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Kind of complex manifold
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
Complex_torus
Doughnut-shaped surface of revolution
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
Torus
Describes the line bundles on a complex torus or complex abelian variety
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
Appell–Humbert_theorem
Projective variety that is also an algebraic group
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
Abelian_variety
Knot which lies on the surface of a torus in 3-dimensional space
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
Torus_knot
Generalisation of Jacobian variety
\operatorname {Alb} (V)} such that any morphism to a complex torus factors uniquely through this map. (The complex torus Alb ( V ) {\displaystyle \operatorname
Albanese_variety
Theory of a class of elliptic curves
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
Complex_multiplication
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
Torus_action
Geometrical object in four-dimensional space
In differential geometry, the Clifford torus is the standard embedding of the 2-torus as a product of circles in Euclidean space R4 (equivalently C2).
Clifford_torus
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
Lattès_map
(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
Abelian_Lie_group
Lie group whose manifold is complex and whose group operation is holomorphic
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
Complex_Lie_group
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
Diagonalizable_group
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
Intermediate_Jacobian
Concept in algebraic geometry
higher-dimensional tori or abelian varieties. Finding criteria for a complex torus of dimension 2 to be a product of two elliptic curves (up to isogeny)
Abelian_surface
the Jacobian variety J(C) of C. In the complex case, again, the result follows since J(C) is a complex torus of dimension 2g. Over a general field, see
Theta_characteristic
Group in algebraic geometry
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
Néron–Severi_group
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
Riemann_form
Common type of fracture in children
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
Torus_fracture
Smooth closed surface with g holes
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
Genus_g_surface
One-dimensional complex manifold
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
Riemann_surface
Complex number whose mapping on a coordinate plane produces a triangular lattice
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
Eisenstein_integer
Riemannian manifold with SU(n) holonomy
of a complex torus of complex dimension 2, which have vanishing first integral Chern class but non-trivial canonical bundle. For a compact complex n {\displaystyle
Calabi–Yau_manifold
Term in mathematics
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
Jacobian_variety
Differentiable manifold
compact torus. It has been shown that every principal torus bundle over a torus is of this form. More generally, a compact nilmanifold is a torus bundle
Nilmanifold
Maximal compact connected Abelian Lie subgroup
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
Maximal_torus
German mathematician (1804–1851)
Riemann theta function for algebraic curves of arbitrary genus. The complex torus associated to a genus g {\displaystyle g} algebraic curve, obtained
Carl_Gustav_Jacob_Jacobi
Describes when a compact Riemann surface is determined by its Jacobian variety
form of a principally polarized abelian variety. In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The same statement
Torelli_theorem
Simplest non-trivial closed knot with three crossings
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
Trefoil_knot
Modular function in mathematics
isomorphism class of elliptic curves. Every elliptic curve E over C is a complex torus, and thus can be identified with a rank 2 lattice; that is, a two-dimensional
J-invariant
Class of mathematical functions
^{2}}}\right).} This series converges locally uniformly absolutely in the complex torus C / Λ {\displaystyle \mathbb {C} /\Lambda } . It is common to use 1
Weierstrass_elliptic_function
Specific algebraic group
In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by G m {\displaystyle \mathbf {G} _{\mathbf {m} }} , G m {\displaystyle
Algebraic_torus
Mathematical function
in u {\displaystyle u} , they factor through a torus – in effect, their domain can be taken to be a torus, just as cosine and sine are in effect defined
Jacobi_elliptic_functions
Concept in string theory
