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 λ of N on T N given by λn(ℇ) = (Ln)*(ℇ), where Ln : N ⟶ N denotes left translation by n and (Ln) denotes the differential map of Ln.
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 Ck 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.
@article{1660,
title = {Left invariant geometry of Lie groups},
journal = {CUBO, A Mathematical Journal},
volume = {6},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/1660}
}
Eberlein, Patrick. Left invariant geometry of Lie groups. CUBO, A Mathematical Journal, Tome 6 (2004) . http://gdmltest.u-ga.fr/item/1660/