Original Article
Kyungpook Mathematical Journal 2006; 46(3): 367-376
Published online September 23, 2006
Copyright © Kyungpook Mathematical Journal.
Symmetry Properties of 3-dimensional D’Atri Spaces
Mohamed Belkhelfa1, Ryszard Deszcz2, Leopold Verstraelen3
1Laboratoire de Physique Quantique de la Matiˇere et Mod´elisations, Math´ematiques - L.P.Q. 3M, Centre Universitaire, Mustapha Stambouli de Mascara, Route de Mamounia, Mascara 29000, Algeria
2Department of Mathematics, Agricultural University of Wroc?law, Grunwaldzka 53, 50-357 Wroc?law, Poland
3Department of Mathematics, Catholic University Leuven, Celestijnenlaan 200 B, B - 3001 Leuven, Belgium
We investigate semi-symmetry and pseudo-symmetry of some 3-dimensional Riemannian manifolds: the D'Atri spaces, the Thurston geometries as well as the $eta$-Einstein manifolds. We prove that all these manifolds are pseudo-symmetric and that many of them are not semi-symmetric.
Keywords: Heisenberg space, D'Atri space, Thurston geometry, semi-symmetry, pseudo-symmetry, $eta$-Einstein manifold, warped product