Torus T^1 - Knowlpedia

the real Lie group $\mathbb T^{1}$ is $U(1)^{1}$ with coordinatewise multiplication; equivalently it is $\mathbb R^{1}/(2\pi\mathbb Z)^{1}$ with addition.

Dimensions and structure

This group has real dimension $1$ . It is compact, connected and abelian. Its universal covering group is additive $\mathbb R^{1}$ . The complex multiplicative group has a different real dimension and is noncompact.

The one-dimensional torus is the circle group by its definition. See $\operatorname{U}(1)$ .

Direct verification

Coordinatewise multiplication and conjugation preserve the unit-circle equations. Each circle contributes one real dimension; the exponential map from Rⁿ has kernel 2πZⁿ.