Generalized geometrical structures of odd dimensional manifolds. (English) Zbl 1163.53051

This paper studies geometrical structures on the phase space of a space-time naturally induced by a metric and a phase connection. The authors first recall some standard structures; then introduce both an almost-cosymplectic-contact structure and an almost-coPoisson-Jacobi structure. These notions generalize the contact structure on a odd dimensional manifold and the Jacobi structure respectively. The general properties of these new structures are then studied. Examples are finally given. In particular, the authors present geometrical dynamical structures on the phase space of a general relativistic particle, regarded as the first jet space of motions in a spacetime.


53D17 Poisson manifolds; Poisson groupoids and algebroids
53D15 Almost contact and almost symplectic manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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[1] de Leon, M.; Tuynman, G. M., A universal model for cosymplectic manifolds, Journal of Geometry and Physics, 20, 77-86 (1996) · Zbl 0861.53026
[3] Janyška, J.; Modugno, M., Classical particle phase space in general relativity, (Differential Geometry and Applications, Proc. Conf.. Differential Geometry and Applications, Proc. Conf., Aug. 28 - Sept. 1, 1995, Brno, Czech Republic (1996), Masaryk University: Masaryk University Brno), 573-602 · Zbl 0862.53024
[4] Janyška, J.; Modugno, M., Relations between linear connections on the tangent bundle and connections on the jet bundle of a fibred manifold, Archivum Mathematicum (Brno), 32, 281-288 (1996), (in honour of I. Kolář) · Zbl 0881.53015
[5] Janyška, J.; Modugno, M., On the graded Lie algebra of quantisable forms, (Kolář, I.; Kowalski, O.; Krupka, D.; Slovák, J., Differential Geometry and Applications, Proc. of the 7th Intern. Conf.. Differential Geometry and Applications, Proc. of the 7th Intern. Conf., Brno, 10-14 August 1998 (1999), Masaryk University), 601-620 · Zbl 0948.58001
[6] Janyška, J.; Modugno, M., Covariant Schrödinger operator, J. Phys. A: Math. Gen., 35, 8407-8434 (2002) · Zbl 1057.81050
[7] Janyška, J.; Modugno, M., Geometric structures of the classical general relativistic phase space, Int. J. Geom. Methods Mod. Phys., 5, 699-754 (2008) · Zbl 1160.53008
[8] Janyška, J.; Modugno, M.; Vitolo, R., Semi-vector spaces and units of measurement (2007), preprint
[9] Kirillov, A. A., Local Lie algebras, Russian Math. Surveys, 31, 55-76 (1976) · Zbl 0357.58003
[10] Kolář, I.; Michor, P. W.; Slovák, J., Natural Operations in Differential Geometry (1993), Springer-Verlag · Zbl 0782.53013
[11] Libermann, P.; Marle, Ch. M., Symplectic Geometry and Analytical Mechanics (1987), Reidel Publ.: Reidel Publ. Dordrecht
[12] Lichnerowicz, A., Les varietés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl., 57, 453-488 (1978) · Zbl 0407.53025
[13] Reeb, G., Sur certaines propriétés topologiques des trajectoires des systèmes dynamiques, Mémoires Acad. Roy. Belgique, 27, 130-194 (1952) · Zbl 0048.32903
[14] Trautman, A., Comparison of Newtonian and relativistic theories of spacetime, (Perspectives in Geometry and Relativity, vol. 42 (1966), Indiana Univ. Press), 413-425
[15] Vinogradov, A. M., An informal introduction to the geometry of jet spaces, Rend. Seminari Fac. Sci. Univ. Cagliari, 58, 301-333 (1988)
[16] Vaisman, I., Lectures on the Geometry of Poisson Manifolds (1994), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0852.58042
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