Previous |  Up |  Next

# Article

Full entry | PDF   (0.4 MB)
Keywords:
Dirac structure; prolongations of vector fields; prolongations of differential forms; Dirac structure of higher order; natural transformations
Summary:
Let $M$ be a smooth manifold. The tangent lift of Dirac structure on $M$ was originally studied by T. Courant in [3]. The tangent lift of higher order of Dirac structure $L$ on $M$ has been studied in [10], where tangent Dirac structure of higher order are described locally. In this paper we give an intrinsic construction of tangent Dirac structure of higher order denoted by $L^{r}$ and we study some properties of this Dirac structure. In particular, we study the Lie algebroid and the presymplectic foliation induced by $L^{r}$.
References:
[1] Cantrijn, F., Crampin, M., Sarlet, W., Saunders, D.: The canonical isomorphism between $T^{k}T^{\ast }$ and $T^{\ast }T^{k}$. C. R. Acad. Sci. Paris Sér. II 309 (1989), 1509–1514. MR 1033091
[2] Courant, T.: Dirac manifolds. Trans. Amer. Math. Soc. 319 (2) (1990), 631–661. DOI 10.1090/S0002-9947-1990-0998124-1 | MR 0998124
[3] Courant, T.: Tangent Dirac Structures. J. Phys. A: Math. Gen. 23 (22) (1990), 5153–5168. DOI 10.1088/0305-4470/23/22/010 | MR 1085863 | Zbl 0715.58013
[4] Courant, T.: Tangent Lie Algebroids. J. Phys. A: Math. Gen. 27 (13) (1994), 4527–4536. DOI 10.1088/0305-4470/27/13/026 | MR 1294955 | Zbl 0843.58044
[5] Gancarzewicz, J., Mikulski, W., Pogoda, Z.: Lifts of some tensor fields and connections to product preserving functors. Nagoya Math. J. 135 (1994), 1–41. MR 1295815 | Zbl 0813.53010
[6] Grabowski, J., Urbanski, P.: Tangent lifts of poisson and related structure. J. Phys. A: Math. Gen. 28 (23) (1995), 6743–6777. DOI 10.1088/0305-4470/28/23/024 | MR 1381143
[7] Kolář, I.: Functorial prolongations of Lie algebroids. Proceedings of the 9th International Conference on Differential Geometry and its Applications, DGA 2004, Prague, Czech Republic, 2005, pp. 301–309. MR 2268942 | Zbl 1114.58010
[8] Kolář, I., Michor, P., Slovák, J.: Natural operations in differential geometry. Springer–Verlag, 1993. MR 1202431 | Zbl 0782.53013
[9] Kouotchop Wamba, P. M., Ntyam, A., Wouafo Kamga, J.: Tangent lift of higher order of multivector fields and applications. to appear.
[10] Kouotchop Wamba, P. M., Ntyam, A., Wouafo Kamga, J.: Tangent Dirac structures of higher order. Arch. Math. (Brno) 47 (2011), 17–22. MR 2813543 | Zbl 1240.53058
[11] Morimoto, A.: Lifting of some type of tensors fields and connections to tangent bundles of $p^{r}$-velocities. Nagoya Math. J. 40 (1970), 13–31. MR 0279720
[12] Ntyam, A., Wouafo Kamga, J.: New versions of curvatures and torsion formulas of complete lifting of a linear connection to Weil bundles. Ann. Polon. Math. 82 (3) (2003), 233–240. DOI 10.4064/ap82-3-4 | MR 2040808
[13] Ntyam, A., Mba, A.: On natural vector bundle morphisms $T^{A}\circ \bigotimes ^{q}_{s}\rightarrow \bigotimes ^{q}_{s}\circ T^{A}$ over $id_{T^{A}}$. Ann. Polon. Math. 96 (3) (2009), 295–301. MR 2534175
[14] Wouafo Kamga, J.: Global prolongation of geometric objets to some jet spaces. International Centre for Theoretical Physics, Trieste, Italy, November 1997.

Partner of