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# Article

 Title: Some properties of tangent Dirac structures of higher order (English) Author: Wamba, P. M. Kouotchop Author: Ntyam, A. Author: Kamga, J. Wouafo Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 48 Issue: 3 Year: 2012 Pages: 233-241 Summary lang: English . Category: math . 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 $L^{r}$. (English) Keyword: Dirac structure Keyword: prolongations of vector fields Keyword: prolongations of differential forms Keyword: Dirac structure of higher order Keyword: natural transformations MSC: 53C15 MSC: 53C75 MSC: 53D05 idMR: MR2995874 DOI: 10.5817/AM2012-3-233 . Date available: 2012-10-03T15:11:37Z Last updated: 2013-09-19 Stable URL: http://hdl.handle.net/10338.dmlcz/142991 . Reference: [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 Reference: [2] Courant, T.: Dirac manifolds.Trans. Amer. Math. Soc. 319 (2) (1990), 631–661. MR 0998124, 10.1090/S0002-9947-1990-0998124-1 Reference: [3] Courant, T.: Tangent Dirac Structures.J. Phys. A: Math. Gen. 23 (22) (1990), 5153–5168. Zbl 0715.58013, MR 1085863, 10.1088/0305-4470/23/22/010 Reference: [4] Courant, T.: Tangent Lie Algebroids.J. Phys. A: Math. Gen. 27 (13) (1994), 4527–4536. Zbl 0843.58044, MR 1294955, 10.1088/0305-4470/27/13/026 Reference: [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. Zbl 0813.53010, MR 1295815 Reference: [6] Grabowski, J., Urbanski, P.: Tangent lifts of poisson and related structure.J. Phys. A: Math. Gen. 28 (23) (1995), 6743–6777. MR 1381143, 10.1088/0305-4470/28/23/024 Reference: [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. Zbl 1114.58010, MR 2268942 Reference: [8] Kolář, I., Michor, P., Slovák, J.: Natural operations in differential geometry.Springer–Verlag, 1993. Zbl 0782.53013, MR 1202431 Reference: [9] Kouotchop Wamba, P. M., Ntyam, A., Wouafo Kamga, J.: Tangent lift of higher order of multivector fields and applications.to appear. Reference: [10] Kouotchop Wamba, P. M., Ntyam, A., Wouafo Kamga, J.: Tangent Dirac structures of higher order.Arch. Math. (Brno) 47 (2011), 17–22. Zbl 1240.53058, MR 2813543 Reference: [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 Reference: [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. MR 2040808, 10.4064/ap82-3-4 Reference: [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 Reference: [14] Wouafo Kamga, J.: Global prolongation of geometric objets to some jet spaces.International Centre for Theoretical Physics, Trieste, Italy, November 1997. .

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