Gradient Ricci solitons with a conformal vector field. (English) Zbl 1397.53060
Summary: We show that a connected gradient Ricci soliton \((M,g,f,\lambda )\) with constant scalar curvature and admitting a non-homothetic conformal vector field \(V\) leaving the potential vector field invariant, is Einstein and the potential function \(f\) is constant. For locally conformally flat case and non-homothetic \(V\) we show without constant scalar curvature assumption, that \(f\) is constant and \(g\) has constant curvature.
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
References:
| [1] | Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects, Mathematical Surveys and Monographs, vol. 135. American Mathematical Society, Providence (2007) · Zbl 1157.53034 |
| [2] | Brozos-Vázquez, M; García-Río, E; Gavino-Fernández, S, Locally conformally flat Lorentzian gradient Ricci solitons, J. Geom. Anal., 23, 1196-1212, (2013) · Zbl 1285.53059 · doi:10.1007/s12220-011-9283-z |
| [3] | Fernández-López, M; García-Río, E, A note on locally conformally flat gradient Ricci solitons, Geom. Dedicata, 168, 1-7, (2014) · Zbl 1284.53046 · doi:10.1007/s10711-012-9815-0 |
| [4] | Fernández-López, M; García-Río, E, On gradient Ricci solitons with constant scalar curvature, Proc. Am. Math. Soc., 144, 369-378, (2016) · Zbl 1327.53057 · doi:10.1090/proc/12693 |
| [5] | Jauregui, JL; Wylie, William, Conformal diffeomorphisms of gradient Ricci solitons and generalized quasi-Einstein manifolds, J. Geom. Anal., 25, 668-708, (2015) · Zbl 1312.53066 · doi:10.1007/s12220-013-9442-5 |
| [6] | Kanai, M, On a differential equation characterizing a Riemannian structure of a manifold, Tokyo J. Math., 6, 143-151, (1983) · Zbl 0534.53037 · doi:10.3836/tjm/1270214332 |
| [7] | Kühnel, W; Rademacher, H-B, Conformal vector fields on pseudo-Riemannian spaces, Differ. Geom. Appl., 7, 237-250, (1997) · Zbl 0901.53048 · doi:10.1016/S0926-2245(96)00052-6 |
| [8] | Okumura, M, Complete Riemannian manifolds and some vector fields, Trans. Am. Math. Soc., 117, 251-275, (1965) · Zbl 0136.17701 · doi:10.1090/S0002-9947-1965-0174022-6 |
| [9] | Perelman, G.: The Entropy Formula for the Ricci Flow and Its Geometric Applications. Preprint. arXiv:0211.1159 · Zbl 1327.53057 |
| [10] | Petersen, P; Wylie, W, On the classification of gradient Ricci solitons, Geom. Topol., 14, 2277-2300, (2010) · Zbl 1202.53049 · doi:10.2140/gt.2010.14.2277 |
| [11] | Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970) · Zbl 0213.23801 |
| [12] | Yano, K; Nagano, T, Einstein spaces admitting a one-parameter group of conformal transformations, Ann. Math. (2), 69, 451-461, (1959) · Zbl 0088.14204 · doi:10.2307/1970193 |
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