Locally homogeneous Riemannian manifolds and Cartan triples
Locally homogeneous and homogeneous Riemannian manifolds have been treated by O. Kowalski, F. Tricerri, the reviewer and other collaborators using homogeneous structures and associated infinitesimal models. One of the remarkable facts of this work is the discovery of locally homogeneous Riemannian manifolds which are not locally isometric to a homogeneous Riemannian space. Such examples already exist in dimension 5. They correspond to the nonregular infinitesimal models while the homogeneous ones are determined by means of regular models. The main aim of this paper, which contains a part of the author's Ph.D. thesis [Locally homogeneous Riemannian and pseudoriemannian manifolds, University of Haifa (1992)], is to develop an alternative algebraic approach to study both classes. This is done by introducing the formalism of Cartan triples. After developing the basic theory, the author also uses his method to determine the local isometry classes of locally homogeneous Riemannian manifolds. For applications of this method concerning the classification of 3- and 4-dimensional simply connected homogeneous Riemannian manifolds, see the author's paper [Pac. J. Math. 173, 511-532 (1996; Zbl 0866.53035)]. The author has extended his method to the pseudo-Riemannian case in [J. Geom. Phys. 17, 59-72 (1995; Zbl 0832.53017)].
locally homogeneous Riemannian manifolds
scientific article; zbMATH DE number 627547