nLab
torus

Definition

The torus is the manifold obtained as the quotient

$$T:={ℝ}^2/{\mathbb{Z}}^2$$T := \mathbb{R}^2 / \mathbb{Z}^2

of the Cartesian plane, regarded as an abelian group, by the subgroup of pairs of integers.

More generally, for $n\in \mathbb{N}$ any natural number, the $n$-torus is

$$T:={ℝ}^n/{\mathbb{Z}}^n.$$T := \mathbb{R}^n / \mathbb{Z}^n \,.

For $n=1$ this is the circle.

Properties

The torus naturally inherits the structure of a Lie group.

Related concepts

• toric variety