localization. This applies when X is toric, meaning that it is acted upon by a complex torus, or at least locally toric. Then one can use the Atiyah–Bott fixed-point
Gromov–Witten_invariant
In differential geometry, the Angenent torus is a smooth embedding of the torus into three-dimensional Euclidean space, with the property that it remains
Angenent_torus
Map from a Lie algebra to its Lie group
\,} that is, the same formula as the ordinary complex exponential. More generally, for complex torus X = C n / Λ {\displaystyle X=\mathbb {C} ^{n}/\Lambda
Exponential_map_(Lie_theory)
general, any point Ω ∈ H g {\displaystyle \Omega \in H_{g}} gives a complex torus X Ω = C g / ( Ω Z g + Z g ) {\displaystyle X_{\Omega }=\mathbb {C} ^{g}/(\Omega
Moduli_of_abelian_varieties
Topics referred to by the same term
the commutator subgroup is abelian Abelianisation Abelian variety, a complex torus that can be embedded into projective space Abelian surface, a two-dimensional
Abelian
Special functions of several complex variables
multi-dimensional periodic systems, such as crystal lattices or points on a torus. Because they are smooth, they allow the study and manipulation of discrete
Theta_function
Conformal field theory on a 2D spacetime
OPE. For example, the torus partition function is invariant under the action of the modular group on the modulus of the torus, equivalently Z ( τ ) =
Two-dimensional conformal field theory
Two-dimensional_conformal_field_theory
Algebraic structure associated with a topological space
The torus is defined as a product of two circles T 2 = S 1 × S 1 {\displaystyle T^{2}=S^{1}\times S^{1}} . The torus has a single path-connected
Homology_(mathematics)
Branch of mathematics
a topologist cannot distinguish a coffee mug from a doughnut. A pliable torus (shaped like a doughnut) can be reshaped to a coffee mug by creating a dimple
Topology
Manifold with Riemannian, complex and symplectic structure
transport. Complex space C n {\displaystyle \mathbb {C} ^{n}} with the standard Hermitian metric is a Kähler manifold. A compact complex torus C / Λ {\displaystyle
Kähler_manifold
^{*}} . This is useful when studying complex tori because the character group of the lattice in a complex torus V / Λ {\displaystyle V/\Lambda } is canonically
Character_group
Pseudometric of complex manifolds
compact complex spaces. Mark Green used Brody's argument to characterize hyperbolicity for closed complex subspaces X of a compact complex torus: X is hyperbolic
Kobayashi_metric
Ruled surface over the projective line
surface Σ n {\displaystyle \Sigma _{n}} can be given an action of the complex torus T = C ∗ × C ∗ {\displaystyle T=\mathbb {C} ^{*}\times \mathbb {C} ^{*}}
Hirzebruch_surface
Fiber bundle whose fibers are projective spaces
Elencwajg, G.; Narasimhan, M. S. (1983), "Projective bundles on a complex torus", Journal für die reine und angewandte Mathematik, 1983 (340): 1–5,
Projective_bundle
Gauge theory providing unifying formalism for integrable systems
has no poles and C = C / Λ {\displaystyle C=\mathbb {C} /\Lambda } a complex torus (with Λ {\displaystyle \Lambda } a 2d lattice). If g = 0 {\displaystyle
Four-dimensional Chern–Simons theory
Four-dimensional_Chern–Simons_theory
Algebraic stack in mathematics
inside of C {\displaystyle \mathbb {C} } , there is an embedding of the complex torus E Λ = C / Λ {\displaystyle E_{\Lambda }=\mathbb {C} /\Lambda } into
Moduli stack of elliptic curves
Moduli_stack_of_elliptic_curves
Disorder of the jaw
An oral torus - also known as: dental torus - is an oral condition in which bony growth occurs in the mouth; there are three locations in which oral tori
Oral_torus
Algebraic variety containing an algebraic torus
algebraic geometry, a toric variety or torus embedding is a kind of algebraic variety that contains an algebraic torus whose group action extends to the whole
Toric_variety
Bony ridge located above the eye sockets of all primates
paleoanthropologists distinguish between frontal torus and supraorbital ridge. In anatomy, a torus is a projecting shelf of bone that unlike a ridge
Brow_ridge
Magnetic confinement device used to produce thermonuclear fusion power
magnetic field inside the torus; a pulsed magnetic field through the hole in the torus induces the axial current in the torus which has a poloidal magnetic
Tokamak
Space of complex matrices with positive definite imaginary part
\Lambda _{\tau }=\mathbb {Z} ^{g}+\tau \mathbb {Z} ^{g}} defines a complex torus A τ = C g / Λ τ . {\displaystyle A_{\tau }=\mathbb {C} ^{g}/\Lambda
Siegel_upper_half-space
Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus. In
Loewner's_torus_inequality
