Geodesic
Differential geometry
Generalized contact bundles
[Sur le fibrés de contact généralisés]

Présenté par : Charles-Michel Marle

Vitagliano, Luca1 ; Wade, Aïssa2
1 DipMat, Università degli Studi di Salerno & Istituto Nazionale di Fisica Nucleare, GC Salerno, via Giovanni Paolo II n
2 Department of Mathematics, Penn State University, University Park, State College, PA 16802, USA
Comptes Rendus. Mathématique, Tome 354 (2016) no. 3, pp. 313-317.

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In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of the integrability condition for generalized contact structures; (3) in light of new results on multiplicative forms and Spencer operators [8], it allows a simple interpretation of the defining equations of a generalized contact structure in terms of Lie algebroids and Lie groupoids.

Dans cette Note, nous proposons une approche des structures de contact généralisées reposant sur les fibrés vectoriels de rang 1. Cette nouvelle approche possède trois principaux avantages : (1) elle inclut toutes les autres approches connues à ce jour ; (2) elle éclaircit la signification géométrique de la condition d'intégrabilité des structures de contact généralisées ; (3) au vu de résultats récents obtenus sur les formes multiplicatives et les opérateurs de Spencer [8], elle permet une interprétation simple des équations définissant une structure généralisée de contact en termes d'algébroïdes et de groupoïdes de Lie.

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DOI : 10.1016/j.crma.2015.12.009

Vitagliano, Luca 1 ; Wade, Aïssa 2

1 DipMat, Università degli Studi di Salerno & Istituto Nazionale di Fisica Nucleare, GC Salerno, via Giovanni Paolo II n
2 Department of Mathematics, Penn State University, University Park, State College, PA 16802, USA 
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Vitagliano, Luca; Wade, Aïssa. Generalized contact bundles. Comptes Rendus. Mathématique, Tome 354 (2016) no. 3, pp. 313-317. doi : 10.1016/j.crma.2015.12.009. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2015.12.009/

[1] Aldi, M.; Grandini, D. Generalized contact geometry and T-duality, J. Geom. Phys., Volume 92 (2015), pp. 78-93 | arXiv

[2] Bruzzo, U.; Rubtsov, V.N. Cohomology of skew-holomorphic Lie algebroids, Theor. Math. Phys., Volume 165 (2010), pp. 1598-1609 | arXiv

[3] Bursztyn, H.; Cabrera, A. Multiplicative forms at the infinitesimal level, Math. Ann., Volume 353 (2012), pp. 663-705 | arXiv

[4] Caseiro, R.; De Nicola, A.; Nunes da Costa, J.M. Jacobi quasi-Nijenhuis algebroids and Courant–Jacobi algebroid morphisms, J. Geom. Phys., Volume 60 (2010), pp. 951-961 | arXiv

[5] Chen, Z.; Liu, Z.-J. Omni-Lie algebroids, J. Geom. Phys., Volume 60 (2010), pp. 799-808 | arXiv

[6] Crainic, M. Generalized complex structures and Lie brackets, Bull. Braz. Math. Soc., Volume 42 (2011), pp. 559-578 | arXiv

[7] Crainic, M.; Salazar, M.A. Jacobi structures and Spencer operators, J. Math. Pures Appl., Volume 103 (2015), pp. 504-521 | arXiv

[8] Crainic, M.; Salazar, M.A.; Struchiner, I. Multiplicative forms and Spencer operators, Math. Z., Volume 279 (2015), pp. 939-979 | arXiv

[9] Grabowski, J. Graded contact manifolds and contact Courant algebroids, J. Geom. Phys., Volume 68 (2013), pp. 27-58 | arXiv

[10] Gualtieri, M. Generalized complex geometry, Ann. Math., Volume 174 (2011), pp. 75-123 | arXiv

[11] Hitchin, N. Generalized Calabi–Yau manifolds, Quart. J. Math., Volume 54 (2003), pp. 281-308 | arXiv

[12] Iglesias-Ponte, D.; Wade, A. Contact manifolds and generalized complex structures, J. Geom. Phys., Volume 53 (2005), pp. 249-258 | arXiv

[13] Kerbrat, Y.; Souici-Benhammadi, Z. Variétés de Jacobi et groupoïdes de contact, C. R. Acad. Sci. Paris, Ser. I, Volume 317 (1993), pp. 81-86

[14] Kosmann-Schwarzbach, Y.; Mackenzie, K.C.H. Differential operators and actions of Lie algebroids, Manchester, 2001 (Contemp. Math.), Volume vol. 315, Amer. Math. Soc., Providence, RI, USA (2002), pp. 213-233 | arXiv

[15] Lê, H.V.; Oh, Y.-G.; Tortorella, A.G.; Vitagliano, L. Deformations of coisotropic submanifolds in abstract Jacobi manifolds | arXiv

[16] Marle, C.M. On Jacobi manifolds and Jacobi bundles, Berkeley, CA, 1989 (Math. Sci. Res. Inst. Publ.), Volume vol. 20, Springer, New York (1991), pp. 227-246

[17] Poon, Y.S.; Wade, A. Generalized contact structures, J. Lond. Math. Soc., Volume 83 (2011) no. 2, pp. 333-352 | arXiv

[18] Sekiya, K. Generalized almost contact structures and generalized Sasakian structures, Osaka J. Math., Volume 52 (2015), pp. 43-59 | arXiv

[19] Sheng, Y. Jacobi quasi-Nijenhuis algebroids, Rep. Math. Phys., Volume 65 (2010), pp. 271-287 | arXiv

[20] Stiénon, M.; Xu, P. Poisson quasi-Nijenhuis manifolds, Commun. Contemp. Math., Volume 270 (2007), pp. 709-725 | arXiv

[21] Vaisman, I. Generalized CRF-structures, Geom. Dedic., Volume 133 (2008), pp. 129-154 | arXiv

[22] Vitagliano, L. Dirac–Jacobi bundles | arXiv

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