Publication: Reduction of Homogeneous Pseudo-Kähler Structures by One-Dimensional Fibers
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2020-08-01
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Carmona Jiménez, J. L.
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We study the reduction procedure applied to pseudo-Kähler manifolds by a one dimensional Lie group acting by isometries and preserving the complex tensor. We endow the quotient manifold with an almost contact metric structure. We use this fact to connect pseudo-Kähler homogeneous structures with almost contact metric homogeneous structures. This relation will have consequences in the class of the almost contact manifold. Indeed, if we choose a pseudo-Kähler homogeneous structure of linear type, then the reduced, almost contact homogeneous structure is of linear type and the reduced manifold is of type C5⊕C6⊕C12 of Chinea-González classification.
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1. Ambrose, W.; Singer, I.M. On homogeneous Riemannian manifolds. Duke Math. J. 1958, 25, 647–669.
2. Kiricenko, V.F. On homogeneous Riemannian spaces with an invariant structure tensor. Sov. Math. Dokl. 1980, 21, 734–737.
3. Tricerri, F.; Tricerri, G.; Vanhecke, L. Homogeneous Structures on Riemannian Manifolds, 1st ed.; Lon. Math. Soc. Lecture Notes Series 83; Cambridge University Press: London, UK, 1983.
4. Abbena, E.; Garbiero, S. Almost hermitian homogeneous structures. Proc. Edinb. Math. Soc. 1988, 31, 375–395.
5. Calvaruso, G.; Castrillón López, M. Pseudo-Riemannian Homogeneous Structures, 1st ed.; Developments in Mathematics 59; Springer: New York, NY, USA, 2019.
6. Castrillón López, M.; Luján, I. Homogeneous structures of linear type on e-Kähler and e- quaternion Kähler manifolds Rev. Mat. Iberoam. 2017, 33, 139–168.
7. Castrillón López, M.; Luján, I. Reduction of homogeneous Riemannian structures. Proc. Edinh. Math. Soc. 2015, 58, 81–106.
8. Chinea, D.; González, C. A Classification of Almost Contact Metric Manifolds. Annali di Matematica Pura ed Applicata 1990, 156, 15–36.
9. Batat, W.; Gadea, P.; Oubiña, J.A. Homogeneous pseudo-Riemannian structures of linear type. J. Geom. Phys. 2011, 60, 745–764.
10. Carmona Jiménez, J.L.; Castrillón López, M. The Ambrose-Singer theorem for general homogeneous spaces with applications to symplectic geometry. arXiv 2020, unpublished. Available online: https://arxiv.org/abs/2001.06254 (accessed on 17 January 2020).
11. Luján. I. Reductive locally homogeneous pseudo-Riemannian manifolds and Ambrose–Singer connections. Diff. Geom. Appl. 2015, 41, 65–90.
12. Blair, D.E. Riemannian Geometry of Contact and Symplectic Manifolds, 1st ed.; Progress in Mathematics 203; Birkhäuser: Basel, Switzerland, 2002.
13. Marsden, J.E.; Ostrowski, J. Symmetries in motion: Geometric foundations of motion control.Nonlinear Sci. Today 1996, 1–21.
14. Palais, R. A Global Formulation of the Lie Theory of Transformation Groups; Mem. Amer. Math. Soc. 22; American Mathematical Society: Providence, RI, USA, 1957.
15. Meessen, P. Homogeneous Lorentzian spaces admitting a homogeneous structure of type T1 + T2. J. Geom. Phys. 2006, 56, 754–761.
16. Castrillón López, M.; Luján, I. Strongly degenerate homogeneous pseudo-Kähler structures of linear type and complex plane waves. J. Geom. Phys. 2013, 73, 1–19.