= Classification of connected abelian Lie groups
{title2=$G\cong\mathbb R^n/\Lambda\cong\mathbb T^r\times\mathbb R^{n-r}$}
For a connected <abelian Lie group>, the <Lie exponential map> is an additive <group homomorphism> and a local <diffeomorphism> at zero. Its image is an open <subgroup>, hence the whole connected <group>. Its kernel is a <discrete additive subgroup of a real vector space>. A basis of that kernel identifies it with $\mathbb Z^r$ inside $\mathbb R^n$, giving the displayed classification. The compact factors record periods of <one-parameter subgroups>.
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