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

TL;DR

This paper introduces symmetric Poisson structures, explores their geometric properties, and establishes a correspondence with Jacobi-Jordan algebras, highlighting new connections between geometry and algebra.

Contribution

It defines symmetric Poisson structures, distinguishes between symmetric and strong types, and links strong symmetric structures to associative Jacobi-Jordan algebras.

Findings

Symmetric Poisson structures correspond to geodesically invariant distributions.

Strong symmetric Poisson structures relate to totally geodesic foliations.

Linear symmetric Poisson structures are equivalent to Jacobi-Jordan algebras.

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.