Exponential map is a local diffeomorphism

Let $G$ be a Lie group with Lie algebra $\mathfrak g$ and exponential map $\exp:\mathfrak g\to G$.

Theorem

There exist open neighborhoods $U\subset \mathfrak g$ of $0$ and $V\subset G$ of the identity element $e$ such that

is a smooth diffeomorphism.

Equivalently, $\exp$ is a local diffeomorphism at $0$.

Key points

• The differential at the origin is the identity map once we identify $T_0\mathfrak g\cong \mathfrak g$ and $T_eG\cong \mathfrak g$: $d(\exp)_0 = \mathrm{id}_{\mathfrak g}.$
• By the inverse function theorem, this implies local invertibility; the local inverse is the logarithm map $\log:V\to U$.

Context

This result supplies canonical local coordinates near the identity and underlies the Baker–Campbell–Hausdorff formula , which describes the group law on $V$ in terms of the Lie bracket on $\mathfrak g$ after transporting multiplication through $\log$ and $\exp`.