I’ll just do a quick finish up on proving basic properties about the exponential map for today, since I’ll be moving on to a different topic after this.
The exponential map actually restricts to a diffeomorphism from some neighborhood of 0 in 
to some neighborhood of 
. Well, if you’ve mastered your basic theorems, then this is no surprise. Last time we showed 
. The differential is the identity map and hence non-singular. Thus by the Inverse Function Theorem, there exists neighborhoods about 0 and 
such that there is a smooth inverse, and hence 
restricted to these is a diffeo.
Probably the most interesting and non-obvious of these facts is the following one. Given another Lie group 
(and Lie algebra 
), and a Lie group homomorphism 
, we get a commutative diagram (which I’ve yet to figure out how to make on wordpress). Anyway it just says that 
. So pushing forward a vector field and then exponentiating it is the same thing as exponentiating first and then doing the group hom.
Let’s do it by showing that 
for all 
. Recall that the LHS is just the one-parameter subgroup generated by 
. Thus by uniqueness, we just need to show that 
is a hom that satisfies 
. It is a composition of homs, so is itself one. Now we compute the derivative: 
.
The last property for today is used quite a bit as well. It says that flowing by a vector field 
is the same thing as right multiplying by 
. In other words, 
.
First note that for any 
, the map 
, satisfies 
by left invariance. Also, left multiplication takes integral curves to integral curves, so this is actually the integral curve of starting at 
. Thus 
.
Now we check what we wanted to show. 
. And we’re done!
Next I’ll talk about infinitesimal generators of Lie group actions, so that I can pull in some stuff I did from the Frobenius theorem.
Update: There is some sort of weird bug where when I hit preview it tells me to save the draft. Then when I hit save draft, it posts. So, sorry if you’ve been getting unedited versions of my posts.
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I am a mathematics graduate student fascinated in how all my interests fit together.
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