Torus

A body obtained from the rotation of a closed circle around an axis lying in the plane of this circle and not intersecting it. The centre of the circle describes a circle, called the axial circle of the torus, its centre is called the centre of the torus. The plane of the axial circle is called the equatorial plane of the torus, and the boundaries of the circles lying on the torus and obtained by its rotation from the given circle are said to be meridians of the 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.

References

[a1]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French) MR0903026 MR0895392 MR0882916 MR0882541 Zbl 0619.53001 Zbl 0606.51001 Zbl 0606.00020
[a2]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) MR0917479 Zbl 0629.53001
[a3]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. 132; 356; 374 MR0346644 Zbl 0181.48101