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 analyze 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). Keywords: Spacetime, phase space, phase connection, Schouten bracket, Froelicher-Nijenhuis bracket, cosymplectic structure, coPoisson structure, contact structure, Jacobi structure, almost-cosymplectic-contact structure, almost-coPoisson-Jacobi structure. 2000 MSC: 53B15, 53B30, 53B50, 53D10, 58A10, 58A32
Generalized geometrical structures of odd dimensional manifolds / M.Modugno; J.Janyska. - In: JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES. - ISSN 0021-7824. - STAMPA. - ***:(2009), pp. 1-28. [10.1016/j.matpur.2008.09.007]
Generalized geometrical structures of odd dimensional manifolds
MODUGNO, MARCO;
2009
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 analyze 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). Keywords: Spacetime, phase space, phase connection, Schouten bracket, Froelicher-Nijenhuis bracket, cosymplectic structure, coPoisson structure, contact structure, Jacobi structure, almost-cosymplectic-contact structure, almost-coPoisson-Jacobi structure. 2000 MSC: 53B15, 53B30, 53B50, 53D10, 58A10, 58A32File | Dimensione | Formato | |
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