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.
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The first author’s work has been partially supported by D.G.I. (Spain) and FEDER Project MTM 2004-06015-C02-01, by a grant AVCiTGRUPOS03/169 and by a Research Grant from Ministerio de Educación y Cultura. The second author’s work has been supported by the grant GA ČR 201/05/2707 and it is part of the research project MSM 0021620839 financed by the Ministry of Education (MŠMT).
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Arias-Marco, T., Kowalski, O. Classification of 4-dimensional homogeneous D’Atri spaces. Czech Math J 58, 203–239 (2008). https://doi.org/10.1007/s10587-008-0014-y
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DOI: https://doi.org/10.1007/s10587-008-0014-y