Geodesic
Classification of 4-dimensional homogeneous D'Atri spaces
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 1, pp. 203-239 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 53B21, 53C21, 53C25, 53C30
Keywords: Riemannian manifold; naturally reductive Riemannian homogeneous space; D’Atri space 
@article{CMJ_2008_58_1_a13,
     author = {Arias-Marco, Teresa and Kowalski, Old\v{r}ich},
     title = {Classification of 4-dimensional homogeneous {D'Atri} spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {203--239},
     year = {2008},
     volume = {58},
     number = {1},
     mrnumber = {2402535},
     zbl = {1174.53024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_1_a13/}
}
TY  - JOUR
AU  - Arias-Marco, Teresa
AU  - Kowalski, Oldřich
TI  - Classification of 4-dimensional homogeneous D'Atri spaces
JO  - Czechoslovak Mathematical Journal
PY  - 2008
SP  - 203
EP  - 239
VL  - 58
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMJ_2008_58_1_a13/
LA  - en
ID  - CMJ_2008_58_1_a13
ER  - 
%0 Journal Article
%A Arias-Marco, Teresa
%A Kowalski, Oldřich
%T Classification of 4-dimensional homogeneous D'Atri spaces
%J Czechoslovak Mathematical Journal
%D 2008
%P 203-239
%V 58
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2008_58_1_a13/
%G en
%F CMJ_2008_58_1_a13
Arias-Marco, Teresa; Kowalski, Oldřich. Classification of 4-dimensional homogeneous D'Atri spaces. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 1, pp. 203-239. http://geodesic.mathdoc.fr/item/CMJ_2008_58_1_a13/

[1] T.  Arias-Marco: The classification of 4-dimensional homogeneous D’Atri spaces revisited. Differential Geometry and its Applications (to appear). | MR | Zbl

[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

[3] E.  Boeckx, L.  Vanhecke, O.  Kowalski: Riemannian Manifolds of Conullity Two. World Scientific, Singapore, 1996. | MR

[4] P.  Bueken, L.  Vanhecke: Three- and four-dimensional Einstein-like manifolds and homogeneity. Geom. Dedicata 75 (1999), 123–136. | DOI | MR

[5] J. E.  D’Atri, H. K.  Nickerson: Divergence preserving geodesic symmetries. J. Differ. Geom. 3 (1969), 467–476. | MR

[6] J. E.  D’Atri, H. K.  Nickerson: Geodesic symmetries in spaces with special curvature tensors. J. Differ. Geom. 9 (1974), 251–262. | MR

[7] G. R.  Jensen: Homogeneous Einstein spaces of dimension four. J. Differ. Geom. 3 (1969), 309–349. | MR | Zbl

[8] S.  Kobayashi, K.  Nomizu: Foundations of Differential Geometry  I. Interscience, New York, 1963. | MR

[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 | Zbl

[10] O.  Kowalski, F.  Prüfer, L.  Vanhecke: D’Atri Spaces. Topics in Geometry. Prog. Nonlinear Differ. Equ. Appl. 20 (1996), 241–284. | MR

[11] J.  Milnor: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21 (1976), 293–329. | DOI | MR | Zbl

[12] H.  Pedersen, P.  Tod: The Ledger curvature conditions and D’Atri geometry. Differ. Geom. Appl. 11 (1999), 155–162. | DOI | MR

[13] F.  Podestà, A.  Spiro: Four-dimensional Einstein-like manifolds and curvature homogeneity. Geom. Dedicata 54 (1995), 225–243. | DOI | MR

[14] I. M.  Singer: Infinitesimally homogeneous spaces. Commun. Pure Appl. Math. 13 (1960), 685–697. | DOI | MR | Zbl

[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 | MR | Zbl