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In this paper, we show how to use the method of Cartan triples (see V. Patrangenaru) in small dimensions. We classify the 3-dimensional simply connected homogeneous Riemannian spaces, and the 4-dimensional simply connected homogeneous Riemannian spaces with 5-dimensional total isometry group. We show that the smallest dimension where locally homogeneous Riemannian manifolds that are not locally isometric to homogeneous Riemannian spaces exist, is 5.

Classifying 3- and 4-dimensional homogeneous Riemannian manifolds by Cartan triples