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Title: Classification of 4-dimensional homogeneous D'Atri spaces (English)
Author: Arias-Marco, Teresa
Author: Kowalski, Oldřich
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 58
Issue: 1
Year: 2008
Pages: 203-239
Summary lang: English
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Category: math
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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. (English)
Keyword: Riemannian manifold
Keyword: naturally reductive Riemannian homogeneous space
Keyword: D’Atri space
MSC: 53B21
MSC: 53C21
MSC: 53C25
MSC: 53C30
idZBL: Zbl 1174.53024
idMR: MR2402535
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Date available: 2009-09-24T11:54:34Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/128255
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