmetric Cartan calculus, the symmetric Schouten bracket, and the Patterson–Walker construction—to interpret these structures geometrically via characteristic distributions, leaf metrics, and leaf connections, and dynamically via Patterson–Walker dynamics on . The authors show that involutive structures induce totally geodesic partitions with leafwise non-degenerate Poisson data, and that strong cases provide leaf Levi-Civita connections; they also establish a deep link between linear symmetric Poisson structures and Jacobi–Jordan algebras, offering a clear algebraic-geometric correspondence. Collectively, the paper lays foundational connections between symmetric bivectors, geodesic invariance, foliations, and algebraic structures, with potential implications for geometry, dynamics, and pseudo-Riemannian or sub-Riemannian contexts.

Abstract

We introduce symmetric Poisson structures as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying an integrability condition analogous to that in usual Poisson geometry. Equivalent conditions in Poisson geometry have inequivalent analogues in symmetric Poisson geometry and we distinguish between symmetric and strong symmetric Poisson structures. We prove that symmetric Poisson structures correspond to locally geodesically invariant distributions together with a characteristic metric, whereas strong symmetric Poisson structures correspond to totally geodesic foliations together with a leaf metric and a leaf connection. We introduce, using the Patterson-Walker metric, a dynamics on the cotangent bundle and show its connection to symmetric Poisson geometry, the parallel transport equation and the Newtonian equation for conservative systems. Finally, we prove that linear symmetric Poisson structures are in correspondence with Jacobi-Jordan algebras, whereas strong symmetric correspond to those that are moreover associative.

Paper Structure

This paper contains 40 sections, 46 theorems, 233 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. tocSymmetric bivector fields and the symmetric Schouten brackettocSymmetric bivector fields and the symmetric Schouten bracket
  3. The symmetric Poisson bracket and the gradient map
  4. Integrability for Poisson structures
  5. Symmetric Cartan calculus SymCartan
  6. The symmetric Schouten bracket
  7. Symmetric Poisson structures
  8. Symmetric Poisson structures
  9. Definition of a symmetric Poisson structure
  10. Strong symmetric Poisson structures
  11. Non-degenerate symmetric Poisson structures
  12. tocGeometric interpretation of symmetric PoissontocGeometric interpretation of symmetric Poisson structures
  13. The characteristic distribution/module of a symmetric bivector field
  14. The characteristic distribution/module of a symmetric Poisson structure
  15. The characteristic metric of a symmetric bivector field
  16. ...and 25 more sections

Key Result

Proposition 2.4

The symmetric Schouten bracket corresponding to a connection $\nabla$ on $M$ is given, for $\mathcal{X}$, $\mathcal{Y}\in\mathfrak{X}^\bullet_\emph{sym}(M)$, by

Figures (3)

  • Figure 1: Regular foliations of the punctured plane.
  • Figure 2: Types of symmetric Poisson structures.
  • Figure 3: Partitions induced by Jacobi-Jordan algebras up to dimension $3$.

Theorems & Definitions (143)

  • Definition
  • Example 2.1: Flat (pseudo-)Riemannian manifolds
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 133 more