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I am trying to learn Lie Algebra with a little knowledge of Differential Geometry. I have read that the exponential map of compact connected Lie groups is always surjective. So I am wondering about the injectivity of the exponential map for compact connected Lie groups. I think it can not be injective and the multiplication of 1-torus with 1-sphere($T^1\times S^1$) is a counter example. Is this true? If so how can I prove it in a proper way?

Grazel
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  • As Tsemo Aristide's answer shows, the easiest (non-trivial) example is the easiest counterexample, $G= S^1$, where the exponential is basically the quotient map $\mathbb R \rightarrow \mathbb R/\mathbb Z$. – Torsten Schoeneberg Apr 29 '20 at 16:03

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If $G$ is compact, then $G$ has a subgroup isomorphic to the circle, therefore the exponential is not injective.

Tsemo Aristide
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  • Maybe exclude $\mathcal{G}={0}, G={1}$ for pedants like me. – Torsten Schoeneberg Apr 29 '20 at 16:02
  • The exponential map is not, in general, a local diffeomorphism. For example, on $SU(2)$, the exponential map has conjugate locus consisting of a $2$-sphere of radius $\pi$ in $\mathfrak{su}(2)\cong \mathbb{R}^3$. (I'm implicitly assuming $SU(2)$ is equipped with a bi-invariant metric scaled in such a way that the Riemanian exponential map coincides with the group exponential map, but this can always be done on a compact Lie group.) It is diffeomophism onto its image when restricted to a small enough neighborhood of $0$. – Jason DeVito - on hiatus Apr 29 '20 at 16:05