On Chapter Xii in Cartan's "Le\c{C}ONS Sur la G\'{E}OM\'{E}TRIE Des Espaces De Riemann"

TL;DR

This paper demonstrates how Cartan's method of adapted frames in Riemannian geometry leads to a classification theorem for homogeneous Riemannian manifolds, with examples in three dimensions illustrating its effectiveness.

Contribution

It clarifies and applies Cartan's method of adapted frames to classify homogeneous Riemannian manifolds, expanding on his original work with explicit 3D examples.

Findings

Classification theorem for homogeneous Riemannian manifolds derived from Cartan's method

Explicit 3D examples of classification provided

Validation of Cartan's approach as a powerful tool in Riemannian geometry

Abstract

One shows that Cartan's method of adapted frames in Chapter XII of his famous treatise of Riemannian geometry, leads to a classification theorem of homogeneous Riemannian manifolds. Examples of classification in 3D dimensions obtained by Cartan are given using this powerful method.