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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
twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus). If the axis of revolution passes through the
Torus
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
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
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
(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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
^{*}} . 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
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
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)
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
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
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
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
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
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
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
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
{\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
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
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
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
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
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)
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
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
Three-holed sphere
−1, and the only other surface with this property is the punctured torus (a torus minus an open disk). The importance of the pairs of pants in the study
Pair_of_pants_(mathematics)
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
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
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
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
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)
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
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
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
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
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
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
Method of describing higher-order polyhedra
square torus, {4,4}1,0 A regular 4x4 square torus, {4,4}4,0 tQ24×12 projected to torus taQ24×12 projected to torus actQ24×8 projected to torus tH24×12
Conway_polyhedron_notation
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
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
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)
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
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
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
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
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
Theorem in geometric topology
ball (which is known in mathematics as the two-dimensional sphere) or of a torus, are two-dimensional. The surface of a ball has trivial fundamental group
Poincaré_conjecture
Mathematics concept
the complex field to the real field, an algebraic torus whose character group consists of the two homomorphisms to GL(1), interchanged by complex conjugation
Mumford–Tate_group
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
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)
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
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
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
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
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
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
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
Theory in algebraic topology
n-torus ( S 1 ) n {\displaystyle (S^{1})^{n}} can be constructed as the CW complex with one 0-cell, n 1-cells, ..., and one n-cell. The chain complex is
Cellular_homology
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
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)
Simple curve of Euclidean geometry
conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are
Circle
Form of differential geometry
{2}{\sqrt {3}}}\cdot \mathrm {area} } for the torus, where the case of equality is attained by the flat torus whose deck transformations form the lattice
Systolic_geometry
Type of algebraic equation
n2-fold covering map from a 2-torus to itself given by the mapping x → n·x on the underlying group) expressed in terms of complex analysis. Modular lambda
Modular_equation
Mathematical conjecture
pair of Lagrangian torus fibrations X , X ^ → B {\displaystyle X,{\hat {X}}\to B} which encodes mirror symmetry is the dual torus fibres of the fibration
SYZ_conjecture
COMPLEX TORUS
COMPLEX TORUS
Girl/Female
Arabic, Muslim
Complex; Zigzag; Curling
Boy/Male
Tamil
Complete
Boy/Male
Tamil
Complete
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : habitational name from any of various places called Copley, for example in County Durham, Staffordshire, and Yorkshire, from the Old English personal name Coppa (apparently a byname for a tall man) or from copp ‘hilltop’ + lēah ‘woodland clearing’.
Girl/Female
Tamil
Complete
Girl/Female
Tamil
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Complete
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Girl/Female
Tamil
Complete
Boy/Male
Indian
Complete
Boy/Male
Indian
Complete
Surname or Lastname
English
English : habitational name from Coppull in Lancashire, recorded in the 13th century as Cophill, from Old English copp ‘peak’ + hyll ‘hill’.English : nickname from Old French curt peil ‘short hair’.Probably an Americanized spelling of German and Jewish Koppel or German and Dutch Kappel.
Girl/Female
Tamil
Complete
Girl/Female
Muslim
Complex, Zigzag, Curling
Girl/Female
Hindu, Indian
Complex
Boy/Male
Tamil
Complete
Girl/Female
Bengali, Indian
Good Complex
Surname or Lastname
English
English : unexplained.Americanized form of German Koppler.
Boy/Male
Tamil
Poornan | பூரà¯à®¨à®¾à®¨
Complete
Poornan | பூரà¯à®¨à®¾à®¨
Girl/Female
Tamil
Shesha Harani | ஷேஷ ஹரணீÂ
Complete
Shesha Harani | ஷேஷ ஹரணீÂ
Girl/Female
Tamil
Complete
Surname or Lastname
English
English : habitational name, probably from Comley in Shropshire or Combley on the Isle of Wight; both are named with Old English cumb ‘valley’ + lēah ‘woodland clearing’.
COMPLEX TORUS
COMPLEX TORUS
Male
English
Brushwood
Boy/Male
Arabic, Muslim
Provider
Boy/Male
Arabic
Good Looking; Handsome
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sikh, Tamil, Telugu, Traditional
Bravery; The Sun of Valour; Lord Hanuman; Praiseworthy; Record; Strong Strong; One who Continue Progress
Girl/Female
Arabic, Hebrew, Indian, Muslim
Great; My Father is the Lord; Yahweh is My Father
Surname or Lastname
English
English : unexplained.The name was brought to Watertown, MA, by John Sawin (b. about 1620 in Boxford, Suffolk, England).
Boy/Male
Hindu, Indian, Punjabi, Sikh
One Absorbed in Righteousness
Surname or Lastname
English
English : variant spelling of Salter.
Boy/Male
Australian, Christian, Greek, Scottish
Defender of Mankind; Similar to Alexander
Boy/Male
Greek Latin Shakespearean
The mythological hero of the Trojan War famous for his valor and manly beauty - his only weak...
COMPLEX TORUS
COMPLEX TORUS
COMPLEX TORUS
COMPLEX TORUS
COMPLEX TORUS
pl.
of Couple-close
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
n.
A complex; an aggregate of parts; a complication.
n.
Two taken together; a pair or couple; especially two lines of verse that rhyme with each other.
n.
One who couples; that which couples, as a link, ring, or shackle, to connect cars.
a.
Finished; ended; concluded; completed; as, the edifice is complete.
a.
Complex, complicated.
a.
That which joins or links two things together; a bond or tie; a coupler.
imp. & p. p.
of Comply
a.
Intricate; entangled; complicated; complex.
a.
Repeatedly compound; made up of complex constituents.
imp. & p. p.
of Compile
adv.
In a complex manner; not simply.
n.
One who complies, yields, or obeys; one of an easy, yielding temper.
a.
One of the pairs of plates of two metals which compose a voltaic battery; -- called a voltaic couple or galvanic couple.
a.
See Couple-close.
imp. & p. p.
of Couple
n.
One who compiles; esp., one who makes books by compilation.
a.
Not complex; uncompounded; simple.
v. t.
To bring to a state in which there is no deficiency; to perfect; to consummate; to accomplish; to fulfill; to finish; as, to complete a task, or a poem; to complete a course of education.