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Let $G$ be a lie group, then it is claimed that the exponential map

$$\exp: Lie(G) \rightarrow G$$

is a smooth map. I want to know - what is the smooth structure of $Lie(G)$?

I am supposing it is identified with $T_eG$ and we show that $T_eG$ is an embedded submanifold of $TG$.

Bryan Shih
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    There is a canonical smooth structure on any finite-dimensional (real) vector space described here: https://math.stackexchange.com/questions/160298/manifold-structure-on-on-a-finite-dimensional-real-vector-space – MaoWao Jan 21 '19 at 12:31

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