In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian...

is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis...

and Albanese varieties). A complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. It can always be...

In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies...

groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus (not to be confused with the complex Lie group C ∗ {\displaystyle...

Z[i] is the Gaussian integer ring, and θ is any non-zero complex number. Any such complex torus has the Gaussian integers as endomorphism ring. It is known...

considers an action of a real or complex torus on a manifold (or an orbifold). A normal algebraic variety with a torus acting on it in such a way that...

and torus admit complex structures but the Möbius strip, Klein bottle and real projective plane do not. Every compact Riemann surface is a complex algebraic...

(real) compact Lie group is a torus; i.e., a Lie group isomorphic to ( S 1 ) h {\displaystyle (S^{1})^{h}} . A connected complex Lie group that is a compact...

itself is orthogonal to that axis. The word "torus" originates from the Latin word "protuberance." Torus fractures are low risk and may cause acute pain...

rational map f = ΘLΘ−1 from the complex sphere to itself such that Θ is a holomorphic map from a complex torus to the complex sphere and L is an affine map...

Stated another way, a torus embedded in R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship...

the definitionpg 30. For a complex torus X = V / Λ {\displaystyle X=V/\Lambda } , where V {\displaystyle V} is a complex vector space of dimension n...

In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g distinct tori: the interior...

compact torus. It has been shown that every principal torus bundle over a torus is of this form, see. More generally, a compact nilmanifold is a torus bundle...

polarized abelian variety, of dimension g, and hence, over the complex numbers, it is a complex torus. If p is a point of C, then the curve C can be mapped to...

algebraic torus (which is not necessarily compact, in contrast to a complex torus). A k-torus is a torus defined over k. The centralizer of a maximal torus is...

of weight 6. The quotient of the complex plane C by the lattice containing all Eisenstein integers is a complex torus of real dimension 2. This is one...

curve. Hodge theory Complex torus Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse...

intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and...

In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by G m {\displaystyle \mathbf {G} _{\mathbf {m} }} , G m {\displaystyle...

this happens are hyperelliptic surfaces, finite quotients of a complex torus of complex dimension 2, which have vanishing first integral Chern class but...

3t\end{aligned}}} The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus ( r − 2 ) 2 + z 2 = 1...

alternatization of its Chern class is the given Riemann form. Furthermore, the complex torus Cg/Λ admits the structure of an abelian variety if and only if there...

group with the real torus given above. In fact, this torus can be equipped with a complex structure, giving the dual complex torus. Explicitly, a line...

Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group G is a compact, connected...

the commutator subgroup is abelian Abelianisation Abelian variety, a complex torus that can be embedded into projective space Abelian surface, a two-dimensional...

Riemann theta function for algebraic curves of arbitrary genus. The complex torus associated to a genus g {\displaystyle g} algebraic curve, obtained...

immersed in Euclidean 3-space). Like the torus, cycles a and b cannot be shrunk while c can be. But unlike the torus, following b forwards right round and...

variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends...