Exponential map is surjective for compact connected Lie group

differential-geometry lie-groups lie-algebras topological-groups

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective?

I tried to find the proof on the internet but most of them are either just a short note or "left as an exercise for reader" with some hints like: use invariant inner product and existence of geodesic but I don't really understand.

So if someone could point out where to find a complete proof of this or give me a more extensive hints on how to start the proof that would be great.

Thank you!


Solution 1:

This is following from the maximal torus theorem. Let $T$ be a maximal torus in the compact, connected Lie group $G$. Then, the $\exp$ map is surjective on $T$. The theorem says that every element in $G$ is contained in some maximal torus and any two maximal tori are conjugate to each other. The surjectivity of $\exp$ on $G$ follows from here.

For reference, Lie groups, Lie algebras, and Representations: An Elementary Introduction 2ed by Brian C. Hall chapter 11 is very useful.

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