Riemannian algebras of lie groups

Abstract

Because many notions of differential geometry is defined by the use of connections on differential manifolds, the study of connections on differential manifolds is important in differential geometry. Many mathematician research the invariant connections on Lie groups and homogeneous spaces by using algebraic language since Lie groups and homogeneous space have relations with Lie algebras. In this thesis, we classify the Riemannian invariant connections of Lie groups by determining the Riemannian algebras. First, we see that Riemannian invariant connections is represented in terms of Riemannian algebras and review the relation between psuedo-Riemannian algebras of Lie groups and symmetric Lie algebras. We classify the symmetric Lie algebras of dimension 4, 5 and determine Riemannian invariant connections of Lie group of dimension 2, 3 by classifying Riemannian algebras of 2, 3-dimensional Lie group. In case of dimension 3, we determine Riemannian algebras by treating the unimodular and nonunimodular cases separately and show the relation between Riemanian algebras of unimodular Lie groups and Jacobi elliptic algebras. Finally, we classify Riemannian algebras of Heisenberg group.