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Mathematical parametrization of vector spaces by another space
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space
Vector_bundle
a 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
Complex_vector_bundle
Complex vector bundle on a complex manifold
mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection
Holomorphic_vector_bundle
Characteristic classes of vector bundles
geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of
Chern_class
Geometric structure
g ) , {\displaystyle (M,g),\,} one defines the spinor bundle to be the complex vector bundle π S : S → M {\displaystyle \pi _{\mathbf {S} }\colon {\mathbf
Spinor_bundle
Defines a notion of parallel transport on a bundle
vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields
Connection_(vector_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
Mathematical operation on vector bundles
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 → X
Dual_bundle
Mathematical technique for vector bundles
technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations
Splitting_principle
Topics referred to by the same term
Generalized complex structure Complex structure deformation Complex vector bundle#Complex structure Complex structure theory in English law Real structure This
Complex_structure
M} of complex dimension n {\displaystyle n} , its tangent bundle as a smooth vector bundle is a real rank 2 n {\displaystyle 2n} vector bundle T M {\displaystyle
Holomorphic_tangent_bundle
Characteristic class for real vector bundles
classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Given a real vector bundle E {\displaystyle
Pontryagin_class
Relates the geometric vector bundles to algebraic projective modules
smooth vector bundles on a smooth manifold (real, complex, or quaternionic). His topological variant is about continuous (real or complex) vector bundles on
Serre–Swan_theorem
Study of vector bundles, principal bundles, and fibre bundles
gauge 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)
Generalization of an orientation of a vector space
orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation
Orientation of a vector bundle
Orientation_of_a_vector_bundle
Vector bundle of rank 1
tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of
Line_bundle
Fiber bundle whose fibers are projective spaces
projective bundle is of the form P ( E ) {\displaystyle \mathbb {P} (E)} for some vector bundle (locally free sheaf) E. Every vector bundle over a variety
Projective_bundle
Vector bundle existing over a Grassmannian
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k {\displaystyle
Tautological_bundle
a 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
Construction for vector bundles
geometry, the determinant line bundle is a construction that assigns, to every vector bundle over a paracompact space, a line bundle. Its name comes from using
Determinant_line_bundle
Generalization of vector bundles
information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under
Coherent_sheaf
Algebraic structure in linear algebra
operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector spaces and complex vector spaces
Vector_space
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
Vector bundle of cotangent spaces at every point in a manifold
mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold
Cotangent_bundle
Concept in algebraic geometry
The pullback of a vector bundle is a vector bundle of the same rank. In particular, the pullback of a line bundle is a line bundle. (Briefly, the fiber
Ample_line_bundle
Vector bundles theorem
Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after Shoshichi
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
smooth vector bundle. Let E be a smooth vector bundle over a smooth manifold M, and let g be a smooth symmetric bilinear form on E. The Clifford bundle of
Clifford_bundle
Mathematics concept
In mathematics, the complex conjugate of a complex vector space V {\displaystyle V\,} is a complex vector space V ¯ {\displaystyle {\overline {V}}} that
Complex conjugate of a vector space
Complex_conjugate_of_a_vector_space
Study of complex manifolds and several complex variables
holomorphic vector bundles often admit solutions to important differential equations arising out of physics such as the Yang–Mills equations. Complex geometry
Complex_geometry
Correspondsnce between Higgs bundles and fundamental group representations
fix a smooth complex vector bundle E {\displaystyle E} . Every Higgs bundle will be considered to have the underlying smooth vector bundle E {\displaystyle
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
Continuous surjection satisfying a local triviality condition
bundle I-bundle Natural bundle Principal bundle Projective bundle Pullback bundle Quasifibration Universal bundle Vector bundle Wu–Yang dictionary Seifert
Fiber_bundle
Smooth manifold
as a vector bundle isomorphism J : T M → T M {\displaystyle J\colon TM\to TM} on the tangent bundle. A manifold equipped with an almost complex structure
Almost_complex_manifold
Set of topological invariants
invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney
Stiefel–Whitney_class
Characteristic class of oriented, real vector bundles
oriented, 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
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
differential geometry, a Clifford module bundle, a bundle of Clifford modules or just Clifford module is a vector bundle whose fibers are Clifford modules,
Clifford_module_bundle
Basic result in the representation theory of Lie groups
sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles. It is built on the earlier
Borel–Weil–Bott_theorem
Structure group sub-bundle on a tangent frame bundle
real vector space is isomorphic to the underlying real space of a complex vector space: it admits a linear complex structure. A real vector bundle admits
G-structure_on_a_manifold
Non-tensorial representation of the spin group
physics, spinors (pronounced "spinner"; /spɪnər/) are elements of a complex vector space that can be associated with Euclidean space. Spinors can be thought
Spinor
Fiber bundle
\mathbb {R} )} to the complex general linear group G L ( n , C ) {\displaystyle \mathrm {GL} (n,\mathbb {C} )} . Decomposing a vector bundle of rank n {\displaystyle
Associated_bundle
positive-definite Hermitian form on each fiber of a complex vector bundle Hermitian matrix, a square matrix with complex entries that is equal to its own conjugate
List of things named after Charles Hermite
List_of_things_named_after_Charles_Hermite
