Some properties of tangent Dirac structures of higher order

P. M. Kouotchop Wamba; A. Ntyam; J. Wouafo Kamga

Archivum Mathematicum (2012)

  • Volume: 048, Issue: 3, page 233-241
  • ISSN: 0044-8753

Abstract

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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 .

How to cite

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Wamba, P. M. Kouotchop, Ntyam, A., and Kamga, J. Wouafo. "Some properties of tangent Dirac structures of higher order." Archivum Mathematicum 048.3 (2012): 233-241. <http://eudml.org/doc/246621>.

@article{Wamba2012,
abstract = {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 $L^\{r\}$.},
author = {Wamba, P. M. Kouotchop, Ntyam, A., Kamga, J. Wouafo},
journal = {Archivum Mathematicum},
keywords = {Dirac structure; prolongations of vector fields; prolongations of differential forms; Dirac structure of higher order; natural transformations; Dirac structure; prolongations of vector fields; prolongations of differential forms; Dirac structure of higher order; natural transformation},
language = {eng},
number = {3},
pages = {233-241},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Some properties of tangent Dirac structures of higher order},
url = {http://eudml.org/doc/246621},
volume = {048},
year = {2012},
}

TY - JOUR
AU - Wamba, P. M. Kouotchop
AU - Ntyam, A.
AU - Kamga, J. Wouafo
TI - Some properties of tangent Dirac structures of higher order
JO - Archivum Mathematicum
PY - 2012
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 048
IS - 3
SP - 233
EP - 241
AB - 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 $L^{r}$.
LA - eng
KW - Dirac structure; prolongations of vector fields; prolongations of differential forms; Dirac structure of higher order; natural transformations; Dirac structure; prolongations of vector fields; prolongations of differential forms; Dirac structure of higher order; natural transformation
UR - http://eudml.org/doc/246621
ER -

References

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  1. Cantrijn, F., Crampin, M., Sarlet, W., Saunders, D., The canonical isomorphism between T k T * and T * T k , C. R. Acad. Sci. Paris Sér. II 309 (1989), 1509–1514. (1989) MR1033091
  2. Courant, T., 10.1090/S0002-9947-1990-0998124-1, Trans. Amer. Math. Soc. 319 (2) (1990), 631–661. (1990) MR0998124DOI10.1090/S0002-9947-1990-0998124-1
  3. Courant, T., 10.1088/0305-4470/23/22/010, J. Phys. A: Math. Gen. 23 (22) (1990), 5153–5168. (1990) Zbl0715.58013MR1085863DOI10.1088/0305-4470/23/22/010
  4. Courant, T., 10.1088/0305-4470/27/13/026, J. Phys. A: Math. Gen. 27 (13) (1994), 4527–4536. (1994) Zbl0843.58044MR1294955DOI10.1088/0305-4470/27/13/026
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  6. Grabowski, J., Urbanski, P., 10.1088/0305-4470/28/23/024, J. Phys. A: Math. Gen. 28 (23) (1995), 6743–6777. (1995) MR1381143DOI10.1088/0305-4470/28/23/024
  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. (2005) Zbl1114.58010MR2268942
  8. Kolář, I., Michor, P., Slovák, J., Natural operations in differential geometry, Springer–Verlag, 1993. (1993) Zbl0782.53013MR1202431
  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. (2011) Zbl1240.53058MR2813543
  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. (1970) MR0279720
  12. Ntyam, A., Wouafo Kamga, J., 10.4064/ap82-3-4, Ann. Polon. Math. 82 (3) (2003), 233–240. (2003) MR2040808DOI10.4064/ap82-3-4
  13. Ntyam, A., Mba, A., On natural vector bundle morphisms T A ∘ ⨂ s q → ⨂ s q ∘ T A over i d T A , Ann. Polon. Math. 96 (3) (2009), 295–301. (2009) MR2534175
  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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