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Keywords:
Riemannian manifolds; curvature tensor; curvature homogeneous spaces
Riemannian manifolds; curvature tensor; curvature homogeneous spaces
Summary:
A six-parameter family is constructed of (algebraic) Riemannian curvature tensors in dimension four which do not belong to any curvature homogeneous space. Also a general method is given for a possible extension of this result.
References:
[CH] Chern S.S.: On the Curvature and Characteristic Classes of a Riemannian manifold. Abh. Math. Sem. Univ. Hamburg 20 (1955), 117-126. MR 0075647
[K1] Kowalski O.: An explicit classification of 3-dimensional Riemannian spaces satisfying $R(X,Y)\cdot R = 0$. Preprint, 1991. MR 1408298
[K2] Kowalski O.: A classification of Riemannian 3-manifolds with constant principal Ricci curvatures $\rho _1 = \rho _2 \ne \rho _3$. To appear in Nagoya Math. J. 132 (1993). MR 1253692
[K3] Kowalski O.: Nonhomogeneous Riemannian 3-manifolds with distinct constant Ricci eigenvalues. Comment. Math. Univ. Carolinae, 34, 3 (1993), 451-457. MR 1243077 | Zbl 0789.53024
[KL] Klinger R.: A Basis that Reduces to Zero as many Curvature Components as Possible. Abh. Math. Sem. Univ. Hamburg 61 (1991), 243-248. MR 1138290 | Zbl 0753.53012
[K-N] Kobayashi S., Nomizu K.: Foundations of Differential geometry I. Interscience Publishers, New York 1963. MR 0152974 | Zbl 0119.37502
[K-P] Kowalski O., Prüfer F.: On Riemannian 3-manifolds with distinct constant Ricci eigenvalues. Preprint, 1993. MR 1289828
[K-T-V1] Kowalski O., Tricerri F., Vanhecke L.: New examples of non-homogeneous Riemannian manifolds whose curvature tensor is that of a Riemannian symmetric space. C. R. Acad. Sci. Paris, Sér. I, 311 (1990), 355-360 MR 1071643 | Zbl 0713.53028
[K-T-V2] Kowalski O., Tricerri F., Vanhecke L.: Curvature homogeneous Riemannian manifolds. J. Math. Pures Appl. 71 (1992), 471 - 501. MR 1193605 | Zbl 0836.53029
[K-T-V3] Kowalski O., Tricerri F., Vanhecke L.: Curvature homogeneous spaces with a solvable Lie group as a homogeneous model. J. Math. Soc. Japan, 44 (1992), 461-484. MR 1167378
[K-V] Kowalski O., Vanhecke L.: Ball-Homogeneous and Disk-Homogeneous Riemannian manifolds. Math. Z. 180 (1982), 429-444. MR 0666999 | Zbl 0476.53023
[N-T] Nicolodi L., Tricerri F.: On two Theorems of I.M. Singer about Homogeneous Spaces. Ann. Global Anal. Geom. 8 (1990), 193-209. MR 1088511 | Zbl 0676.53058
[M] Milnor J.: Curvatures of left invariant metrics on Lie groups. Adv. in Math. 21 (1976), 293-329. MR 0425012 | Zbl 0341.53030
[SE] Sekigawa K.: On some 3-dimensional Riemannian manifolds. Hokkaido Math. J. 2 (1973), 259-270. MR 0353204 | Zbl 0266.53034
[SI] Singer I.M.: Infinitesimally homogeneous spaces. Comm. Pure Appl. Math. 13 (1960), 685-697. MR 0131248 | Zbl 0171.42503
[SI-TH] Singer I.M., Thorpe J.A.: The curvature of 4-dimensional Einstein spaces. In: Global Analysis (Papers in honor of K. Kodaira, pp. 355-366) Princeton, New Jersey, Princeton University Press 1969. MR 0256303 | Zbl 0199.25401
[S-T] Spiro A., Tricerri F.: 3-dimensional Riemannian metrics with prescribed Ricci principal curvatures. Preprint, 1993. MR 1327884
[T] Tsukada T.: Curvature homogeneous hypersurfaces immersed in a real space form. Tôhoku Math. J. 40 (1988), 221-244. MR 0943821 | Zbl 0651.53037
[T-V] Tricerri F., Vanhecke L.: Curvature tensors on almost Hermitian manifolds. Trans. Amer. Math. Soc. 267 (1981), 365-398. MR 0626479 | Zbl 0484.53014
[YA] Yamato K.: A characterization of locally homogeneous Riemannian manifolds of dimension three. Nagoya Math. J. 123 (1991), 77-90. MR 1126183
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