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Article

Keywords:
Riemannian manifold; curvature homogeneous spaces; homogeneous spaces
Summary:
We prove that a four-dimensional, connected, simply connected and complete Riemannian manifold which is curvature homogeneous up to order two is a homogeneous Riemannian space.
References:
[1] Bérard Bergery L.: Les espaces homogènes riemanniens de dimension $4$. Géométrie riemannienne en dimension 4, Séminaire A. Besse, Cedic, Paris, 1981, 40-60. MR 0769130
[2] Derdziński A.: preprint.
[3] Gromov M.: Partial differential equations. Ergeb. Math. Grenzgeb. 3 Folge 9, Springer-Verlag, Berlin, Heidelberg, New York, 1987.
[4] Jensen G.: Homogeneous Einstein spaces in dimension four. J. Differential Geom. 3 (1969), 309-349. MR 0261487
[5] Kowalski O.: A note to a theorem by K. Sekigawa. Comment. Math. Univ. Carolinae 30 (1989), 85-88. MR 0995705 | Zbl 0679.53043
[6] Kowalski O., Tricerri F., Vanhecke L.: Exemples nouveaux de variétés riemanniennes non- homogènes dont le tenseur de courbure est celui d'un espace symétrique riemannien. C.R. Acad. Sci. Paris Sér. I 311 (1990), 355-360. MR 1071643
[7] Kowalski O., Tricerri F., Vanhecke L.: Curvature homogeneous Riemannian manifolds. J. Math. Pures Appl., to appear. MR 1193605 | Zbl 0836.53029
[8] Kowalski O., Tricerri F., Vanhecke L.: Curvature homogeneous spaces with a solvable Lie group as homogeneous model. to appear. MR 1167378 | Zbl 0762.53031
[9] Sekigawa K.: On the Riemannian manifolds of the form $B\times _f F$. Kōdai Math. Sem. Rep. 26 (1975), 343-347. MR 0438253 | Zbl 0304.53019
[10] Sekigawa K.: On some $3$-dimensional curvature homogeneous spaces. Tensor N.S. 31 (1977), 87-97. MR 0464115 | Zbl 0356.53016
[11] Singer M.I.: Infinitesimally homogeneous spaces. Comm. Pure Appl. Math. 13 (1960), 685-697. MR 0131248 | Zbl 0171.42503
[12] Takagi H.: On curvature homogeneity of Riemannian manifolds. Tôhoku Math. J. 26 (1974), 581-585. MR 0365417 | Zbl 0302.53022
[13] Tricerri F., Vanhecke L.: Curvature homogeneous Riemannian manifolds. Ann. Sci. Ecole Norm. Sup. 22 (1989), 535-554. MR 1026749 | Zbl 0698.53033
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