Generalized Ricci Solitons on Twisted Products

Abstract

An h-almost Ricci soliton is a generalization of the Ricci soliton. In this paper we study Ricci solitons and h-almost Ricci solitons on twisted (and warped) product manifolds. First, we obtain some results about Ricci solitons on twisted products. Then we generalize them to h- almost Ricci solitons.

Keywords

• Ricci soliton,h-almost Ricci soliton,concurrent vector field,twisted product,warped product,Robertson-Walker space-time

References

1. [1] A.L. Besse, Einstein Manifolds, Spring-Verlag, Berlin, (1987).
2. [2] H.D. Cao, Geometry of Ricci solitons, Chin. Ann. Math. Ser. B, 27(2), (2006), 121-142.
3. [3] H.D. Cao, Recent progress on Ricci solitons, arXiv:0908.2006v1.
4. [4] B.Y. Chen, Pseudo-Riemannian geometry, d-invariants and applications,World Scientific Publishing Co. Pte. Ltd,Usa, (2011).
5. [5] B.Y. Chen, SH. Deshmukh, Ricci solitons and concurrent vector fields, arXiv:1407.2790.
6. [6] S. Deshmukh, F.R. Al-Solamy, Conformal vector fields on a Riemannian manifold, Balkan Journal of Geometryand Its Applications, 19(2), (2014), 86-93.
7. [7] J.N. Gomes, Q. Wang, C. Xia, On the h-almost Ricci soliton. arXiv:1411.6416v2.
8. [8] R.S. Hamilton, Three manifolds with positive Ricci curvature, J. Diff. Geom., 17, (1982),255-306.

9. [9] R.S. Hamilton, The Ricci flow on surfaces, Contemporary Mathematics, 71, (1988), 237-261.
10. [10] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press Limited, London,(1983).
11. [11] S. Pigola, M. Rigoli, M. Rimoldi, A.G. Setti, Ricci almost solitons, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10,(2011), 757-799.
12. [12] M. Sanchez, Lorentzian manifolds admitting a Killing vector field, Nonlinear Analysis, 30, (1997), 643-654.
13. [13] S. Shenawy, Ricci solitons on warped product manifolds, arXiv:1508.02794.
14. [14] S. Shenawy B. Unal, 2-Killing vector fields on warped product manifolds, Int. J. Math, 26, (2015), [17 pages].