Classification of 4-dimensional homogeneous D'Atri spaces

@article{Arias2008,
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.},
author = {Arias-Marco, Teresa, Kowalski, Oldřich},
journal = {Czechoslovak Mathematical Journal},
keywords = {Riemannian manifold; naturally reductive Riemannian homogeneous space; D’Atri space; Riemannian manifold; naturally reductive Riemannian homogeneous space; D'Atri space},
language = {eng},
number = {1},
pages = {203-239},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Classification of 4-dimensional homogeneous D'Atri spaces},
url = {http://eudml.org/doc/31208},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Arias-Marco, Teresa
AU - Kowalski, Oldřich
TI - Classification of 4-dimensional homogeneous D'Atri spaces
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 1
SP - 203
EP - 239
AB - 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.
LA - eng
KW - Riemannian manifold; naturally reductive Riemannian homogeneous space; D’Atri space; Riemannian manifold; naturally reductive Riemannian homogeneous space; D'Atri space
UR - http://eudml.org/doc/31208
ER -