Classifies holomorphic vector bundles over the complex projective line
theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over C P 1 {\displaystyle \mathbb
Birkhoff–Grothendieck_theorem
Differential form on a manifold which is permitted to have complex coefficients
Thus the spaces Ω0,1 and Ω1,0 determine complex vector bundles on the complex manifold. The wedge product of complex differential forms is defined in the
Complex_differential_form
Describes a periodicity in the homotopy groups of classical groups
for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity
Bott_periodicity_theorem
versions of bundle maps depending on the specific types of fiber bundles involved—for example, smooth bundles, vector bundles, or principal bundles—and on
Bundle_map
Mathematics concept
topological space with an involution. A real vector bundle over a real space X is defined to be a complex vector bundle E over X that is also a real space, such
KR-theory
Type of differentiable manifold
{\displaystyle M} of dimension n is called parallelizable if there exist smooth vector fields { V 1 , … , V n } {\displaystyle \{V_{1},\ldots ,V_{n}\}} on the
Parallelizable_manifold
Fiber bundle whose fibers are group torsors
principal bundle is the frame bundle F ( E ) {\displaystyle F(E)} of a vector bundle E {\displaystyle E} , which consists of all ordered bases of the vector space
Principal_bundle
Characteristic class in algebraic topology
{\displaystyle \operatorname {td} (E)} where E {\displaystyle E} is a complex vector bundle on a topological space X {\displaystyle X} , it is usually possible
Todd_class
Superconductivity theory
Ginzburg–Landau functional can be formulated in the general setting of a complex vector bundle over a compact Riemannian manifold. This is the same functional
Ginzburg–Landau_theory
Mathematical theory
Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature
Chern–Weil_homomorphism
connection ∇ {\displaystyle \nabla } is a connection on a Hermitian vector bundle E {\displaystyle E} over a smooth manifold M {\displaystyle M} which
Hermitian_connection
Metric on a determinant line bundle
determinant line bundle of a certain family of differential operators parametrised by the space of unitary connections on a complex vector bundle over a compact
Quillen_metric
Mathematical concept
account). Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is
Complex_projective_space
In mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection. Let π :
Flat_vector_bundle
Combination of higher category theory with Chern–Weil theory
equivalent ways to describe the k {\displaystyle k} -th Chern class of complex vector bundles of rank n {\displaystyle n} , which is as a: (1-categorical) natural
∞-Chern–Weil_theory
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
Mathematical concept
form V ⊕ V∗, every complex structure on a vector space is isomorphic to one of the form V ⊕ V. Using these structures, the tangent bundle of an n-manifold
Symplectic_vector_space
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
Property of a differential manifold that includes complex structures
twisted) generalized complex structures. Consider an N-manifold M. The tangent bundle of M, which will be denoted T, is the vector bundle over M whose fibers
Generalized_complex_structure
Concept in differential geometry
tangent bundle TM.) The bundle of spinors πS: S → M over M is then the complex vector bundle associated with the corresponding principal bundle πP: P →
Spin_structure
Japanese mathematician
hypersurface of complex projective space. Kobayashi is also notable for having proved that a hermitian–Einstein metric on a holomorphic vector bundle over a compact
Shoshichi_Kobayashi
Broad concept generalizing scalars in mathematics and physics
three-dimensional linear algebra and vector calculus Vector bundle, a topological construction that makes precise the idea of a family of vector spaces parameterized
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Manifold
is complex (which is what the chart definition says). Tensoring the tangent bundle with the complex numbers we get the complexified tangent bundle, on
Complex_manifold
Concept in algebraic geometry
bundle Ω {\displaystyle \Omega } on V {\displaystyle V} . Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle T
Canonical_bundle
Elliptic differential operators in geometry mathematics
be a compact, oriented manifold equipped with a metric. Let E be a vector bundle over M equipped with a fiber metric and a compatible connection, ∇ {\displaystyle
Laplace operators in differential geometry
Laplace_operators_in_differential_geometry
Algebraic operation on coordinate vectors
numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the scalar product of two vectors is the dot product of their
Dot_product
geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles. The concept of stability
Stable_principal_bundle
mathematics, an adjoint bundle is a vector bundle naturally associated with any smooth principal bundle. The fibers of the adjoint bundle carry a Lie algebra
Adjoint_bundle
Assignment of a tensor continuously varying across a region of space
the fiber is a vector space and the tensor bundle is a special kind of vector bundle. (There are vector bundles that are not tensor bundles: the Möbius band
Tensor_field
Assignment of a vector to each point in a subset of Euclidean space
setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are
Vector_field
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space)
Hopf_fibration
Special type of principal bundle
} Unlike the associated vector bundle, a complex plane bundle, the adjoint vector bundle is a orientable real vector bundle of third rank. Also since
Principal_SU(2)-bundle
Mathematical result in differential geometry
be a complex manifold of (complex) dimension n with a holomorphic vector bundle V. We let the vector bundles E and F be the sums of the bundles of differential
Atiyah–Singer_index_theorem
Concept in topology (mathematics)
mathematics, a weak Lie algebra bundle ξ = ( ξ , p , X , θ ) {\displaystyle \xi =(\xi ,p,X,\theta )\,} is a vector bundle ξ {\displaystyle \xi \,} over
Lie_algebra_bundle
Math/physics concept
covariant derivative. A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial
Connection_form
or exceptional line of a vector bundle over projective space is a projective line in projective space where the vector bundle has exceptional behavior
Jumping_line
F(BG+,X)G). Example: ℤ/2 acts on the complex K-theory KU by taking the conjugate bundle of a complex vector bundle. Then KUhℤ/2=KO, the real K-theory.
