Torus

Definition

A torus is a particular kind of Riemann surface. it is a Riemann surface obtained as the quotient of $\mathbb {C}$ by a lattice.

A torus is the same as an elliptic curve, though the latter description is typically used when we want to study it as an algebraic object. Topologically, a torus is a topological torus: a product of circles.

Not all tori are isomorphic as Riemann surfaces: in fact, two tori are isomorphic as Riemann surfaces iff the corresponding lattices are related by a scaling and rotation.