SOME RESULTS ON CONCIRCULAR VECTOR FIELDS AND THEIR APPLICATIONS TO RICCI SOLITONS

Title & Authors
SOME RESULTS ON CONCIRCULAR VECTOR FIELDS AND THEIR APPLICATIONS TO RICCI SOLITONS
CHEN, BANG-YEN;

Abstract
A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies $\small{{\nabla}X^v={\mu}X}$ for any vector X tangent to M, where $\small{{\nabla}}$ is the Levi-Civita connection and $\small{{\mu}}$ is a non-trivial function on M. A smooth vector field $\small{{\xi}}$ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: $\small{\frac{1}{2}L_{\xi}g+Ric={\lambda}g}$, where $\small{L_{\xi}g}$ is the Lie-derivative of the metric tensor g with respect to $\small{{\xi}}$, Ric is the Ricci tensor of (M, g) and $\small{{\lambda}}$ is a constant. A Ricci soliton (M, g, $\small{{\xi}}$, $\small{{\lambda}}$) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field is a concircular vector field. In the first part of this paper we determine Riemannian manifolds which admit a concircular vector field. In the second part we classify Ricci solitons with concircular potential field. In the last part we prove some important properties of Ricci solitons on submanifolds of a Riemannian manifold equipped with a concircular vector field.
Keywords
concircular vector field;Ricci soliton;submanifolds;Einstein manifold;concircular potential field;concurrent vector field;concircular curvature tensor;
Language
English
Cited by
1.
Some Results About Concircular and Concurrent Vector Fields On Pseudo-Kaehler Manifolds, Journal of Physics: Conference Series, 2016, 766, 012034
2.
Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves, International Journal of Geometric Methods in Modern Physics, 2016, 13, 02, 1650015
3.
Rigidity of (m,ρ)-quasi Einstein manifolds, Mathematische Nachrichten, 2017
References
1.
A. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987.

2.
B.-Y. Chen, Geometry of Submanifolds, Marcel Dekker, New York, 1973.

3.
B.-Y. Chen, Pseudo-Riemannian Geometry, ${\delta}$-invariants and Applications, World Scientific, Hackensack, NJ, 2011.

4.
B.-Y. Chen, A simple characterization of generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 46 (2014), no. 12, Art. 1833, 5 pp.

5.
B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, 2nd Edition, World Scientific, Hackensack, NJ, 2015.

6.
B.-Y. Chen and S. Deshmukh, Geometry of compact shrinking Ricci solitons, Balkan J. Geom. Appl. 19 (2014), no. 1, 13-21.

7.
B.-Y. Chen and S. Deshmukh, Classification of Ricci solitons on Euclidean hypersurfaces, Internat. J. Math. 25 (2014), no. 11, 1450104, 22 pp.

8.
B.-Y. Chen and S. Deshmukh, Ricci solitons and concurrent vector fields, Balkan J. Geom. Appl. 20 (2015), no. 1, 14-25.

9.
B.-Y. Chen and K. Yano, On submanifolds of submanifolds of a Riemannian manifold, J. Math. Soc. Japan 23 (1971), no. 3, 548-554.

10.
J. T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J. (2) 61 (2009), no. 2, 205-212.

11.
J. T. Cho and M. Kimura, Ricci solitons of compact real hypersurfaces in Kahler manifolds, Math. Nachr. 284 (2011), no. 11-12, 1385-1393.

12.
J. T. Cho and M. Kimura, Ricci solitons on locally conformally flat hypersurfaces in space forms, J. Geom. Phys. 62 (2012), no. 8, 1882-1891.

13.
A. Fialkow, Conformals geodesics, Trans. Amer. Math. Soc. 45 (1939), no. 3, 443-473.

14.
R. S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7-136, Int. Press, Cambridge, MA, 1995.

15.
S. Hiepko, Eine innere Kennzeichnung der verzerrten Produkte, Math. Ann. 241 (1979), no. 3, 209-215.

16.
J. Morgan and G. Tian, Ricci Flow and the Poincare Conjecture, Clay Mathematics Monographs, 5, Cambridge, MA, 2014.

17.
G. Perelman, The Entropy Formula For The Ricci Flow And Its Geometric Applications, arXiv math/0211159.

18.
Ya. L. Sapiro, Geodesic fields of directions and projective path systems, Mat. Sb. N.S. 36 (78) (1955), 125-148.

19.
H. Takeno, Concircular scalar field in spherically symmetric space-times I, Tensor 20 (1967), no. 2, 167-176.

20.
K. Yano, Concircular geometry. I. Concircular transformations, Proc. Imp. Acad. Tokyo 16 (1940), 195-200.