Local exponential chart
= Local exponential chart
{title2=$\log:\exp(U)\longrightarrow U\subseteq\mathfrak g$}
Since the differential of the <Exponential map of a Lie group> at zero is the identity, the <inverse function theorem> makes it a <diffeomorphism> on a sufficiently small neighborhood of zero. Its local inverse is a logarithm near the group identity. It need not extend to a global logarithm, even for the real <general linear group>.