S1×S1

S1×S1, often called the two-dimensional torus or T^2, is the Cartesian product of two circles. As a topological space it consists of pairs (θ, φ) with θ and φ in S1, and it inherits a natural product topology. It can be realized as the quotient R^2/Z^2, equivalently the unit square with opposite edges identified, giving a compact, connected, and orientable 2-manifold of genus 1. The Euler characteristic is 0.

Geometrically, S1×S1 carries a natural Riemannian metric as the product of the standard circle metrics, making it a flat 2-manifold in that sense. However, the common embedding of the torus in R^3 as a doughnut-shaped surface has nonuniform Gaussian curvature, illustrating the difference between intrinsic flat metrics and extrinsic curvature.

As a Lie group, S1×S1 is a compact abelian Lie group, isomorphic to the product of two copies of the circle. Its fundamental group is Z×Z, and its homology groups are H0 ≅ Z, H1 ≅ Z^2, and H2 ≅ Z. The cohomology ring reflects two independent 1-forms generating H1. When viewed as a complex torus, it can be modeled as C/Λ for a lattice Λ, endowing it with a natural complex-analytic structure of genus 1.

S1×S1 is used as a fundamental example in topology, geometry, and dynamical systems, illustrating concepts such as product manifolds, compact Lie groups, and toroidal geometry. It also serves as a model for the configuration space of two independent angular coordinates and as a basic object in the study of torus actions and moduli.