Geometry of Ricci solitons admitting a new geometric vector field

Document Type : Original Article

Authors

Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran

Abstract

In the present paper, we introduce a new geometric vector field (it will be called semi-Killing field) on semi-Riemannaian manifolds. A complete classification of semi-Killing vector fields on 3-dimensional Walker manifolds will be derived. Then, we study Ricci solitons admitting this new vector field (called semi-Killing vector field) as their potential. In Riemannain setting, we prove that Ricci solitons with semi-Killing potential vector field are Einstein. Our results show that such Lorentzian solitons have constant scalar curvature. Finally, application of this new structure in theoretical physics has been investigated.

Keywords

Main Subjects

References

[1] H.-D. Cao, Geometry of Ricci solitons, Chinese Ann. Math. Ser. B, 27 (2006), pp. 121–142.
[2] M. Chaichi, E. Garc´ıa-R´ıo, and M. E. Vazquez-Abal ´ , Three-dimensional Lorentz manifolds admitting a parallel null vector field, J. Phys. A, 38 (2005), pp. 841–850.
[3] B.-Y. Chen and S. Deshmukh, Geometry of compact shrinking Ricci solitons, Balkan J. Geom. Appl., 19 (2014), pp. 13–21.
[4] B. Chow and D. Knopf, The Ricci flow: an introduction, vol. 110 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2004.
[5] B. Chow, P. Lu, and L. Ni, Hamilton’s Ricci flow, vol. 77 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006.
[6] A. Derdzinski, A Myers-type theorem and compact Ricci solitons, Proc. Amer. Math. Soc., 134 (2006), pp. 3645–3648.
[7] A. Derdzinski ´ , Ricci solitons, Wiad. Mat., 48 (2012), pp. 1–32.
[8] S. Deshmukh, Jacobi-type vector fields on Ricci solitons, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55(103) (2012), pp. 41–50.
[9] S. Deshmukh, H. Alodan, and H. Al-Sodais, A note on Ricci solitons, Balkan J. Geom. Appl., 16 (2011), pp. 48–55.
[10] M. Fernandez-L ´ opez and E. Garc ´ ´ıa-R´ıo, A remark on compact Ricci solitons, Math. Ann., 340 (2008), pp. 893–896.
[11] A. Naber, Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math., 645 (2010), pp. 125–153.
[12] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002. arXiv.
[13] A. G. Walker, Canonical form for a Riemannian space with a parallel field of null planes, Quart. J. Math. Oxford Ser. (2), 1 (1950), pp. 69–79.
AUT Journal of Mathematics and Computing
Volume 6, Issue 4
2025
Pages 361-370

Files

Share

How to cite

Statistics