A note on Codazzi tensors. (English) Zbl 1319.53036
Summary: We discuss a gap in A. L. Besse’s book [Einstein manifolds. Reprint of the 1987 edition. Berlin: Springer (2008; Zbl 1147.53001)], recently pointed out by G. Merton in [Proc. Am. Math. Soc. 141, No. 9, 3265–3273 (2013; Zbl 1282.53013)], which concerns the classification of Riemannian manifolds admitting a Codazzi tensors with exactly two distinct eigenvalues. For such manifolds, we prove a structure theorem, without adding extra hypotheses and then we conclude with some application of this theory to the classification of three-dimensional gradient Ricci solitons.
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
References:
| [1] | Besse, A.L.: Einstein Manifolds. Springer, Berlin (2008) · Zbl 1147.53001 |
| [2] | Cao, H.-D., Chen, Q.: On locally conformally flat gradient steady Ricci solitons. Trans. Am. Math. Soc. 364, 2377-2391 (2012) · Zbl 1245.53038 · doi:10.1090/S0002-9947-2011-05446-2 |
| [3] | Chen, B.-L.: Strong uniqueness of the Ricci flow. J. Diff. Geom. 82, 363-382 (2009) · Zbl 1177.53036 |
| [4] | 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 II. In: Proceedings of Analytic Aspects, Mathematical Surveys and Monographs, vol. 144. American Mathematical Society, Providence (2008) · Zbl 1157.53035 |
| [5] | Eminenti, M., La Nave, G., Mantegazza, C.: Ricci solitons: the equation point of view. Manuscripta Math. 127(3), 345-367 (2008) · Zbl 1160.53031 · doi:10.1007/s00229-008-0210-y |
| [6] | Gromoll, D., Klingenberg, W., Meyer, W.: Riemannsche Geometrie im Großen. Lecture notes in Mathematics, vol. 55. Springer, Berlin (1975) · Zbl 0293.53001 |
| [7] | Merton, G.: Codazzi tensors with two eigenvalue functions. Proc. Am. Math. Soc. 141, 3265-3273 (2013) · Zbl 1282.53013 · doi:10.1090/S0002-9939-2013-11616-3 |
| [8] | Zhang, Z.-H.: On the completeness of gradient Ricci solitons. Proc. Am. Math. Soc. 137(8), 2755-2759 (2009) · Zbl 1176.53046 · doi:10.1090/S0002-9939-09-09866-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
