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Mathematical operation on vector bundles
mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. The dual bundle of a vector bundle π : E →
Dual_bundle
Line of gamepads developed by Sony
console's bundle was updated to include DualShock, and phase out the original PlayStation controller and the Dual Analog Controller. The DualShock is the
DualShock
Mathematical parametrization of vector spaces by another space
dual vector bundle E* is the Hom bundle Hom(E, R × X) of bundle homomorphisms of E and the trivial bundle R × X. There is a canonical vector bundle isomorphism
Vector_bundle
\operatorname {Pic} ^{0}(A)} is called a degree 0 line bundle on A. To A one then associates a dual abelian variety Av (over the same field), which is the
Dual_abelian_variety
Vector bundle existing over a Grassmannian
the dual of the hyperplane bundle or Serre's twisting sheaf O P n ( 1 ) {\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(1)} . The hyperplane bundle is
Tautological_bundle
Vector bundle of cotangent spaces at every point in a manifold
cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle
Cotangent_bundle
Tangent spaces of a manifold
vector field on M {\displaystyle M} , and the dual bundle to T M {\displaystyle TM} is the cotangent bundle, which is the disjoint union of the cotangent
Tangent_bundle
Concept in mathematics
The conormal bundle is defined as the dual bundle to the normal bundle. It can be realised naturally as a sub-bundle of the cotangent bundle (of M {\displaystyle
Normal_bundle
line bundle, then E ⊗ O = E for any E. Example: E ⊗ E∗ is canonically isomorphic to the endomorphism bundle End(E), where E∗ is the dual bundle of E.
Tensor_product_bundle
Principal bundle associated to a vector bundle
In mathematics, a frame bundle is a principal fiber bundle F ( E ) {\displaystyle F(E)} associated with any vector bundle E {\displaystyle E} . The fiber
Frame_bundle
Partial differential equations whose solutions are instantons
of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the
Yang–Mills_equations
Defines a notion of parallel transport on a bundle
gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify
Connection_(vector_bundle)
Assignment of a tensor continuously varying across a region of space
fields in physics) and V ∗ = T ∗ M {\displaystyle V^{*}=T^{*}M} is its dual bundle, the cotangent space (whose sections are called 1 forms, or covariant
Tensor_field
Groot dual Dual abelian variety Dual basis in a field extension Dual bundle Dual curve Dual (category theory) Dual graph Dual group Dual object Dual pair
List_of_dualities
complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the
Complex_vector_bundle
Generalization of vector bundles
that the canonical bundle K P n {\displaystyle K_{\mathbb {P} ^{n}}} (the dual of the determinant line bundle of the tangent bundle) is isomorphic to O
Coherent_sheaf
Study of vector bundles, principal bundles, and fibre bundles
theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused
Gauge_theory_(mathematics)
Vector bundle of rank 1
sends the fibers of L {\displaystyle L} to the fibers of the dual of the tautological bundle. More specifically, suppose that s 0 , … , s r {\displaystyle
Line_bundle
Characteristic classes of vector bundles
c_{t}(E)=\sum _{i=0}^{r}c_{i}(E)t^{i}} the Chern polynomial of it. For the dual bundle E ∗ {\displaystyle E^{*}} of E {\displaystyle E} , c i ( E ∗ ) = ( −
Chern_class
Complex vector bundle on a complex manifold
tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle. By
Holomorphic_vector_bundle
Theorem in algebraic geometry
duality in topology, with the canonical line bundle replacing the orientation sheaf. The Serre duality theorem is also true in complex geometry more
Serre_duality
Citizenship in multiple countries held by the same person
multiple citizenship can affect these categories. As a bundle of rights, dual citizenship (a) enables dual citizens by granting special privileges, (b) affects
Multiple_citizenship
Formulation of classical mechanics using momenta
fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T∗Et, which comes equipped
Hamiltonian_mechanics
Differentiable manifold
operator mapping locally defined functions to local sections of the dual bundle L ∗ {\displaystyle L^{*}} . This is the ∂ b ¯ {\displaystyle {\overline
CR_manifold
Concept in algebraic geometry
The tangent bundle of a smooth scheme over k can be defined as the dual bundle, TX = (Ω1)*. Smoothness is a geometric property, meaning that for any
Smooth_scheme
Mathematical concept
one-dimensional space, denoted by O(H), and called the hyperplane bundle. The dual bundle is denoted O(−H), and the kth tensor power of O(H) is denoted by
Complex_projective_space
In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the
Clifford_bundle
Scheme parametrizing flags in the fibers of a vector bundle
define flag bundles using chains of locally free quotients rather than subbundles. The two conventions are equivalent after passing to dual bundles and reversing
Flag_bundle
Expression that may be integrated over a region
