Einstein warped product spaces on Lie groups

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DOI:

https://doi.org/10.56754/0719-0646.2403.0485

Abstract

We consider a compact Lie group with bi-invariant metric, coming from the Killing form. In this paper, we study Einstein warped product space, \(M = M_1 \times_{f_1} M_2\) for the cases, \((i)\) \(M_1\) is a Lie group \((ii)\) \(M_2\) is a Lie group and \((iii)\) both \(M_1\) and \(M_2\) are Lie groups. Moreover, we obtain the conditions for an Einstein warped product of Lie groups to become a simple product manifold. Then, we characterize the warping function for generalized Robertson-Walker spacetime, \((M = I \times_{f_1} G_2, - dt^2 + f_1^2 g_2)\) whose fiber \(G_2\), being semi-simple compact Lie group of \(\dim G_2>2\), having bi-invariant metric, coming from the Killing form.

Keywords

Einstein space , warped product , Lie group , bi-invariant metric , Killing form
  • Pages: 485–500
  • Date Published: 2022-12-21
  • Vol. 24 No. 3 (2022)

L. J. Alías, A. Romero and M. Sánchez, “Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes”, Gen. Relativity Gravitation, vol. 27, no. 1, pp. 71–84, 1995.

A. L. Besse, Einstein manifolds, Berlin: Springer-Verlag, 2007.

R. Bishop and B. O‘Neill, “Manifolds of negative curvature”, Trans. Am. Math. Soc., vol. 145, pp. 1–49, 1969.

N. Bokan, T. Å ukilović and S. Vukmirović, “Lorentz geometry of 4-dimensional nilpotent Lie groups”, Geom. Dedicata, vol. 177, pp. 83–102, 2015.

B. Y. Chen, “Twisted product CR-submanifolds in Kaehler manifolds”, Tamsui Oxf. J. Math. Sci., vol. 16, no. 2, pp. 105–121, 2000.

B. Y. Chen, “Geometry of warped product CR-submanifolds in Kaehler manifolds”, Monatsh. Math., vol. 133, no. 3, pp. 177–195, 2001.

B. Y. Chen, “Geometry of warped products as Riemannian submanifolds and related problems”, Soochow J. Math., vol. 28, no. 2, pp. 125–156, 2002.

Ì. Ciftci, “A generalization of Lancret‘s theorem”, J. Geom. Phys., vol. 59, no. 12, pp. 1597–1603, 2009.

F. Dobarro and E. Lami Dozo, “Scalar curvature and warped products of Riemann manifolds”, Trans. Amer. Math. Soc., vol. 303, no. 1, pp. 161–168, 1987.

P. E. Ehrlich, Y. T. Jung and S. B. Kim, “Constant scalar curvatures on warped product manifolds”, Tsukuba J. Math, vol. 20, no. 1, pp. 239–256, 1996.

M. Fernández-López, E. García-Río, D. N. Kupeli and B. Ìnal, “A curvature condition for a twisted product to be a warped product”, Manuscripta Math., vol. 106, no. 2, pp. 213–217, 2001.

J. Gallier and J. Quaintance, Differential geometry and Lie groups: a computational perspective, Geometry and Computing 12, Cham: Springer, 2020.

C. He, P. Petersen and W. Wylie, “On the classification of warped product Einstein metrics”, Comm. Anal. Geom., vol. 20, no. 2, pp. 271–311, 2012.

C. He, P. Petersen and W. Wylie, “Warped product Einstein metrics over spaces with constant scalar curvature”, Asian J. Math., vol. 18, no. 1, pp. 159–189, 2014.

C. He, P. Petersen and W. Wylie, “Warped product Einstein metrics on homogeneous spaces and homogeneous Ricci solitons”, J. Reine Angew. Math., vol. 707, pp. 217–245, 2015.

D. Kim and Y. Kim, “Compact Einstein warped product spaces with nonpositive scalar curvature”, Proc. Amer. Math. Soc., vol. 131, no. 8, pp. 2573–2576, 2003.

J. Lauret, “Homogeneous nilmanifolds of dimensions 3 and 4”, Geom. Dedicata, vol. 68, no. 2, pp. 145–155, 1997.

J. Lauret, “Degenerations of Lie algebras and geometry of Lie groups”, Differential Geom. Appl., vol. 18, no. 2, pp. 177–194, 2003.

J. Meléndez and M. Hernández, “A note on warped products”, J. Math. Anal. Appl., vol. 508, no. 2, pp. 161–168, 2022.

J. Milnor, “Curvatures of left invariant metrics on Lie groups”, Advances in Math., vol. 21, no. 3, pp. 293–329, 1976.

M. T. Mustafa, “A non-existence result for compact Einstein warped products”, J. Physics A., vol. 38, no. 47, pp. L791–L793, 2005.

O. Zeki Okuyucu, I. Gök, Y. Yayli and N. Ekmekci, “Slant helices in three dimensional Lie groups”, Appl. Math. Comput., vol. 221, pp. 672–683, 2013.

B. O‘Neill, Semi-Riemannian geometry with applications to relativity, New York: Academic Press, 1983.

S. Pahan, B. Pal and A. Bhattacharyya, “On Ricci flat warped products with a quarter- symmetric connection”, J. Geom., vol. 107, no. 3, pp. 627–634, 2016.

B. Pal and P. Kumar, “Compact Einstein multiply warped product space with nonpositive scalar curvature”, Int. J. Geom. Methods Mod. Phys., vol. 16, no. 10, 14 pages, 2019.

B. Pal and P. Kumar, “On Einstein warped product space with respect to semi symmetric metric connection”, Hacet. J. Math. Stat., vol. 50, no. 50, pp. 1477–1490, 2021.

P. Petersen, Riemannian geometry, Graduate Texts in Mathematics 171, New York: Springer, 2006.

R. Ponge, H. Reckziegel, “Twisted products in pseudo-Riemannian geometry”, Geom. Dedicata, vol. 48, no. 1, pp. 15–25, 1993.

J. Rahmani, “Métriques de Lorentz sur les groupes de Lie unimodulaires, de dimension trois”, J. Geom. Phys., vol. 9, no. 3, pp. 295–302, 1992.

M. Rimoldi, “A remark on Einstein warped products”, Pacific J. Math., vol. 252, no. 1, pp. 207–218, 2011.

M. Sánchez, “On the geometry of generalized Robertson-Walker spacetimes: curvature and Killing fields”, J. Geom. Phys., vol. 31, no. 1, pp. 1–15, 1999.

B. Ìnal, “Multiply warped products”, J. Geom. Phys., vol. 34, no. 3–4, pp. 287–301, 2000.

B. Ìnal, “Doubly warped products”, Differential Geom. Appl., vol. 15, no. 3, pp. 253–263, 2001.

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Published

2022-12-21

How to Cite

[1]
B. Pal, S. Kumar, and P. Kumar, “Einstein warped product spaces on Lie groups”, CUBO, vol. 24, no. 3, pp. 485–500, Dec. 2022.

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