Group exponentials and general group of diffeomorphisms

Question

I read on the wiki page (http://en.wikipedia.org/wiki/Exponential_map_%28Lie_theory%29) that the group exponential is not a local diffeomorphism at all points. Can someone give me an example?

Answer 1

The exponential map, although it is a local diffeomorphism on some neighborhood of $0$ in the Lie algebra $\mathfrak{g}$, need not be a local diffeomorphism near all points of $\mathfrak{g}$. Take for example the Lie algebra $\mathfrak{su}(2)$ : all points at distance $π$ from $0$ are sent to the south pole of $S_3≃SU(2)$, so the exponential fails to be locally injective near any of those points.