G-spectrum
Concept in differential geometry
holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of these cases, the
Holonomy
TP{\xrightarrow {d\pi }}\pi ^{*}TM\to 0} of vector bundles over P {\displaystyle P} , where the vertical bundle V P {\displaystyle VP} is the kernel of d
Atiyah_algebroid
Concept in mathematics
any fiber bundle associated to P {\displaystyle P} via the associated bundle construction. In particular, on any associated vector bundle the principal
Connection_(principal_bundle)
Concept in differential geometry
almost complex structure and the fundamental form are integrable, then we have a Kähler structure. A Hermitian metric on a complex vector bundle E over
Hermitian_manifold
Restriction of electrical impulse flow in the heart's bundle branches
A bundle branch block is a partial or complete interruption in the flow of electrical impulses in either of the bundle branches of the heart's electrical
Bundle_branch_block
Branch of algebraic topology
K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general)
Topological_K-theory
Kind of complex manifold
number 2n, where n is the complex dimension of M. All such complex structures can be obtained as follows: take a lattice Λ in a vector space V isomorphic to
Complex_torus
Topics referred to by the same term
mathematician Chern class, a type of characteristics class associated to complex vector bundles; named after Shiing-Shen Chern Chern, Russia, several inhabited
Chern_(disambiguation)
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
unitary group U in Bott's sense has a classifying space BU for complex vector bundles (see Classifying space for U(n)). A deeper application coming from
Kuiper's_theorem
Infinitesimal version of Lie groupoid
In mathematics, a Lie algebroid is a vector bundle A → M {\displaystyle A\rightarrow M} together with a Lie bracket on its space of sections Γ ( A ) {\displaystyle
Lie_algebroid
Euclidean space without distance and angles
the principal bundle P→πM) is an affine space for the vector space of ad(P)-valued 1-forms, where ad(P) is the associated adjoint bundle. For any non-empty
Affine_space
Construct allowing differentiation of tangent vector fields of manifolds
the simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry
Affine_connection
Mathematical object
of the monoid of complex vector bundles on X. Also, K 1 ( X ) {\displaystyle K^{1}(X)} is the group corresponding to vector bundles on the suspension
Spectrum_(topology)
Concept in category theory
of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for
Fibred_category
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
orthonormal k-tuples of vectors in ℝn. It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold Vk(ℂn)
Stiefel_manifold
Theorem in algebraic geometry
varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide
Serre_duality
Tool to track locally defined data attached to the open sets of a topological space
between vector bundles and locally free sheaves of 𝒪X-modules. This paradigm applies to real vector bundles, complex vector bundles, or vector bundles in
Sheaf_(mathematics)
In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X: p:Gd(E)→X such that the fiber p−1(x)=Gd(Ex)
Grassmann_bundle
Special type of principal bundle
\mathbb {Z} )} , which is an isomorphism over CW complexes. Principal bundles also have an adjoint vector bundle, which is trivial for principal U ( 1 ) {\displaystyle
Principal_U(1)-bundle
On the Euler characteristic of a holomorphic vector bundle on a compact complex manifold
Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle E on a compact complex manifold X, to calculate the holomorphic Euler characteristic
Hirzebruch–Riemann–Roch theorem
Hirzebruch–Riemann–Roch_theorem
Category with direct sums and certain types of kernels and cokernels
If X is a topological space, then the category of all (real or complex) vector bundles on X is not usually an abelian category, as there can be monomorphisms
Abelian_category
Manifold upon which it is possible to perform calculus
principal bundle made up of the set of all frames over M. The frame bundle is useful because tensor fields on M can be regarded as equivariant vector-valued
Differentiable_manifold
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COMPLEX VECTOR-BUNDLE
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