December  2020, 12(4): 553-562. doi: 10.3934/jgm.2020017

A family of multiply warped product semi-Riemannian Einstein metrics

1. 

Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India

* Corresponding author: Buddhadev Pal

Received  January 2020 Revised  June 2020 Published  December 2020 Early access  July 2020

Fund Project: The Second author is supported by UGC JRF of India, Ref. No: 1269/(SC)(CSIR-UGC NET DEC. 2016)

In this paper, we characterize multiply warped product semi -Riemannian manifolds when the base is conformal to an $ n $-dimensional pseudo-Euclidean space. We prove some conditions on warped product semi- Riemannian manifolds to be an Einstein manifold which is invariant under the action of an $ (n-1) $-dimensional translation group. After that we apply this result for the case of Ricci-flat multiply warped product space when the fibers are Ricci-flat. We also discuss the existence of infinitely many Ricci-flat multiply warped product spaces under the same action with null like vector.

Citation: Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017
References:
[1]

J. K. Beem and P. Ehrlich, Global Lorentzian Geometry, 1$^{st}$ edition, Markel-Deccer, New York, 1981.

[2]

J. K. Beem, P. Ehrlich and K. Easley, Global Lorentzian Geometry, 2$^{nd}$ edition, Markel-Deccer, New York, 1996.

[3]

A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb., 3, Berlin, Heidelberg, New York, Springer Verlag, 1987. doi: 10.1007/978-3-540-74311-8.

[4]

R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49.  doi: 10.1090/S0002-9947-1969-0251664-4.

[5]

P. G. C. Bonfim and R. Pina, Quasi-Einstein manifolds with structure of warped product, arXiv: 1905.02006. doi: 128.84.4.27/1905.02006.

[6]

M. Brózos-Vázquez, E. García-Río and R. Vázquez-Lorenzo, Some remarks on locally conformally flat static space-times, Journal of Mathematical Physics, 46 (2005), 022501. doi: 10.1063/1.1832755.

[7]

M. Brozos-VázquezE. Garcia-Rio and R. Vázquez-Lorenzo, Complete locally conformally flat manifolds of negative curvature, Pac. J. Math., 226 (2006), 201-219.  doi: 10.2140/pjm.2006.226.201.

[8]

M. Brozos-Vázquez, E. Garcia-Rio and R. Vázquez-Lorenzo, Warped product metrics and locally conformally flat structures, Matemática Contemporânea, in: SBM, 28 (2000), 91–110.

[9]

Q. Chen and C. He, On Bach flat warped product Einstein manifolds, Pac. J. Math., 265 (2013), 313-326.  doi: 10.2140/pjm.2013.265.313.

[10]

J. Choi, Multiply warped products with nonsmooth metrics, Journal of Mathematical Physics, 41 (2000), 8163. doi: 10.1063/1.1287432.

[11]

F. Dobarro and B. Unal, $\ddot{U}nal$, Curvature of multiply warped products, Journal of Geometry and Physics, 55 (2005), 75-106.  doi: 10.1016/j.geomphys.2004.12.001.

[12]

D. Dumitru, On multiply Einstein warped products, Scientific Annals of the Alexandru Loan Cuza University of Laşi (New Series). Mathematics, Tomul LXII, 2016, F.1.

[13]

F. Gholami, F. Darabi and A. Haji-Badali, Multiply-warped product metrices and reduction of Einstein equations, International Journal of Geometric Methods in Modern Physics, 14 (2017), 1750021. doi: 10.1142/S0219887817500219.

[14]

C. HeP. Petersen and W. Wylie, On the classification of warped product Einstein metrics, Comm. Anal. Geom., 20 (2012), 271-312.  doi: 10.4310/CAG.2012.v20.n2.a3.

[15]

B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.

[16]

S. PahanB. Pal and A. Bhattacharyya, On Ricci flat warped products with a quarter-symmetric connection, J. Geom., 107 (2016), 627-634.  doi: 10.1007/s00022-015-0301-3.

[17]

S. Pahan, B. Pal and A. Bhattacharyya, On Einstein warped products with a quarter-symmetric connection, International Journal of Geometric Methods in Modern Physics, 14 (2017), 1750050. doi: 10.1142/S0219887817500505.

