Lie group geometry: Riemann and Ricci tensors and normal forms of Lie algebras

Abstract

In the context of the connection discovered in a preceding paper between left-invariant objects (both geometric and dynamical) defined on a Lie group and the algebra of right automorphisms (the dual algebra), we consider the representation of the main geometric characteristics via this algebra and the corresponding metric form. These characteristics are shown to be constant (independent of a point) and defined only by the structure constants of the dual algebra and the coefficients of the metric form. Due to this connection, it is possible to introduce the concept of normal forms of a Lie algebra. Reducing any algebra and any metric to normal form in fact consists in reducing two quadratic forms to canonical form: first, the metric is reduced to the sum of squares of linear differential forms, and then the constant matrix characterizing the Ricci tensor is reduced to diagonal form (with the principal curvatures appearing on the diagonal). It turns out that there are only two different normal forms for three-dimensional Lie algebras, each depending on three parameters associated with three principal curvatures in the general case.