Real torus (${\mathbb{T}}^n={\mathbb{R}}^n/{\mathbb{Z}}^n$)

A real torus is the compact Lie group obtained by identifying real vectors differing by integer vectors, with addition induced from Euclidean addition. It is a finite product of circle groups and has normalized Haar measure. Dimension two is the familiar surface torus, but toral dynamics and Fourier characters naturally use arbitrary finite dimension.