×

Pseudo-projective curvature tensor on warped product manifolds and its applications in space-times. (English) Zbl 1493.53053

Summary: In this paper we study the pseudo-projective curvature tensor on warped product manifolds. We obtain some significant results of the pseudo-projective curvature tensor on warped product manifolds in terms of its base and fiber manifolds. Moreover, we derive some interesting results which describe the geometry of base and fiber manifolds for a pseudo-projectively at warped product manifold. Lastly, we study the pseudo-projective curvature tensor on generalized Robertson-Walker space-times and standard static space-times.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] D. Allison, Energy conditions in standard static spacetimes, General Relativity and Gravitation 20 (1988), 115-122. · Zbl 0638.53059
[2] D. Allison, Geodesic completeness in static space-times, Geometriae Dedicata 26 (1988), 85-97. · Zbl 0644.53063
[3] D. E. Allison and B.Ünal, Geodesic structure of standard static space-times, Journal of Geometry and Physics 46 (2003), 193-200. · Zbl 1034.53040
[4] A. L. Besse, Einstein manifolds (Classics in Mathematics), Springer-Verlag, Berlin, (2008).
[5] N. Bhunia, S. Pahan and A. Bhattacharyya, Application of hyper-generalized quasi-Einstein spacetimes in general relativity, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences 91 (2021), 297-307. · Zbl 1490.53051
[6] R. L. Bishop and B. O’Neill, Geometry of slant submnaifolds, Transactions of the American Mathematical Society 145 (1969), 1-49. · Zbl 0191.52002
[7] F. Dobarro and B.Ünal, Curvature of multiply warped products, Journal of Geometry and Physics 55 (2005), 75-106. · Zbl 1089.53049
[8] Y. Dogru, Hypersurfaces satisfying some curvature conditions on pseudo projec-tive curvature tensor in the semi-Euclidean space, Mathematical Sciences and Applications E-Notes 2 (2014), 99-105. · Zbl 1439.53020
[9] J. L. Flores and M. Sánchez, Geodesic connectedness and conjugate points in GRW space-times, Journal of Geometry and Physics 36, (2000), 285-314. · Zbl 0979.53043
[10] J. P. Jaiswal and R. H. Ojha, On weakly pseudo-projectively symmetric mani-folds, Differential Geometry-Dynamical Systems 12 (2010), 83-94. · Zbl 1200.53029
[11] H. G. Nagaraja and G. Somashekhara, On pseudo projective curvature tensor in Sasakian manifolds, Int. J. Contemp. Math. Sciences 6 (2011), 1319-1328. · Zbl 1252.53058
[12] D. Narain, A. Prakash and B. Prasad, A pseudo projective curvature tensor on a Lorentzian para-Sasakian manifold, Analele Stiintifice ale Universitatii Al I Cuza din Iasi -Matematica 55 (2009), 275-284. · Zbl 1199.53040
[13] S. Nölker, Isometric immersions of warped products, Differential Geometry and its Applications 6 (1996), 1-30. · Zbl 0881.53052
[14] B. O’Neill, Semi-Riemannian geometry with applications to relativity, Pure and Applied Mathematics, Academic Press. Inc., New York, (1983), 336-341. · Zbl 0531.53051
[15] B. Prasad, A pseudo-projective curvature tensor on a Riemannian manifold, Bulletin of Calcutta Mathematical Society 94 (2002), 163-166. · Zbl 1028.53016
[16] M. Sánchez, On the geometry of generalized Robertson-Walker space-times: cur-vature and Killing fields, Journal of Geometry and Physics 31 (1999), 1-15. · Zbl 0964.53046
[17] M. Sánchez, On the geometry of generalized Robertson-Walker space-times: Geodesics, General Relativity and Gravitation 30 (1998), 915-932. · Zbl 1047.83509
[18] S. Shenawy and B.Ünal, 2-Killing vector fields on warped product manifolds, International Journal of Mathematics 26 (2015), 1550065(1)-1550065(17). · Zbl 1334.53077
[19] S. Shenawy and B.Ünal, The W2-curvature tensor on warped product manifolds and applications, International Journal of Geometric Methods in Modern Physics 13 (2016), 1650099(1)-1650099(14). · Zbl 1345.53076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.