Scorza applies the term "abelian variety" to complex tori. 1921 Solomon Lefschetz shows that any complex torus with Riemann matrix satisfying the necessary
Timeline_of_abelian_varieties
Mathematical question in algebraic geometry
classical case, over the complex number field, has received most of the attention, and then an abelian variety A is simply a complex torus of a particular type
Schottky_problem
Branch of mathematics
resembles Euclidean space. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle
Algebraic_topology
Type of motion that is approximately periodic
motion on a torus that never comes back to the same point. This behavior can also be called quasiperiodic evolution, dynamics, or flow. The torus may be a
Quasiperiodic_motion
{\displaystyle G=\langle x,y|xyx^{-1}y^{-1}\rangle .} Then the presentation complex for G is a torus, obtained by gluing the opposite sides of a square, the 2-cell
Presentation_complex
Surface of genus g Torus Double torus 3-sphere, S3 3-torus, T3 Poincaré homology sphere SO(3) ≅ RP3 Solid Klein bottle Solid torus Whitehead manifold
List_of_manifolds
Three-holed sphere
The only other orientable surface with this property is the punctured torus (a torus minus an open disk). This subsection assumes orientability, analogous
Pair_of_pants_(mathematics)
American mathematician (born 1945)
and Sperber considered holomorphic functions from the n-fold complex torus to the complex numbers, defined by a Laurent polynomial. They introduced the
Steven_Sperber
Cohomology with real coefficients computed using differential forms
Betti number of a 2 {\displaystyle 2} -torus is two. More generally, on an n {\displaystyle n} -dimensional torus T n {\displaystyle T^{n}} , one can consider
De_Rham_cohomology
and no curves. K. Hasegawa gives a list of all complex 2-dimensional solvmanifolds; these are complex torus, hyperelliptic surface, Kodaira surface and Inoue
Inoue_surface
Non-orientable mathematical surface
image of the other, yield a fundamental region of the torus. The universal cover of both the torus and the Klein bottle is the plane R2. The fundamental
Klein_bottle
Representation of mathematical space
triangulation of the torus. The projective plane P 2 {\displaystyle \mathbb {P} ^{2}} admits a triangulation (see CW-complexes) One can show that differentiable
Triangulation_(topology)
Manifold equipped with a quaternionic structure
-dimensional compact torus. It is remarkable that any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus. Hypercomplex
Hypercomplex_manifold
Russian manual docking system for Soyuz and Progress spacecraft
TORU (Tele-robotically Operated Rendezvous Unit, Russian: Телеоператорный Режим Управления, lit. 'Teleoperator Control Mode') is a manual docking system
TORU
Term in geometric topology
{\displaystyle a(0;0,1)=\{z\in \mathbb {C} :0<|z|<1\}} in the complex plane. By extending to the torus the twisting map ( e i θ , t ) ↦ ( e i ( θ + 2 π t ) ,
Dehn_twist
Two-dimensional manifold
as a 'closed' surface. The two-dimensional sphere, the two-dimensional torus, and the real projective plane are examples of closed surfaces. The Möbius
Surface_(topology)
Knot invariant
sufficiently large crossing numbers, non-alternating torus knots will have a lower ropelength than alternating torus knots. This is seen empirically even at low
Ropelength
Class of magnetic fusion energy devices
bumpy torus is a class of magnetic fusion energy devices that consist of a series of magnetic mirrors connected end-to-end to form a closed torus. It is
Bumpy_torus
Abstraction useful in the construction and triangulation of topological spaces
Hatcher's Algebraic Topology. Consider the Δ-set structure given to the torus in the figure, which has one vertex, three edges, and two 2-simplices. The
Delta_set
Figure-eight-shaped curve
century AD. Proclus considered the cross-sections of a torus by a plane parallel to the axis of the torus. As he observed, for most such sections the cross
Lemniscate
United States government initiative
digital twin of the Lab's primary fusion experiment, the National Spherical Torus Experiment-Upgrade, and a new computational infrastructure called STELLAR-AI
Genesis_Mission
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
infinity"). Each torus is the stereographic projection of the inverse image of a circle of latitude of the 2-sphere. (Topologically, a torus is the product
Hopf_fibration
In algebraic geometry, a GKM variety is a complex algebraic variety equipped with a torus action that meets certain conditions. The concept was introduced
GKM_variety
Linear accelerator
