Flat torus

A flat torus is a torus with a Riemannian metric of zero Gaussian curvature. It can be formed by identifying points in the Euclidean plane that differ by vectors in a lattice.

Lattice: A discrete subgroup of a vector space generated by linearly independent vectors. Its two generators specify the translations used to form a flat torus.
Riemannian metric: A smoothly varying inner product on tangent spaces that defines lengths and angles on a manifold. A flat torus is defined by equipping its underlying surface with this structure.
Standard torus: A doughnut-shaped surface of revolution embedded in three-dimensional Euclidean space. Its induced metric has varying curvature, unlike the everywhere-flat metric of a flat torus.
Periodic boundary conditions: Boundary conditions that identify opposite edges of a domain so fields repeat periodically. They model functions on a flat torus using a lattice parallelogram.
Euclidean plane: The two-dimensional plane with its standard Euclidean distance and angle. A flat torus inherits its metric from the plane before lattice points are identified.
Torus: A surface topologically equivalent to the product of two circles. The lattice quotient has torus topology even though its local geometry is Euclidean.
Sphere: The set of points at a fixed distance from a center in Euclidean space. A sphere cannot carry a smooth metric of zero curvature everywhere while retaining its usual topology.
Fourier analysis on the torus: The study of periodic functions through their expansion in discrete Fourier modes. Lattice geometry determines the allowed frequencies and spectrum of the flat torus.
Quotient space: A space formed by treating specified points of another space as equivalent. Identifying lattice translates turns the plane into a compact torus.
Flat manifold: A Riemannian manifold whose sectional curvature vanishes everywhere. Flat tori are compact two-dimensional examples of this broader geometric class.

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