Rigidity Results On Generalized m-Quasi Einstein Manifolds with Associated Affine Killing Vector Field.

Year 2023, Volume: 16 Issue: 1, 266 - 271, 30.04.2023

Rahul Poddar , Balasubramanian Subramanian , R. Sharma

https://doi.org/10.36890/iejg.1286128

Abstract

We study a non-trivial generalized $m$-quasi Einstein manifold $M$ with finite $m$ and associated divergence-free affine Killing vector field, and show that $M$ reduces to an $m$-quasi Einstein manifold. In addition, if $M$ is complete, then it splits as the product of a line and an $(n-1)$-dimensional negatively Einstein manifold. Finally, we show that the same result holds for a complete non-trivial $m$-quasi Einstein manifold $M$ with finite $m$ and associated affine Killing vector field.

Keywords

Generalized $m$-quasi Einstein manifold , affine Killing vector field , Einstein manifold , Ricci almost soliton

References

• [1] Barros, A. and Ribeiro, E.: Integral formulae on quasi-Einstein manifolds and applications, Glasg. Math. J. 54, 213-223 (2012).
• [2] Barros, A. and Ribeiro, E.: Characterizations and integral formulae for generalized m-quasi-Einstein metrics, Bull. Braz. Math. Soc., New Series 45, 325-341 (2014).
• [3] Case, J., Shu, Y. and Wei, G.: Rigidity of quasi-Einstein metrics, Diff. Geom. Appl. 29, 93-100 (2011).
• [4] Catino, G.: Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Zeits. 271, 751-756 (2012).
• [5] Ghosh, A. & Sharma, R.: Sasakian metric as a Ricci soliton and related results, Journal of Geometry and Physics 75, 1-6 (2014).
• [6] He, C., Petersen, P. Wylie, W.: On the classification of warped product Einstein metrics, ArXiv Preprint ArXiv: 1010.5488, (2010).
• [7] Pigola, S., Rigoli, M., Rimoldi, M. and Setti, A.G.: Ricci almost solitons, Ann.Della Scuola Norm. Sup. Di Pisa-Classe Di Sci. 10, 757-799 (2011).
• [8] Sharma, R.: Some results on almost Ricci solitons and geodesic vector fields, Beitr. Alg. Geom. 59, 289-294 (2018).
• [9] Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117, 251-275 (1965).
• [10] Yano, K.: Integral formulas in Riemannian geometry, Marcel Dekker, 1970.