T_{p}^{*}(M)} is its dual space. This space is naturally isomorphic[clarification needed] to the fiber at p {\displaystyle p} of the dual bundle of the k {\displaystyle
Differential_form
Concept in algebraic geometry
it is the line bundle of holomorphic n {\displaystyle n} -forms on V {\displaystyle V} . This is the dualising object for Serre duality on V {\displaystyle
Canonical_bundle
Isomorphism between the tangent and cotangent bundles of a manifold
musical isomorphisms show that a vector bundle on a smooth manifold is (non-canonically) isomorphic to its dual. Abstractly, the musical isomorphisms can
Musical_isomorphism
algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth
Dualizing_module
Auxiliary functions used to probe equations, distributions, and weak formulations
useful, for example, when defining weak or distributional sections of the dual bundle, differential forms with distributional coefficients, or weak formulations
Test_function
Construct in differenital geometry
\cdot )} of a vector space and its dual, which is intrinsic to any vector bundle. The latter is a function on the bundle of endomorphisms End ( E ) = E ⊗
Metric_connection
x 1 , … , x k {\displaystyle x_{1},\dots ,x_{k}} , the Segre for the dual bundle E ∨ {\displaystyle E^{\vee }} which has Chern roots − x 1 , … , − x k
Segre_class
Moduli space of the Yang–Mills equations
\operatorname {SU} (n)} -bundles. The self-dual Yang–Mills moduli space of a principal SO ( 3 ) {\displaystyle \operatorname {SO} (3)} -bundle P ↠ X {\displaystyle
Yang–Mills_moduli_space
Series of Chinese camera drones
II Pro RTK V2 Drone Rugged Bundle". Autelpilot. Retrieved 12 July 2026. "Autel Robotics EVO II Dual 640T RTK V2 Rugged Bundle". Autelpilot. Retrieved 12
Autel_Evo
(February 26, 2008). "Destination PlayStation News: MGS PS3 Bundle, Kratos PSP and DualShock3 Release Date". PlayStation.Blog. Archived from the original
PlayStation_3_models
Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations:
Hermitian Yang–Mills connection
Hermitian_Yang–Mills_connection
complex vector bundle T M ⊗ C {\displaystyle TM\otimes \mathbb {C} } , and their duals may be taken. The holomorphic cotangent bundle is the dual of the holomorphic
Holomorphic_tangent_bundle
Type of vector bundle
In mathematics, a Higgs bundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle
Higgs_bundle
System of partial differential equations used in Higgs field theory
differential equations for a connection and Higgs field on a vector bundle or principal bundle over a Riemann surface, written down by Nigel Hitchin in 1987
Hitchin's_equations
Construction in differential topology
differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to
Jet_bundle
Concept in algebraic geometry
an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to
Ample_line_bundle
In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X: p : G d ( E ) → X {\displaystyle
Grassmann_bundle
Short exact sequence of sheaves on projective space
sum of the dual of the Serre twisting sheaf. The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter
Euler_sequence
Operation on fibered manifolds
T*X are the cotangent bundles of Y, respectively, and V*Y → Y is the dual bundle to VY → Y, called the vertical cotangent bundle. This splitting is given
Connection_(fibred_manifold)
Concept in mathematics
transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal G-connection on a principal G-bundle P {\displaystyle
Connection_(principal_bundle)
Connects homology and cohomology groups for oriented closed manifolds
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of
Poincaré_duality
Projective variety that is also an algebraic group
variety to its dual that is symmetric with respect to double-duality for abelian varieties and for which the pullback of the Poincaré bundle along the associated
Abelian_variety
Various mathematical dualites
In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory
Koszul_duality
Concept in mathematics
is simpler for a vector bundle (i.e., a variety corresponding to a locally free sheaf of constant rank). We say a vector bundle E on an algebraic variety
Equivariant_sheaf
Exterior algebraic map taking tensors from p forms to n-p forms
can play a role in differential geometry when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms. This
Hodge_star_operator
Aspect of heart function
the right atrium to the atrioventricular node, along the bundle of His, and through the bundle branches to Purkinje fibers in the walls of the ventricles
Cardiac_conduction_system
Concept in algebraic geometry
geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described
Nef_line_bundle
Principle in mathematical optimization
Claude (1993). "14 Duality for Practitioners". Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren
Duality_(optimization)
Generalization of Riemannian manifolds
the geodesic flow on the tangent bundle. Taking the convex dual, it defines a cogeodesic flow on the cotangent bundle. A version of the Huygens–Fresnel
Finsler_manifold
Gaming-focused smartphone manufacturer
Golden Saga Limited Edition - Snapdragon 8 Elite, Dual-Layer Gold Vapor Chambers, 24GB+1TB Gaming Phone Bundle". REDMAGIC (Global). Retrieved 27 June 2026.
