Vector Flow - Vector Flow in Lie Group Theory

Vector Flow in Lie Group Theory

Relevant concepts: (exponential map, infinitesimal generator, one-parameter group)

Every left-invariant vector field on a Lie group is complete. The integral curve starting at the identity is a one-parameter subgroup of G. There are one-to-one correspondences

{one-parameter subgroups of G} ⇔ {left-invariant vector fields on G} ⇔ g = T_eG.

Let G be a Lie group and g its Lie algebra. The exponential map is a map exp : g → G given by exp(X) = γ(1) where γ is the integral curve starting at the identity in G generated by X.

• The exponential map is smooth.
• For a fixed X, the map t ↦ exp(tX) is the one-parameter subgroup of G generated by X.
• The exponential map restricts to a diffeomorphism from some neighborhood of 0 in g to a neighborhood of e in G.
• The image of the exponential map always lies in the connected component of the identity in G.