Exponential map of a Lie group

Summary

Let $G$ be a Lie group with Lie algebra $\mathfrak{g} = T_eG$. The exponential map sends $X \in \mathfrak{g}$ to the value at time $1$ of the one-parameter subgroup with initial velocity $X$. For matrix groups it is the matrix exponential. It restricts to a diffeomorphism from a neighbourhood of $0$ onto a neighbourhood of the identity. So it carries information from the Lie algebra to the group. It is the main tool for passing between Lie groups and Lie algebras, and it differs in general from the exponential map of Riemannian geometry.