Self similar solutions of the Yamabe flow and gradient Einstein-type manifolds in the Finsler geometry

Subject Areas : هندسه

Mohamad Yar Ahmadi ¹ (Department of Mathematics , Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz , , Ahvaz , Iran)
Neda Izadian ² (Department of Mathematics , Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz , , Ahvaz , Iran)
Sina Hedayatian ³ (Department of Mathematics,, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, , Ahvaz, Iran)

Received: 2022-05-15 Accepted : 2022-12-03 Published : 2024-01-18

Keywords: منیفلد انیشتین‌گونه, توپولوژی‌گونه متناهی, متر فینسلر, سالیتون یامابه, شار یامابه,

Abstract :

In the present work, the concepts of Finslerian Yamabe flow equation and Yamabe solitons are investigated. First of all, by using the local one-parameter group of diffeomorphism relevant to Yamabe soliton’s vector fields, we find a group of Finslerian metrics as solutions to the Finslerian Yamabe flow. In the other words, a Finslerian Yamabe soliton is a solution of the Finslerian Yamabe flow. These solutions are self-similar solutions of Finslerian Yamabe flow equation and have interesting geometric and physical properties. Furthermore, the notion of extended gradient Einstein-type manifolds is studied on Finsler spaces. Moreover, we show that by considering either the Ricci tensor is bounded from below and injectivity radius is bounded away from zero or the Ricci tensor is bounded from above, then the complete extended gradient Einstein-type Finslerian manifold is of finite topological type structure. Indeed, this manifold is homeomorghic to the interior of a compact manifold with boundary.

References:


[1] R. S. Hamilton. The Ricci flow on surfaces. In: Mathematics and general relativity (Santa Cruz, CA, 1986), Volume 71 of Contemp. Math. 237-262, Amer. Math. Soc. 1988

[2] B. Bidabad, M. Yar Ahmadi. On complete Yamabe solitons. Advances in Geometry. 1 18: 101-104 (2018)

[3] L. Ma, L. Cheng, A. Zhu. Extending Yamabe flow on complete Riemannian manifolds. Bull. Sci. Math. 136: 882-891 (2012)

[4] S. Deshmukh, B. Y. Chen. A note on Yamabe solitons. Balkan of Geometry and Its Applications. 23 1: 37-43 (2018)

[5] S. Y. Hsu. A note on compact gradient Yamabe solitons. J. Math. Anal. Appl. 388: 725-726 (2012)

[6] B. Bidabad, M. Yar Ahmadi.
On Quasi-Einstein Finsler spaces.
Bulletin of the Iranian Mathematical Society, 40 no. 4: 921-930 (2014)

[7] G. Catino. On the geometry of gradient Einstein-type manifolds. Pacific Journal of Mathematics 286. 1: 39-67 (2016)

[8] J. Milnor. A note on curvature and fundamental group. J. Differential Geometry, 2: 1-7 (1968)

[9] D. Gromoll, W. T. Meyer. Examples of complete manifolds with positive Ricci curvature. Journal of
Differential Geometry, 21 2: 195-211 (1985)

[10] F. Q. Fang, J. W. Man, Z. L. Zhang. Complete gradient shrinking Ricci solitons have finite topological type. Comptes Rendus Mathematique, Vol.
346: 653-656 (2008)

[11] M. Yar Ahmadi, S. Hedayatian. Finite topological type of complete Finsler gradient shrinking Ricci solitons. Turk J Math. 45: 2419-2426 (2021)

[12] M. Yar Ahmadi. On the gradient Finsler Yamabe solitons. AUT Journal of Mathematics an computing. 1(2): 229-233 (2020)

[13] B. Bidabad, M. Yar Ahmadi. On complete Finslerian Yamabe solitons. Differential Geometry and it’s Application. 66: 52-60 (2019)

[14] B. Bidabad, M. Yar Ahmadi. On compact Ricci solitons in Finsler geometry. Comptes Rendus Mathematique. 353(11): 1023-1027 (2015)

[15] B. Bidabad, M. Yar Ahmadi. Complete Ricci solitons on Finsler manifolds. Science China Mathematics. 61 (10): 1825-1832 (2018)

[16] Z. Shen. Riemann-Finsler geometry with applications to information geometry. Chinese. Annals. Math. 27: 73-94 (2006)

[17] H. Akbar-Zadeh. Initiation To Global Finslerian Geometry. Volume 68. Elsevier Science, 2006

[18] B. Chow, P. Lu, L. Ni. Hamiltons Ricci Flow. Graduate Studies in Mathematics, Vol. 77. Providence, RI, USA: Science Press, 2006.

[19] W. Wylie, Complete shrinking Ricci solitons have finite fundamental group, Proceedings of the AMS 136 5: 1803-1806 (2008)
[20] Z. Shen. On complete manifolds of nonnegative kth-Ricci curvature. Transactions of the American Mathematical Society, 338 1: 289--310 (1993)