# 12. Smooth Actions of Lie Groups

We define the *exponential map* $ \exp \colon\mathfrak{g} \to G$ of a Lie group and prove it is a diffeomorphism on a neighbourhood of $0 \in \mathfrak{g}$.

The reason for the name is that when $G$ is a matrix Lie group, the exponential map is given by matrix exponentiation $A \mapsto e^A$.

We then look a Lie groups *acting* on manifolds.

If you think back to your introductory course on group theory, you will no doubt remember that one of the most important reasons to study groups is because we can let them act on sets. In fact, the notion of a group acting on a set is arguably more fundamental than that of a group itself, and certainly historically, the idea of a “transformation group” (i.e. an action of a group on a given set) predates that of an abstract group. One might even go so far as to say that groups are interesting precisely *because* of their actions.

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