Classification of 4-dimensional homogeneous D’Atri spaces
- 1 March 2008
- journal article
- Published by Institute of Mathematics, Czech Academy of Sciences in Czechoslovak Mathematical Journal
- Vol. 58 (1), 203-239
- https://doi.org/10.1007/s10587-008-0014-y
Abstract
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.
Keywords
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