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

Generalized geometrical structures of odd dimensional manifolds