Induced polynomial structures on generalized geometry
Induced polynomial structures on generalized geometry
In this paper, we study different geometric structures that can be defined as section endomorphisms of the generalized tangent bundle TM := TM ⊕ T^∗M → M . This vector bundle admits some structures that arise canonically and other that can be induced from geometric structures defined on the manifold. We comment some well-known examples and present new structures, focusing on the polynomial structures that can be induced in the generalized tangent bundle.
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- [1] Blaga AM, Crasmareanu M. A class of almost tangent structures in generalized geometry. Balkan Journal of Geometry and its Applications 2014; 19 (2): 23-35.
- [2] Etayo F, Santamaría R. $(J^2 = ±1)$-metric manifolds. Publicationes Mathematicae Debrecen 2000; 57 (3-4): 435-444.
- [3] Etayo F, Santamaría R. Distinguished connections on $(J^2 = ±1)$-metric manifolds. Archivum Mathematicum 2016; 52 (3): 159-203. doi: 10.5817/AM2016-3-159
- [4] Fernández-Culma EA, Godoy Y, Salvai M. Generalized complex and paracomplex structures on product manifolds. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A Matematicas 2020; 114 (3): Paper No. 154, 16. doi: 10.1007/s13398-020-00887-3
- [5] Gualtieri M. Generalized complex geometry. Annals of Mathematics 2011; 174 (1): 75-123. doi: 10.4007/annals.2011.174.1.3
- [6] Hitchin N. Generalized Calabi-Yau manifolds. The Quarterly Journal of Mathematics 2003; 54 (3): 281-308. doi: 10.1093/qjmath/54.3.281
- [7] Hull C, Lindström U. The generalised complex geometry of (p, q) Hermitian geometries. Communications in Mathematical Physics 2020; 375 (1): 479-494. doi: 10.1007/s00220-019-03488-3
- [8] Ida C, Manea A. On the integrability of generalized almost para-Norden and para-Hermitian structures. Mediterranean Journal of Mathematics 2017; 14 (4): Paper No. 173, 21. doi: 10.1007/s00009-017-0975-x
- [9] Milnor JW, Stasheff JD. Characteristic classes. Princeton, NJ, USA: Princeton University Press, 1974.
- [10] Nannicini A. Calibrated complex structures on the generalized tangent bundle of a Riemannian manifold. Journal of Geometry and Physics 2006; 56 (6): 903-916. doi: 10.1016/j.geomphys.2005.05.006
- [11] Nannicini A. Almost complex structures on cotangent bundles and generalized geometry. Journal of Geometry and Physics 2010; 60 (11): 1781-1791. doi: 10.1016/j.geomphys.2010.06.004
- [12] Nannicini A. Special Kähler manifolds and generalized geometry. Differential Geometry and its Applications 2013; 31 (2): 230-238. doi: 10.1016/j.difgeo.2013.01.007
- [13] Nannicini A. Generalized geometry of Norden manifolds. Journal of Geometry and Physics 2016; 99: 244-255. doi: 10.1016/j.geomphys.2015.10.011
- [14] Nannicini A. Norden structures on cotangent bundles. Bollettino dell’Unione Matematica Italiana 2019; 12 (1-2): 165-175. doi: 10.1007/s40574-018-0173-1
- [15] Poor WA. Differential geometric structures. Mineola, NY, USA: McGraw-Hill, 1981.
- [16] Vaisman I. Geometry of big-tangent manifolds. Publicationes Mathematicae Debrecen 2015; 86 (1-2): 213-243. doi: 10.5486/PMD.2015.7085
- [17] Wade A. Dirac structures and paracomplex manifolds. Comptes Rendus Mathematique 2004; 338 (11): 889-894. doi: 10.1016/j.crma.2004.03.031