Some results in $\eta$-Ricci soliton and gradient $\rho$-Einstein soliton in a complete Riemannian manifold
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient $\rho$-Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient $\rho$-Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost $\eta$-Ricci soliton.
Abstract
The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient $\rho$-Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient $\rho$-Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost $\eta$-Ricci soliton.
참고문헌 (19)
- W. Ambrose, "A theorem of Myers", Duke Math. J, 24, pp.345~348 (1957)[DOI]
- C. Aquino, "Some Applications of the Hodge-de Rham Decomposition to Ricci Solitons", Results in Mathematics, 60(1-4) (2011), Springer Science and Business Media LLC[DOI]
- Abdênago Barros, "A note on rigidity of the almost Ricci soliton", Archiv der Mathematik, 100(5), pp.481~490 (2013), Springer Science and Business Media LLC[DOI]
- A. Barros, "Some characterizations for compact almost Ricci solitons", Proceedings of the American Mathematical Society, 140(3), pp.1033~1040 (2012), American Mathematical Society (AMS)[DOI]
- A. M. Blaga, "Almost-Ricci solitons in(LCS)n-manifolds", Bull. Belg. Math. Soc. Simon Stevin, 25(5), pp.641~653 (2018)[DOI]
- M. Brozos-Vazquez, "Half conformally at gradient Ricci almost solitons", Proc. A, 472(2189), pp.20160043~ (2016)[DOI]
- Esteban Calviño-Louzao, "Homogeneous Ricci almost solitons", Israel Journal of Mathematics, 220(2), pp.531~546 (2017), Springer Science and Business Media LLC[DOI]
- Giovanni Catino, "Generalized quasi-Einstein manifolds with harmonic Weyl tensor", Mathematische Zeitschrift, 271(3-4), pp.751~756 (2012), Springer Science and Business Media LLC[DOI]
- G. Catino, "The Ricci-Bourguignon flow", Pacic J. Math, 287(2), pp.337~370 (2017)[DOI]
- Giovanni Catino, "On the geometry of gradient Einstein-type manifolds", Pacific Journal of Mathematics, 286(1), pp.39~67 (2017), Mathematical Sciences Publishers[DOI]
- Giovanni Catino, "Gradient Einstein solitons", Nonlinear Analysis, 132, pp.66~94 (2016), Elsevier BV[DOI]
- J. T. Cho, "Ricci solitons and real hypersurfaces in a complex space form", Tohoku Math. J. (2), 61(2), pp.205~212 (2009)[DOI]
- R. Greene, "On the subharmonicity and plurisubharmonicity of geodesically convex functions", Indiana University Mathematics Journal, 22(7), pp.641~ (1973), Indiana University Mathematics Journal[DOI]
- R. S. Hamilton, "Three-manifolds with positive Ricci curvature", J. Dierential Geom, 17(2), pp.255~306 (1982)[DOI]
- G. Perelman, "The entropy formula for the Ricci ow and its geometric applications, arXiv:math", 0211159, 2002
- P. Petersen, "Riemannian Geometry" (2006), Springer
- S. Pigola, "Ricci almost solitons", Ann. Sc. Norm. Super. Pisa Cl. Sci., 10(4), pp.757~799 (2011)
- Ramesh Sharma, "Almost Ricci solitons and K-contact geometry", Monatshefte für Mathematik, 175(4), pp.621~628 (2014), Springer Science and Business Media LLC[DOI]
- K. Yano, "Integral Formulas in Riemannian Geometry, Pure and Applied Mathematics, No. 1" (1970), Marcel Dekker, Inc