Redmagic
Kind of complex manifold
the dual complex torus (or Abelian variety). From the construction of the dual complex torus, it is suggested that there should exist a line bundle P {\displaystyle
Complex_torus
Nonconceptual awareness and interconnectedness
Nirvanic experiences – Found in Buddhism, where one perceives the self as a bundle of fleeting states rather than a fixed entity. Kevala experiences – Found
Nondualism
g and let ω g {\displaystyle \omega _{g}} be its relative dualizing sheaf. The Hodge bundle is the pushforward of this sheaf, i.e., Λ g = π ∗ ω g {\displaystyle
Hodge_bundle
Mathematical conjecture
fibre of the SYZ fibration, this Lagrangian section is mirror dual to a choice of line bundle structure supported on each torus fibre of the mirror manifold
SYZ_conjecture
Concept from algebraic geometry
nodal curves. The Hodge bundle is then defined as the direct image of a relative dualizing sheaf. As mentioned above, the dualizing sheaf exists for all
Dualizing_sheaf
Mathematical operation
1-forms on N {\displaystyle N} (the linear space of sections of the cotangent bundle) to the space of 1-forms on M {\displaystyle M} . This linear map is known
Pullback (differential geometry)
Pullback_(differential_geometry)
Handheld game console
worldwide between 2004 and 2005. The "DS" in the name is an initialism for "Dual Screen", reflecting the system's most distinctive feature: two LCD screens
Nintendo_DS
Concept in mathematics
In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension
Banach_bundle
Section of a certain line bundle
a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume
Density_on_a_manifold
2010 video game
original on 2012-08-26. Retrieved 2012-08-20. 2014-05-20, Chaos;Head Dual bundles Noah and Love Chu Chu for PS Vita Archived 2014-05-20 at the Wayback
Chaos;Head_Love_Chu_Chu!
Gopakumar–Vafa duality is a duality in string theory, hence a correspondence between two different theories, in this case between Chern–Simons theory and
Gopakumar–Vafa_duality
Describes the line bundles on a complex torus or complex abelian variety
giving the dual complex torus. Explicitly, a line bundle on T = V / Λ {\displaystyle T=V/\Lambda } may be constructed by descent from a line bundle on V {\displaystyle
Appell–Humbert_theorem
Math/physics concept
{e} )e_{k}.} If θ = {θi | i = 1, 2, ..., n}, denotes the dual basis of the cotangent bundle, such that θi(ej) = δij (the Kronecker delta), then the connection
Connection_form
{P}}^{S_{n}}\cong {\mathcal {O}}} ) Procesi bundles which are dual to each other. The first construction of a Procesi bundle was given by American mathematician
Procesi_bundle
Four-dimensional analog of the icosahedron
[citation needed] The bundle of 12 Clifford parallel decagon fibers is divided into a bundle of 12 left pentagon fibers and a bundle of 12 right pentagon
600-cell
Equations describing magnetic monopoles
{\displaystyle G} -bundle over a 3-manifold M {\displaystyle M} , Φ {\displaystyle \Phi } is a section of the corresponding adjoint bundle, d A {\displaystyle
Bogomolny_equations
1998 video game
September 25. They were sold both individually and as a $24.99 bundle. The PC port is based on the Dual Shock ver. and includes high-resolution character models
Resident_Evil_2
Characteristic class of oriented, real vector bundles
real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth
Euler_class
Process in mathematics
see Pullback (differential geometry) Pullback (cohomology) The pullback bundle is an example that bridges the notion of a pullback as precomposition, and
Pullback
Medical device for artificially stimulating heart contractions
heart muscle and potentially shorter life expectancy. His bundle pacing (HBP) and left bundle branch area pacing (LBBAP) leads to a more natural or perfectly
Pacemaker
vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may
Stable_vector_bundle
Formalization in mathematical topos theory
certain fibre bundles on manifolds: namely bundles of jets (see also jet bundle). The second insight is that the operation of assigning a bundle of jets to
Synthetic differential geometry
Synthetic_differential_geometry
Correspondsnce between Higgs bundles and fundamental group representations
Donaldson, Simon K. (1985). "Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundle". Proceedings of the London Mathematical
