Abstract
We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields \(h:=\frac{1}{2}\pounds _\xi \varphi \) and \(\ell := R(\cdot ,\xi )\xi \), emphasizing analogies and differences with respect to the contact metric case. Certain identities involving \(\xi \)-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost CR structure \((\mathcal {H}(M), J, \theta )\) corresponding to almost contact pseudo-metric manifold M to be CR manifold. Finally, we prove that a contact pseudo-metric manifold \((M, \varphi ,\xi ,\eta ,g)\) is Sasakian pseudo-metric if and only if the corresponding nondegenerate almost CR structure \((\mathcal {H}(M), J)\) is integrable and J is parallel along \(\xi \) with respect to the Bott partial connection.
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References
Alegre, P.: Semi-invariant submanifolds of Lorentzian Sasakian manifolds. Demonstr. Math. 44(2), 391–406 (2011)
Alegre, P.: Slant submanifolds of Lorentzian Saakian and Para Sasakian manifolds. Taiwan J. Math. 17(3), 897–910 (2013)
Alegre, P., Carriazo, A.: Semi-Riemannian generalized Sasakian-space-forms. Bull. Malays. Math. Sci. Soc. (2015). https://doi.org/10.1007/s40840-015-0215-0
Bejancu, A., Duggal, K.L.: Real hypersurfaces of indefinite Kaehler manifolds. Int. J. Math. Math. Sci. 16, 545–556 (1993)
Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203. Birkhäuser, Boston (2010)
Calvaruso, G.: Contact Lorentzian manifolds. Differ. Geom. Appl. 29, 541–551 (2011)
Calvaruso, G., Perrone, D.: Contact pseudo-metric manifolds. Differ. Geom. Appl. 28(2), 615–634 (2010)
Calvaruso, G., Perrone, D.: Erratum to: Contact pseudo-metric manifolds [Differential Geom. Appl. 28 (2010), 615–634]. Differ. Geom. Appl. 31(6), 836–837 (2013)
Calvaruso, G., Perrone, D.: H-contact semi-Riemannian manifolds. J. Geom. Phys. 71, 11–21 (2013)
Carriazo, A., Pérez-García, M.J.: Slant submanifolds in neutral almost contact pseudo-metric manifolds. Differ. Geom. Appl. 54, 71–80 (2017)
Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on \(CR\) Manifolds. Progress in Mathematics, vol. 246. Birkhäuser, Basel (2007)
Duggal, K.L.: Space time manifolds and contact structures. Int. J. Math. Math. Sci. 13, 545–554 (1990)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)
Perrone, A.: Some results on almost paracontact metric manifolds. Mediterr. J. Math. 13, 3311–3326 (2016)
Perrone, D.: Curvature of K-contact semi-Riemannian manifolds. Can. Math. Bull. 57(2), 401–412 (2014)
Perrone, D.: Contact pseudo-metric manifolds of constant curvature and CR geometry. Results Math. 66, 213–225 (2014)
Perrone, D.: Remarks on Levi harmonicity of contact semi-Riemannian manifolds. J. Korean Math. Soc. 51(5), 881–895 (2014)
Perrone, D.: On the standard nondegenerate almost CR structure of tangent hyperquadric bundles. Geom. Dedic. 185, 15–33 (2016)
Perrone, D.: Contact semi-Riemannian structures in CR geometry: some aspects. Axioms 8(1), 6 (2019)
Rovenskii, V.: Foliation on Riemannian Manifolds and Submanifolds. Birkhäuser, Boston (1998)
Sharma, R.: Notes on contact metric manifolds. Ulam Q. 3(1), 27–33 (1995)
Takahashi, T.: Sasakian manifold with pseudo-Riemannian metrics. Tôhoku Math. J. 21, 271–290 (1969)
Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc. 314, 349–379 (1989)
Wang, Y., Liu, X.: Almost Kenmotsu pseudo-metric manifolds. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) (2014). https://doi.org/10.2478/aicu-2014-0030
Welyczko, J.: Para-CR structures on almost paracontact metric manifolds. J. Appl. Anal. 20(2), 105–117 (2014)
Acknowledgements
The authors would like to thank the reviewer for careful and thorough reading of this manuscript and thankful for helpful suggestions towards the improvement of this paper. The second author (D.M.N.) is grateful to University Grants Commission, New Delhi (Ref. No.: 20/12/2015(ii)EU-V) for financial support in the form of Junior Research Fellowship.
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Venkatesha, V., Naik, D.M. & Tripathi, M.M. Certain results on almost contact pseudo-metric manifolds. J. Geom. 110, 41 (2019). https://doi.org/10.1007/s00022-019-0498-7
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DOI: https://doi.org/10.1007/s00022-019-0498-7