A symmetric cousin of Poisson geometry

23 Mar 2026. Talk by Filip Moučka at Poisson Geometry and its relatives: a thematic day at CRM, Barcelona.

Abstract: Poisson geometry is a natural extension of symplectic geometry that encodes Hamiltonian dynamics, Lie algebras, and singular foliations via skew-symmetric bivector fields. In this talk, I will explore a symmetric counterpart of this framework by introducing symmetric Poisson structures, defined as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying a natural compatibility condition. These new geometric structures extend (pseudo-)Riemannian geometry and describe locally geodesically invariant distributions endowed with a metric along them. In particular, they include totally geodesic foliations equipped with a metric and a compatible connection on each leaf. I will explain how symmetric Poisson geometry is closely related to the Patterson–Walker metric, a split-signature metric on the cotangent bundle that is analogous to the canonical symplectic form. Finally, I will present several examples of symmetric Poisson structures, with special emphasis on the linear case, which is in one-to-one correspondence with real finite-dimensional Jacobi–Jordan algebras.