# Torus

The surface of the torus with as radius vector, in the Cartesian coordinates of the Euclidean space $E ^{3}$ , $$r = a \ \mathop{\rm sin}\nolimits \ u \mathbf k + l (1 + \epsilon \ \cos \ u) ( \mathbf i \ \cos \ v + \mathbf j \ \mathop{\rm sin}\nolimits \ v)$$ ( here $(u,\ v)$ are the intrinsic coordinates, $a$ is the radius of the rotating circle, $l$ is the radius of the axial circle, and $\epsilon = a/l$ is the eccentricity), is often also called a torus. Its line element is $$ds ^{2} = a ^{2} \ du ^{2} + l ^{2} (1 + \epsilon \ \cos \ u) ^{2} \ dv ^{2} ,$$ and its curvature is $K = ( \cos \ u)/al (1 + \epsilon \ \cos \ u)$ . A torus is a special case of a surface of revolution and of a canal surface.
From the topological point of view, a torus is the product of two circles, and therefore a torus is a two-dimensional closed manifold of genus one. If this product is metrizable, then it can be realized in $E ^{4}$ or in the elliptic space $El ^{3}$ as a Clifford surface; its equation in $E ^{4}$ , for example, is $$x _{1} ^{2} + x _{2} ^{2} = a ^{2} , x _{3} ^{2} + x _{4} ^{2} = b ^{2} .$$ A natural generalization of a torus is a multi-dimensional torus, i.e. the topological product of several copies of the circle $S$ , i.e. of the manifold of all complex numbers equal to one in modulus. The natural smoothness on $S$ determines on the torus the structure of a smooth manifold, and the natural multiplicative structure induces on the torus the structure of a connected compact commutative real Lie group. These latter groups play an important part in the theory of Lie groups and they are also called tori (see Lie group, compact; Maximal torus; Cartan subgroup). An even-dimensional torus admits a complex structure; when certain conditions are satisfied such a structure transforms the torus into an Abelian variety (see also Complex torus). In the structure theory of algebraic groups, a torus, like a real Lie group, has an important analogue, an algebraic torus (see also Algebraic group; Linear algebraic group). An algebraic torus is not a torus itself (if the ground field is that of the complex numbers), but presents a subgroup that is a torus and onto which it can be contracted (as a topological space). More accurately, an algebraic torus, as a Lie group, is isomorphic to the product of a certain torus and several copies of the multiplicative group of positive real numbers.