[18]

M. L. D. Sousa and R. Pina, A family of warped product semi-Riemannian Einstein metrics, Differential Geometry and its Applications, 50 (2017), 105-115.  doi: 10.1016/j.difgeo.2016.11.004.

[19]

B. Unal, Multiply warped products, J. Geom. Phys., 34 (2000), 287-301.  doi: 10.1016/S0393-0440(99)00072-8.

show all references

References:
[1]

J. K. Beem and P. Ehrlich, Global Lorentzian Geometry, 1$^{st}$ edition, Markel-Deccer, New York, 1981.

[2]

J. K. Beem, P. Ehrlich and K. Easley, Global Lorentzian Geometry, 2$^{nd}$ edition, Markel-Deccer, New York, 1996.

[3]

A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb., 3, Berlin, Heidelberg, New York, Springer Verlag, 1987. doi: 10.1007/978-3-540-74311-8.

[4]

R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49.  doi: 10.1090/S0002-9947-1969-0251664-4.

[5]

P. G. C. Bonfim and R. Pina, Quasi-Einstein manifolds with structure of warped product, arXiv: 1905.02006. doi: 128.84.4.27/1905.02006.

[6]

M. Brózos-Vázquez, E. García-Río and R. Vázquez-Lorenzo, Some remarks on locally conformally flat static space-times, Journal of Mathematical Physics, 46 (2005), 022501. doi: 10.1063/1.1832755.

[7]

M. Brozos-VázquezE. Garcia-Rio and R. Vázquez-Lorenzo, Complete locally conformally flat manifolds of negative curvature, Pac. J. Math., 226 (2006), 201-219.  doi: 10.2140/pjm.2006.226.201.

[8]

M. Brozos-Vázquez, E. Garcia-Rio and R. Vázquez-Lorenzo, Warped product metrics and locally conformally flat structures, Matemática Contemporânea, in: SBM, 28 (2000), 91–110.

[9]

Q. Chen and C. He, On Bach flat warped product Einstein manifolds, Pac. J. Math., 265 (2013), 313-326.  doi: 10.2140/pjm.2013.265.313.

[10]

J. Choi, Multiply warped products with nonsmooth metrics, Journal of Mathematical Physics, 41 (2000), 8163. doi: 10.1063/1.1287432.

[11]

F. Dobarro and B. Unal, $\ddot{U}nal$, Curvature of multiply warped products, Journal of Geometry and Physics, 55 (2005), 75-106.  doi: 10.1016/j.geomphys.2004.12.001.

[12]

D. Dumitru, On multiply Einstein warped products, Scientific Annals of the Alexandru Loan Cuza University of Laşi (New Series). Mathematics, Tomul LXII, 2016, F.1.

[13]

F. Gholami, F. Darabi and A. Haji-Badali, Multiply-warped product metrices and reduction of Einstein equations, International Journal of Geometric Methods in Modern Physics, 14 (2017), 1750021. doi: 10.1142/S0219887817500219.

[14]

C. HeP. Petersen and W. Wylie, On the classification of warped product Einstein metrics, Comm. Anal. Geom., 20 (2012), 271-312.  doi: 10.4310/CAG.2012.v20.n2.a3.

[15]

B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.

[16]

S. PahanB. Pal and A. Bhattacharyya, On Ricci flat warped products with a quarter-symmetric connection, J. Geom., 107 (2016), 627-634.  doi: 10.1007/s00022-015-0301-3.

[17]

S. Pahan, B. Pal and A. Bhattacharyya, On Einstein warped products with a quarter-symmetric connection, International Journal of Geometric Methods in Modern Physics, 14 (2017), 1750050. doi: 10.1142/S0219887817500505.

[18]

M. L. D. Sousa and R. Pina, A family of warped product semi-Riemannian Einstein metrics, Differential Geometry and its Applications, 50 (2017), 105-115.  doi: 10.1016/j.difgeo.2016.11.004.

[19]

B. Unal, Multiply warped products, J. Geom. Phys., 34 (2000), 287-301.  doi: 10.1016/S0393-0440(99)00072-8.

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