Exponential map - Knowlpedia

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}=T_eG$ . For each $X\in\mathfrak{g}$ , let $X^L$ denote the left-invariant vector field on $G$ determined by $X$ (so $X^L_e=X$ ). Let $\gamma_X:\mathbb R\to G$ be the unique solution of $\gamma_X\prime(t)=X^L_{\gamma_X(t)}$ , $\gamma_X(0)=e$ . A left-invariant vector field is complete, so this curve is defined for every real $t$ .

The (Lie group) exponential map is

\exp:\mathfrak{g}\to G,\qquad \exp(X)=\gamma_X(1).

Local properties

It satisfies $\exp(0)=e$ and its differential at $0$ is the identity map on $\mathfrak{g}$ . Moreover, $\exp$ is a local diffeomorphism near $0\in\mathfrak{g}$ (so it provides coordinates near the identity in $G$ ).

Examples

Matrix groups. For $G\subset \mathrm{GL}(n,\mathbb{R})$ a matrix Lie group, $\exp$ agrees with the matrix exponential $X\mapsto \sum_{k\ge0}\frac{1}{k!}X^k$ (and lands in $G$ when $X\in\mathfrak{g}$ ).
Additive group. For $G=\mathbb{R}^n$ under addition, $\mathfrak{g}\cong\mathbb{R}^n$ and $\exp$ is the identity map.
Circle group. For $G=S^1\subset\mathbb{C}$ , $\mathfrak{g}\cong\mathbb{R}$ and $\exp(t)=e^{it}$ (up to the chosen identification of $\mathfrak{g}$ with $\mathbb{R}$ ).