DeTurck, Dennis M.; Koiso, Norihito Uniqueness and non-existence of metrics with prescribed Ricci curvature. (English) Zbl 0556.53026 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 351-359 (1984). The authors give conditions under which the Ricci tensor uniquely determines the underlying Levi-Civita connection and find restrictions on a doubly covariant tensor to be the Ricci tensor of some Riemannian metric. The results extend a result of R. Hamilton who showed that the standard metric on \(S^ n\) is uniquely determined by its Ricci tensor [see The Ricci curvature equation, preprint, MSRI (1983)]. Reviewer: G.Thorbergsson Cited in 2 ReviewsCited in 21 Documents MSC: 53C20 Global Riemannian geometry, including pinching 53B20 Local Riemannian geometry Keywords:Ricci tensor; Levi-Civita connection PDF BibTeX XML Cite \textit{D. M. DeTurck} and \textit{N. Koiso}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 351--359 (1984; Zbl 0556.53026) Full Text: DOI Numdam EuDML OpenURL References: [1] A. Besse, Géométrie riemannien en dimension 4, Cedic Fernand Nathan, Paris, 1981. [2] Bourguignon, J.; Karcher, H., « curvature operators: pinching estimates and geometric examples », Ann. Scient. Ec. Norm. Sup., t. 11, 4^{e} série, 71-92, (1978) · Zbl 0386.53031 [3] Calabi, E., « extremal Kähler metrics », (Yau, S. T., Seminar on Differential Geometry, (1982), Princeton U. Press), 259-290, Annals of Math Study 102 [4] Calabi, E., « Métriques Kählériennes et fibrés holomorphes », Ann. Sci. Ec. Norm. Sup., t. 12, 4^{e} série, 269-294, (1979) · Zbl 0431.53056 [5] DeTurck, D., « metrics with prescribed Ricci curvature », (Yau, S. T., Seminar on Differential Geometry, (1982), Princeton U. Press), 525-537, Annals of Math Study 102 [6] Eells, J.; Lemaire, L., « A report on harmonic maps », Bull. London Math. Soc., t. 10, 1-68, (1978) · Zbl 0401.58003 [7] R. Hamilton, The Ricci curvature equation, preprint, MSRI Berkeley, 1983. [8] Kobayashi, S.; Nomizu, K., Foundations of differential geometry, vol. 1, (1963), Wiley-Interscience New York · Zbl 0119.37502 [9] Kobayashi, S.; Nomizu, K., Foundations of differential geometry, vol. 2, (1969), Wiley-Interscience New York · Zbl 0175.48504 [10] Milnor, J., « curvatures of left-invariant metrics on Lie groups », Adv. in Math., t. 21, 293-329, (1976) · Zbl 0341.53030 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.