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
Concept in quantum mechanics
Wave–particle duality is the concept in quantum mechanics that fundamental entities of the universe, like photons and electrons, exhibit particle or wave
Wave–particle_duality
Extends Poincaré duality beyond oriented manifolds
non-degenerate. The oriented Poincaré duality is contained in this statement, as understood from the fact that the orientation bundle o(M) is trivial if the manifold
Twisted_Poincaré_duality
Topics referred to by the same term
symplectic geometry Lagrangian system, a pair consisting of a smooth fiber bundle and a Lagrangian density Lagrangian mechanics, a formulation of classical
Lagrangian
curvature forms of ample line bundles (also known as positive line bundles). Let L be a holomorphic Hermitian line bundle on a complex manifold, ∂ ¯ :
Positive_form
Generalization of a vector bundle
line bundle L, then Spec X R {\displaystyle \operatorname {Spec} _{X}R} is the total space of the dual of L. More generally, given a vector bundle (finite-rank
Cone_(algebraic_geometry)
French applied mathematician
(1993). "XII Abstract duality for practitioners". Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren
Claude_Lemaréchal
Cable assembly containing one or more optical fibers that are used to carry light
properties. Individual coated fibers (or fibers formed into ribbons or bundles) then have a tough resin buffer layer or core tube(s) extruded around them
Fiber-optic_cable
Algebraic topology theory
structure of a fibre bundle over the classifying space B G {\displaystyle BG} . This is the associated X {\displaystyle X} -bundle obtained from the action
Equivariant_cohomology
Branch of geometry
structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently
Contact_geometry
characteristic form related to the orientation line bundle, useful to formulate Twisted Poincaré duality Picard group whose connected component, the Jacobian
Flat_vector_bundle
Mathematical structure in non-Riemannian differential geometry
of two compatible Lie algebroids defined on dual vector bundles. Lie bialgebroids are the vector bundle version of Lie bialgebras. A Lie algebroid consists
Lie_bialgebroid
Digital circuit without clock cycles
experimented with different asynchronous design styles: a bundled data XAP, a 1-of-4 XAP, and a 1-of-2 (dual-rail) XAP (2003?) ARM-compatible processor (2003?)
Asynchronous_circuit
Ryzen 1000 desktop CPUs: Socket: AM4. All the CPUs support DDR4-2666 in dual-channel mode. All the CPUs support 24 PCIe 3.0 lanes. 4 of the lanes are
List_of_AMD_Ryzen_processors
\mathbb {R} .} If M is oriented; i.e., the orientation sheaf of the tangent bundle of M is literally trivial, then the above reduces to the usual integration
Orientation_sheaf
Manifold upon which it is possible to perform calculus
The tangent bundle (or more precisely its sheaf of sections) can be identified with the sheaf of morphisms of OM into the ring of dual numbers. The category
Differentiable_manifold
Electric pickup truck
marketed as the "Cyberbeast" with an EPA range of 320 miles (515 km) and a dual-motor AWD model with a range of 350 miles (565 km). As of 2026[update], the
Tesla_Cybertruck
Formula for spinors
Given a spin structure on a pseudo-Riemannian manifold M and a spinor bundle S, the Lichnerowicz formula states that on a section ψ of S, D 2 ψ = ∇ ∗
Lichnerowicz_formula
Device used with games or entertainment systems
of the Xbox One. This controller was bundled with the console on launch, and was removed from the default bundle in June 2014. Sony's EyeToy similarly
Game_controller
Genterprise Genterprise 20080920080925September 25, 2008 (JP) Undercover: Dual Motives Sproing Interactive Media DTP Entertainment, Eidos Interactive 20080420080425April
List of Nintendo DS games (Q–Z)
List_of_Nintendo_DS_games_(Q–Z)
DUAL BUNDLE
DUAL BUNDLE
Boy/Male
Indian, Punjabi, Sikh
Kind
Surname or Lastname
English
English : variant of Dale (from the Old Kentish form del) or a habitational name from Deal in Kent, named with this word.Americanized spelling of German Diel or Diehl.Dutch (de Ruyter) : variant spelling (17th century) of De Ruiter
Boy/Male
Arabic, Christian, Egyptian, Hindu, Indian, Kannada, Marathi, Muslim, Telugu
Asked for
Girl/Female
Indian
Dual, Second
Boy/Male
Shakespearean
Love's Labours Lost' A constable.
