jacobrhcp
Oct18-08, 12:19 PM
1. The problem statement, all variables and given/known data
Let G be a Lie group with unit e. For every g in G let Lg: G -> G, h-> gh be left multiplication.
a) Prove that the map G x TeG -> TG, (g,v) -> De Lg (v) is a diffeomorphism, and conclude G has a basis of vector fields.
b) Prove for every v in TeG the is a unique vector field V on G that is left invariant (in the sense that the pullback of Lg on V is just V for all g in G) with Ve=v
for the original problem statement, see
https://www.math.uu.nl/people/looijeng/difftop06eng.pdf
Questions about this
Lg (v) is just (gv), but I'm not sure how De transforms (gv). It is defined as the map that transforms a curve tangent to a point b to a curve tangent to e(b) on some other chart in G. If I understood this right, then De would send (gv) to the tangent vector at the origin in the direction of gv? - this last sentence of my own I did not fully understand.
if someone could help clarify the problem a bit, not so much as give a solution... that would be nice.
Let G be a Lie group with unit e. For every g in G let Lg: G -> G, h-> gh be left multiplication.
a) Prove that the map G x TeG -> TG, (g,v) -> De Lg (v) is a diffeomorphism, and conclude G has a basis of vector fields.
b) Prove for every v in TeG the is a unique vector field V on G that is left invariant (in the sense that the pullback of Lg on V is just V for all g in G) with Ve=v
for the original problem statement, see
https://www.math.uu.nl/people/looijeng/difftop06eng.pdf
Questions about this
Lg (v) is just (gv), but I'm not sure how De transforms (gv). It is defined as the map that transforms a curve tangent to a point b to a curve tangent to e(b) on some other chart in G. If I understood this right, then De would send (gv) to the tangent vector at the origin in the direction of gv? - this last sentence of my own I did not fully understand.
if someone could help clarify the problem a bit, not so much as give a solution... that would be nice.