Ricci Solitons on Multiply Warped Product Manifolds

Year 2020, Volume: 13 Issue: 2, 152 - 159, 15.10.2020

Dilek Açıkgöz Kaya , Leyla Onat

https://doi.org/10.36890/iejg.777046

Cited By: 6

Abstract

In the last years, many researchers focused to Einstein metrics and some similar structures that are called Einstein type metrics. One of the most famous among these are Ricci solitons. On the besides this, there is a special interest on the relation between warped product
manifolds and Ricci solitons. So, in this paper, we study Ricci solitons with the structure of multiply warped product manifolds. We also study Ricci solitons equipped with the concurrent vector fields on multiply warped product manifolds.

Keywords

Ricci soliton, gradient Ricci soliton, warped product, multiply warped product

References

• 1. Besse, A.L.: Einstein Manifolds, Classics in Mathematics, Springer-Verlag, Berlin (2008).
• 2. Bishop, R.L. and O'Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145, 1-49 (1969).
• 3. Cao, H-D. and Zhou,D.: On complete gradient shrinking Ricci solitons. Journal of Differential Geometry 85, 175-186 (2010).
• 4. Chen, B-Y.: Some results on concircular vector fields and their applicationsto Ricci solitons. Bull. Korean Math. Soc. 52 No. 5, 1535-1547 (2015).
• 5. Dobarro, F., Ünal, B.: Curvature of multiply warped products. J. Geom. Phys. 55, 75-106 (2005).
• 6. Feitosa, F.E.S.,Freitas Filho A.A., Gomes, J.N.V.: On the construction of gradient Ricci soliton warped product. Nonlinear Analysis 161, 30(43) (2017).
• 7. Fernandez-Lopez, M., Garcia-Rio, E.: Rigidity of shrinking Ricci solitons. Mathematische Zeitschrift, 269, 461-466 (2011).
• 8. Günsen, S., Onat, L. and Açıkgöz Kaya, D.: The warped product manifold as gradient Ricci soliton and relation to its components. C. R. Acad. Bulg. Sci. 72(8), 1015-1023.
• 9. Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (2), 255-306 (1982).
• 10. Karaca, F. and Ozgur, C: Gradient Ricci Solitons on Multiply Warped Product Manifolds. Filomat 32:12, 4221-4228 (2018).
• 11. Mantica, C.A., Shenawy, S. and Unal, B.: Ricci Solitons on Singly Warped Product Manifolds and Applications. arXiv:1508.02794v2.
• 12. Munteanu, O.,Sesum, N.: On Gradient Ricci Solitons. Journal of Geometric Analysis 23, 539-561 (2013) .
• 13. Petersen, P., and Wylie, W,:Rigidity of gradient Ricci solitons. Pasific Journal of Mathematics, 241, 329-345 (2009) .
• 14. Petersen, P., and Wylie, W,: On gradient Ricci solitons with symmetry. Proc. Amer. Math. Soc. 137, 2085-2092 (2009).
• 15. Petersen, P., and Wylie, W,: On the classification of gradient Ricci solitons.Geom. Topol.14, 2277-2300 (2010).
• 16. Sousa, M.L., and Pina, R.: Gradient Ricci Solitons with Structure of Warped Product. Results Math 71, 825-840 (2017).
• 17. Ünal, B.: Multiply warped products. J. Geom. Phys. 34, 287{301 (2000). MR1762779 and DOI 10.1016/S0393-0440(99)00072-8