Male
English
Variant spelling of English Dougal, DUGAL means "black stranger."Â
Surname or Lastname
English
English : of uncertain origin; possibly an altered form of Irish Doyle. Compare Dyal.Name found among people of Indian origin in Guyana and Trinidad : altered spelling of Dayal. This spelling is found in Indian names occasionally when -dial is the final element of a compound personal name.
Boy/Male
African, Arabic, Punjabi
Brave and Courageous
Girl/Female
Hindu, Indian
Shaking
Girl/Female
Arabic, Muslim
Prayer
Surname or Lastname
French
French : topographic name from Old French du val ‘from the valley’ (from Latin vallis).English : variant of Duvall 1.
Boy/Male
French
Of the valley.
Girl/Female
Muslim
Prayer
Boy/Male
Hindu
Dear one
Surname or Lastname
English
English : of uncertain origin; possibly an altered form of Irish Doyle. Compare Dial.Indian : variant spelling of Dayal.
Girl/Female
Indian
Loved by Everyone
Boy/Male
American, Australian, French, Jamaican
Of the Valley; Combination of the Prefix Du and Val
Boy/Male
Scottish
Dark-skinned stranger.
Girl/Female
Tamil
Dual, Second
Boy/Male
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Telugu
Dear One; Beloved
DUAL BUNDLE
DUAL BUNDLE
Boy/Male
American, Australian, British, English, French, Hebrew, Spanish
Supplanter; He who Supplants
Girl/Female
Hindu, Indian, Marathi
One who Gives Joy
Girl/Female
Hindi
Lily.
Boy/Male
Gaelic Celtic Irish
White.
Girl/Female
Indian
A Place Where Shri Lord Krishna Grew Up
Surname or Lastname
English
English : variant spelling of Collier.
Biblical
that drinks; thread
Boy/Male
Indian
Crystal
Male
Japanese
(里) Japanese name MINORU means "truth."
Boy/Male
Indian, Sikh
Light
DUAL BUNDLE
DUAL BUNDLE
DUAL BUNDLE
DUAL BUNDLE
DUAL BUNDLE
n.
Wood of the pine or fir; as, a floor of deal.
n.
In Shetland and Orkney, a freehold; property held by udal, or allodial, right.
n.
Specifically: To distribute, as cards, to the players at the commencement of a game; as, to deal the cards; to deal one a jack.
v. t.
To make dull, stupid, or sluggish; to stupefy, as the senses, the feelings, the perceptions, and the like.
a.
Obtuse; dull.
n.
A part or portion; a share; hence, an indefinite quantity, degree, or extent, degree, or extent; as, a deal of time and trouble; a deal of cold.
v. i.
To become dull or stupid.
superl.
Furnishing little delight, spirit, or variety; uninteresting; tedious; cheerless; gloomy; melancholy; depressing; as, a dull story or sermon; a dull occupation or period; hence, cloudy; overcast; as, a dull day.
n.
An instrument, formerly much used for showing the time of day from the shadow of a style or gnomon on a graduated arc or surface; esp., a sundial; but there are lunar and astral dials. The style or gnomon is usually parallel to the earth's axis, but the dial plate may be either horizontal or vertical.
n.
The division of a piece of timber made by sawing; a board or plank; particularly, a board or plank of fir or pine above seven inches in width, and exceeding six feet in length. If narrower than this, it is called a batten; if shorter, a deal end.
v. t.
To survey with a dial.
superl.
Not bright or clear to the eye; wanting in liveliness of color or luster; not vivid; obscure; dim; as, a dull fire or lamp; a dull red or yellow; a dull mirror.
v. i.
To contend (with); to treat (with), by way of opposition, check, or correction; as, he has turbulent passions to deal with.
a.
Dull; stupid.
a.
Dull.
a.
Heavy; dull.
a.
Dull; stupid.
a.
Expressing, or consisting of, the number two; belonging to two; as, the dual number of nouns, etc. , in Greek.
a.
Heavy; dull.
v. t.
To measure with a dial.