On Einstein warped product space with respect to semi symmetric metric connection

Abstract

In this paper, we study Einstein warped product space with respect to semi symmetric metric connection. During this study we establish some results on curvature, Ricci and scalar tensors with respect to semi symmetric metric connection and second order semi symmetric metric connection. In the last section, we investigate under what conditions, if $M$ is an Einstein warped space with nonpositive scalar curvature and compact base with respect to semi symmetric metric connection then $M$ is simply a Riemannian product space.

Keywords

• Einstein manifold
• semi symmetric metric connection
• Warped product space
• Ricci tensor
• Hessian tensor
• Ricci identity

References

1. [1] N.S. Agashe and M.R. Chafle, A semi-symmetric non-metric connection in a Rie- mannian manifold, Indian J. Pure Appl. Math. 23 (6), 399–409, 1992.
2. [2] R. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Am. Math. Soc. 145, 1–49, 1969.
3. [3] A.L. Besse, Einstein manifolds. Ergeb. Math. Grenzgeb., 3. Folge, Bd. 10. Berlin, Heidelberg, New York, Springer-Verlag. 1987.
4. [4] F. Dobarro and B, Ünal, Curvature of multiply warped products, J. Geom. Phys. 55 (1), 75–106, 2005.
5. [5] D. Dumitru, On Compact Einstein Warped Products, Annals Of Spiru Haret Univer- sity: Mathematics-Informatics Series, Bucharest, Romania, 2011.
6. [6] A. Friedmann and J.A. Schouten, Über die Geometrie der halbsymmetrischen Uber- tragungen, Math. Z. 21, 211–223, 1924.
7. [7] F.E.S. Feitosa, A.A.F. Filho and J.N.V. Gomes, On the construction of gradient Ricci soliton warped product, Nonlinear Analysis, 161, 30–43, 2017.
8. [8] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer-Verlag, Berlin, 1987.

9. [9] H.A. Hayden, Subspace of a space with torsion, Proc. Lond. Math. Soc. II Series, 34, 27–50, 1932.
10. [10] D.S. Kim, Compact Einstein Warped Product Spaces, Trends Math. 5 (2), 1–5, 2002.
11. [11] D.S. Kim and Y.H. Kim, On compact Einstein warped product spaces with nonpositive scalar curvature , Proc. Amer. Math. Soc. 131 (8), 2573–2576, 2003.
12. [12] M.T. Mustafa, A non-existence result for compact Einstein warped products, J. Phys. A: Math. Gen. 38, L791–L793, 2005.
13. [13] B. O’Neill, Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics 103, Academic Press, Inc., New York, 1983.
14. [14] S. Pahan, B. Pal and A. Bhattacharyya, Multiply warped product on quasi-Einstein manifold with a semi-symmetric metric connection, Analele Universitatii Oradea Fasc. Matematica XXIV (1), 171–183, 2017.
15. [15] Q. Qu and Y. Wang, Multiply warped products with a quarter-Symmetric connection, J. Math. Anal. Appl. 431 (2), 955–987, 2015.
16. [16] K. Yano, On semi-symmetric connection, Rev. Roumaine Math Pures App. 15, 1579– 1586, 1970.