16 Sep 2024. Poster by Filip Moučka at the meeting of the Spanish Network of Geometry and Physics, Barcelona.
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. Geometrically, 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 present several classes of examples of such structures. In particular, we show that linear symmetric Poisson structures are in correspondence with Jacobi-Jordan algebras, whereas strong symmetric correspond to those that are moreover associative.