Left invariant geometry of Lie groups

Abstract

In this article we investigate the geometry of a Lie group N with a left invariant metric, particularly in the case that N is 2-step nilpotent. Our primary interest will be in properties of the geodesic flow, but we describe a more general framework for studying left invariant functions and vector fields on the tangent bundle TN. Here we consider the natural left action Î» of N on T N given by Î»_n(ℇ) = (L_n)*(ℇ), where L_n : N âŸ¶ N denotes left translation by n and (L_n) denotes the differential map of L_n.

For convenience all manifolds in this article are assumed to be connected and C^∞ unless otherwise specified. Many of the assertions remain valid true for manifolds that are not connected and are C^k for a small integer k.

We assume that the reader has a familiarity with manifold theory and with the basic concepts of Lie groups and their associated Lie algebras of left invariant vector fields.