Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
Riemannian manifold; naturally reductive Riemannian homogeneous space; D’Atri space
Riemannian manifold; naturally reductive Riemannian homogeneous space; D’Atri space
Summary:
The property of being a D’Atri space (i.e., a space with volume-preserving symmetries) is equivalent to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold $(M,g)$ satisfying the first odd Ledger condition is said to be of type $\mathcal {A}$. The classification of all 3-dimensional D’Atri spaces is well-known. All of them are locally naturally reductive. The first attempts to classify all 4-dimensional homogeneous D’Atri spaces were done in the papers by Podesta-Spiro and Bueken-Vanhecke (which are mutually complementary). The authors started with the corresponding classification of all spaces of type $\mathcal {A}$, but this classification was incomplete. Here we present the complete classification of all homogeneous spaces of type $\mathcal {A}$ in a simple and explicit form and, as a consequence, we prove correctly that all homogeneous 4-dimensional D’Atri spaces are locally naturally reductive.
References:
[1] T. Arias-Marco: The classification of 4-dimensional homogeneous D’Atri spaces revisited. Differential Geometry and its Applications (to appear). MR 2293639 | Zbl 1121.53026
[2] L. Bérard Bergery: Les espaces homogènes riemanniens de dimension 4. Géométrie riemannienne en dimension 4, L. Bérard Bergery, M. Berger, C. Houzel (eds.), CEDIC, Paris, 1981. (French) MR 0769130
[3] E. Boeckx, L. Vanhecke, O. Kowalski: Riemannian Manifolds of Conullity Two. World Scientific, Singapore, 1996. MR 1462887
[4] P. Bueken, L. Vanhecke: Three- and four-dimensional Einstein-like manifolds and homogeneity. Geom. Dedicata 75 (1999), 123–136. DOI 10.1023/A:1005060208823 | MR 1686754
[5] J. E. D’Atri, H. K. Nickerson: Divergence preserving geodesic symmetries. J. Differ. Geom. 3 (1969), 467–476. MR 0262969
[6] J. E. D’Atri, H. K. Nickerson: Geodesic symmetries in spaces with special curvature tensors. J. Differ. Geom. 9 (1974), 251–262. MR 0394520
[7] G. R. Jensen: Homogeneous Einstein spaces of dimension four. J. Differ. Geom. 3 (1969), 309–349. MR 0261487 | Zbl 0194.53203
[8] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry I. Interscience, New York, 1963. MR 0152974
[9] O. Kowalski: Spaces with volume-preserving symmetries and related classes of Riemannian manifolds. Rend. Semin. Mat. Univ. Politec. Torino, Fascicolo Speciale (1983), 131–158. MR 0829002 | Zbl 0631.53033
[10] O. Kowalski, F. Prüfer, L. Vanhecke: D’Atri Spaces. Topics in Geometry. Prog. Nonlinear Differ. Equ. Appl. 20 (1996), 241–284. MR 1390318
[11] J. Milnor: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21 (1976), 293–329. DOI 10.1016/S0001-8708(76)80002-3 | MR 0425012 | Zbl 0341.53030
[12] H. Pedersen, P. Tod: The Ledger curvature conditions and D’Atri geometry. Differ. Geom. Appl. 11 (1999), 155–162. DOI 10.1016/S0926-2245(99)00026-1 | MR 1712123
[13] F. Podestà, A. Spiro: Four-dimensional Einstein-like manifolds and curvature homogeneity. Geom. Dedicata 54 (1995), 225–243. DOI 10.1007/BF01265339 | MR 1326728
[14] I. M. Singer: Infinitesimally homogeneous spaces. Commun. Pure Appl. Math. 13 (1960), 685–697. DOI 10.1002/cpa.3160130408 | MR 0131248 | Zbl 0171.42503
[15] Z. I. Szabó: Spectral theory for operator families on Riemannian manifolds. Proc. Symp. Pure Maths. 54 (1993), 615–665.
[16] K. P. Tod: Four-dimensional D’Atri-Einstein spaces are locally symmetric. Differ. Geom. Appl. 11 (1999), 55–67. DOI 10.1016/S0926-2245(99)00024-8 | MR 1702467 | Zbl 0930.53028
Search
Partner of