McLean, H. S. (14 January 1991). "Quasistatic compression of a compact torus". Physical Review Letters. 66 (2): 165–168. Bibcode:1991PhRvL..66..165M
Plasma_railgun
Algebraic curve in mathematics
Weierstrass functions are naturally defined on a torus T = C/Λ. This torus may be embedded in the complex projective plane by means of the map z ↦ [ 1 :
Elliptic_curve
formal parameter is zero it degenerates to a pinched torus, and when it is nonzero it is a torus). The j-invariant of the Tate curve is given by a power
Tate_curve
2006 film directed by Kenji Kamiyama
Ghost in the Shell: Stand Alone Complex – Solid State Society (Japanese: 攻殻機動隊 STAND ALONE COMPLEX Solid State Society, Hepburn: Kōkaku Kidōtai Sutando
Ghost in the Shell: Stand Alone Complex – Solid State Society
Ghost_in_the_Shell:_Stand_Alone_Complex_–_Solid_State_Society
Universal construction of a complex Lie group from a real Lie group
K contains a maximal torus of G, so has maximal rank. On the other hand, the centralizer of the subgroup generated by the torus S of elements exp tT is
Complexification_(Lie_group)
Number of "holes" of a surface
genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. The genus of a connected, orientable surface is an integer
Genus_(mathematics)
Relation between genus, degree, and dimension of function spaces over surfaces
case is a Riemann surface of genus g = 1 {\displaystyle g=1} , such as a torus C / Λ {\displaystyle \mathbb {C} /\Lambda } , where Λ {\displaystyle \Lambda
Riemann–Roch_theorem
Roughly, the number of k-dimensional holes on a topological surface
the number of two-dimensional "voids" or "cavities". Thus, for example, a torus has one connected surface component so b0 = 1, two "circular" holes (one
Betti_number
Toroidal polyhedron with 7 faces
geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. The tetrahedron and the Szilassi polyhedron
Szilassi_polyhedron
Three dimensional analogue of uniformization conjecture
the mapping torus of an Anosov map of a torus has a finite volume solv structure, but its JSJ decomposition cuts it open along one torus to produce a
Geometrization_conjecture
Region between two concentric circles
of redirect targets Spherical shell – Three-dimensional geometric shape Torus – Doughnut-shaped surface of revolution Haunsperger, Deanna; Kennedy, Stephen
Annulus_(mathematics)
Cohomology theory for Lie algebras
assumption, as this is the first cohomology group of the n {\displaystyle n} -torus viewed as an abelian group, and R n {\displaystyle \mathbb {R} ^{n}} can
Lie_algebra_cohomology
Study of the topology of a complex manifold
torus. Then, the Picard-Lefschetz formula reads w j ( γ ) = γ − δ {\displaystyle w_{j}(\gamma )=\gamma -\delta } if the j {\displaystyle j} -th torus
Picard–Lefschetz_theory
Building in Beijing, China
southwestern corner of Chaoyang Park in Beijing. The centre features a torus-like design. The building's sculptural shape originates from the "Möbius
Phoenix_Center
an algebraically closed field K {\displaystyle K} with a split maximal torus T {\displaystyle T} then its root datum is a quadruple ( X ∗ , Φ , X ∗
Root_datum
Theorem in differential topology
This is a consequence of the Poincaré–Hopf theorem. In the case of the torus, the Euler characteristic is 0; and it is possible to "comb a hairy doughnut
Hairy_ball_theorem
Magnetic field around the Jovian system
large torus around the planet. Jupiter's magnetic field forces the torus to rotate with the same angular velocity and direction as the planet. The torus in
Magnetosphere_of_Jupiter
Symmetric bipartite cubic graph with 16 vertices and 24 edges
embedding into a triple torus (genus 3 torus) that is a regular map having four 12-gonal faces, and is the Petrie dual of the double torus embedding described
Möbius–Kantor_graph
Electricity generation by nuclear fusion
approach. This method drives hot plasma around in a magnetically confined torus, with an internal electric current. When completed, ITER will become the
Fusion_power
Subgroup of a root system's isometry group
given a torus T < G (which need not be maximal), the Weyl group with respect to that torus is defined as the quotient of the normalizer of the torus N = N(T)
Weyl_group
Whitney umbrella Châtelet surfaces Dupin cyclides, inversions of a cylinder, torus, or double cone in a sphere Gabriel's horn Right circular conoid Roman surface
List of complex and algebraic surfaces
List_of_complex_and_algebraic_surfaces
R(G)} is K 0 G ( ∗ ) {\displaystyle K_{0}^{G}(*)} . Given an algebraic torus T ≅ G m k {\displaystyle \mathbb {T} \cong \mathbb {G} _{m}^{k}} a finite-dimensional
Equivariant algebraic K-theory
Equivariant_algebraic_K-theory
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