The n-torus and its exponential lattice

Integer translations preserve the standard smooth charts, so addition and negation descend to smooth operations on the quotient; hence [F1] gives an $n$-dimensional Lie group with tangent space ${\mathbb{R}}^n$ at the identity.

For $X\in {\mathbb{R}}^n$, the curve $t\mapsto [tX]$ is a one-parameter subgroup with initial velocity $X$. By [F2], its time-one point is ${\exp }_{{\mathbb{T}}^n}(X)=[X]$. This equals the identity exactly when $X\in {\mathbb{Z}}^n$.

At $n=0$ the torus and kernel are trivial; at $n=1$ this is the circle quotient. The lattice is discrete but no nondegeneracy is asserted. There is no metric, endpoint issue, or biconditional beyond the direct kernel calculation. The assumed ${\mathrm{A}\mathrm{C}}_{\omega }$ is used by [F2] through its stated supplier chain, and the fixed integer lattice adds no choice.