= Exponential map of a Lie group
{c}
{title2=$\exp:\mathfrak g\to G$}
= Lie exponential map
{c}
{synonym}
For any <Lie group>, the <Exponential map of a Lie group> is defined by $\exp(X)=\gamma_X(1)$, where $\gamma_X$ is the <one-parameter subgroup> with initial <tangent vector> $X$. For a <Matrix Lie group>, it is the <matrix exponential> $\exp X=\sum_{n\geq0}X^n/n!$. Its <differential> at zero is the identity, so the <inverse function theorem> supplies a <local exponential chart>. A small transformation can therefore be encoded by a <vector> in the <Lie algebra>. See https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec07.pdf[MIT 18.745, Lecture 7, Definition 7.2 and Theorem 7.5].
Local invertibility makes no promise about a global inverse. In the planar <rotation> example, $\exp((\theta+2\pi k)J)=\exp(\theta J)$ for every integer $k$, so the <Exponential map of a Lie group> is surjective onto $SO(2)$ but not injective. Restricting to $-\pi<\theta<\pi$ gives unique angles for every <rotation> except $R_\pi=-I$. The omitted half-turn exposes the cut needed for this single angle chart.
Surjectivity can also fail, even in a connected <Lie group>. The <matrix> $A=\operatorname{diag}(-2,-1/2)$ lies in the <special linear group> $SL_2(\mathbb R)$ but has no real <matrix logarithm>: any logarithm must commute with $A$, hence preserve each of its distinct real <eigenvectors>, where exponentiating a real scalar gives a positive number. See the <exponential-surjectivity obstruction from distinct negative eigenvalues>. These two examples distinguish two different global obstacles: repeated parameters and unreachable elements. The <Exponential map of a Lie group> remains locally effective in both settings.
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