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 = TeG.
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.
Read more about this topic: Vector Flow
Famous quotes containing the words flow, lie, group and/or theory:
“The joys of the evil flow away like a torrent.”
—Jean Racine (1639–1699)
“There is none who does not lie hourly in the respect he pays to false appearance.”
—Henry David Thoreau (1817–1862)
“For me, as a beginning novelist, all other living writers form a control group for whom the world is a placebo.”
—Nicholson Baker (b. 1957)
“Osteopath—One who argues that all human ills are caused by the pressure of hard bone upon soft tissue. The proof of his theory is to be found in the heads of those who believe it.”
—H.L. (Henry Lewis)