Generalized geometrical structures of odd dimensional manifolds

TL;DR

This paper introduces generalized geometric structures on odd-dimensional manifolds, linking them to physical phase spaces in general relativity, and explores their algebraic and geometric properties.

Contribution

It defines new almost--cosymplectic--contact and almost--coPoisson--Jacobi structures, extending classical geometric frameworks and analyzing their relations and applications in spacetime physics.

Findings

Established conditions for metric and connection to produce cosymplectic and coPoisson structures.

Connected geometric structures to phase space models of relativistic particles.

Illustrated structures in both Galilei and Einstein spacetimes.

Abstract

We define an almost--cosymplectic--contact structure which generalizes cosymplectic and contact structures of an odd dimensional manifold. Analogously, we define an almost--coPoisson--Jacobi structure which generalizes a Jacobi structure. Moreover, we study relations between these structures and analyse the associated algebras of functions. As examples of the above structures, we present geometrical dynamical structures of the phase space of a general relativistic particle, regarded as the 1st jet space of motions in a spacetime. We describe geometric conditions by which a metric and a connection of the phase space yield cosymplectic and dual coPoisson structures, in case of a spacetime with absolute time (a Galilei spacetime), or almost--cosymplectic--contact and dual almost--coPoisson--Jacobi structures, in case of a spacetime without absolute time (an Einstein spacetime).