Knowlpedia

Definition

Flat torus

A Euclidean space modulo a full-rank lattice, with its quotient distance and local Euclidean coordinates.

For a Λ⊂Rn\Lambda\subset\mathbb R^n, the flat torus Rn/Λ\mathbb R^n/\Lambda consists of classes [x]=x+Λ[x]=x+\Lambda with distance

d([x],[y])=min⁡λ∈Λ∣x−y−λ∣.d([x],[y])=\min_{\lambda\in\Lambda}|x-y-\lambda|.

Discreteness makes the minimum well-defined, and the expression is independent of representatives. It is a metric: zero distance means equal classes, symmetry follows from −Λ=Λ-\Lambda=\Lambda, and the Euclidean triangle inequality descends to the quotient.

Local coordinates and normalization

Balls smaller than half the shortest nonzero lattice length have injective Euclidean lifts; still smaller balls preserve all pairwise Euclidean distances. These charts specify smooth functions and the local Euclidean geometry. A compact fundamental parallelepiped maps onto the torus. The unit torus is Tn=Rn/Zn\mathbb T^n=\mathbb R^n/\mathbb Z^n; an angular convention uses Rn/(2πZ)n\mathbb R^n/(2\pi\mathbb Z)^n. The related emphasizes its group structure instead of this metric.