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mathematics, a bundle map (or bundle morphism) is a function that relates two fiber bundles in a way that respects their internal structure. Fiber bundles are mathematical
Bundle_map
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 X
Vector_bundle
Continuous surjection satisfying a local triviality condition
B.} The map π , {\displaystyle \pi ,} called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space
Fiber_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
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in
Hopf_fibration
Concept in algebraic geometry
canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle
Canonical_bundle
Mathematical operation
linear space of sections of the cotangent bundle) to the space of 1-forms on M {\displaystyle M} . This linear map is known as the pullback (by ϕ {\displaystyle
Pullback (differential geometry)
Pullback_(differential_geometry)
Generalization of an orientation of a vector space
the vector space Ex and one demands that each trivialization map (which is a bundle map) ϕ U : π − 1 ( U ) → U × R n {\displaystyle \phi _{U}:\pi ^{-1}(U)\to
Orientation of a vector bundle
Orientation_of_a_vector_bundle
Vector bundle of rank 1
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent
Line_bundle
Fiber bundle whose fibers are group torsors
In the mathematical area of topology, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product
Principal_bundle
Right inverse of a fiber bundle map
bundle over a base space, B {\displaystyle B} : π : E → B {\displaystyle \pi \colon E\to B} then a section of that fiber bundle is a continuous map,
Section_(fiber_bundle)
Linear approximation of smooth maps on tangent spaces
obvious manner, a bundle map (in fact a vector bundle homomorphism) from the tangent bundle of M {\displaystyle M} to the tangent bundle of N {\displaystyle
Pushforward_(differential)
Tangent spaces of a manifold
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself.
Tangent_bundle
Collection of maps
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a continent or region of Earth. Advances in astronomy have also resulted
Atlas
Fiber bundle induced by a map of its base space
mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle π : E → B {\displaystyle
Pullback_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
Concept in mathematics
is called the vertical bundle of P {\displaystyle P} . It follows that ω {\displaystyle \omega } determines uniquely a bundle map v : T P → V {\displaystyle
Connection_(principal_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
classifying space BG, such that every bundle with the given structure group G over M is a pullback by means of a continuous map M → BG. When the definition of
Universal_bundle
M a vector bundle on M, then a metric on E is a bundle map k : E ×M E → M × R from the fiber product of E with itself to the trivial bundle with fiber
Bundle_metric
vector bundle with an additional structure, the complex structure. By definition, a complex structure is a bundle map between a real vector bundle E and
Complex_vector_bundle
Generalization of a fiber bundle
and p : E → B is a map. E is called the total space B is the base space of the bundle p is the projection This definition of a bundle is quite unrestrictive
Bundle_(mathematics)
Construction in differential topology
associated with the derivatives of maps, particularly those associated with the calculus of variations. Consequently, the jet bundle is now recognized as the correct
Jet_bundle
Differential geometry topic
tangent bundle TM. In the case where M = R n {\displaystyle M=\mathbf {R} ^{n}} , the tangent bundle is trivialized (so the Grassmann bundle becomes a map to
Gauss_map
Isomorphism between the tangent and cotangent bundles of a manifold
isomorphism) is an isomorphism between the tangent bundle T M {\displaystyle \mathrm {T} M} and the cotangent bundle T ∗ M {\displaystyle \mathrm {T} ^{*}M} of
Musical_isomorphism
Structure group sub-bundle on a tangent frame bundle
the structure group of a G-bundle B is choosing an H-bundle whose image is B. The inducing map from H-bundles to G-bundles is in general neither onto
G-structure_on_a_manifold
Way to create new manifolds out of disk bundles
disk bundles. It was first described by John Milnor (1956) and subsequently used extensively in surgery theory to produce manifolds and normal maps with
Plumbing_(mathematics)
{\displaystyle P'\to T'} is a bundle map (i.e., forms a commutative diagram) that is compatible with the equivariant maps P → X {\displaystyle P\to X}
Quotient_stack
Mathematical concept
complex line bundles. Equivalently it accounts for the first Chern class. This can be seen heuristically by looking at the fiber bundle maps S 1 ↪ S 2 n
Complex_projective_space
Typically linear operator defined in terms of differentiation of functions
multi-index α, P α ( x ) : E → F {\displaystyle P^{\alpha }(x):E\to F} is a bundle map, symmetric on the indices α. The kth order coefficients of P transform
Differential_operator
Set of topological invariants
of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney classes
Stiefel–Whitney_class
Concept in differential geometry
PSO(E) to a principal bundle PSpin(E) under the action of the spin group Spin(n), by which we mean that there exists a bundle map ϕ {\displaystyle \phi
Spin_structure
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
group G (which may be a topological or Lie group), an equivariant bundle is a fiber bundle π : E → B {\displaystyle \pi \colon E\to B} such that the total
Equivariant_bundle
Most general completion of a commutative square given two morphisms with same codomain
bundles: given a bundle map π : E → B and a continuous map f : X → B, the pullback (formed in the category of topological spaces with continuous maps)
Pullback_(category_theory)
Term in mathematics
of induced maps. Note that the tangent bundle TJ of J is the trivial bundle J × R and there is a canonical cross-section ι of this bundle such that ι(t)
Integral_curve
Infinitesimal version of Lie groupoid
groupoid gives rise to a Lie algebroid, which is the vertical bundle of the source map restricted at the units. However, unlike Lie algebras, not every
Lie_algebroid
Fiber bundle
theory of fiber bundles with a structure group G {\displaystyle G} (a topological group) allows an operation of creating an associated bundle, in which the
Associated_bundle
Fiber bundle whose fibers are projective spaces
projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a Pn-bundle if it is locally
Projective_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
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
Mathematical technique for vector bundles
( E ) {\displaystyle Y=Fl(E)} , called the flag bundle associated to E {\displaystyle E} , and a map p : Y → X {\displaystyle p\colon Y\rightarrow X}
Splitting_principle
Complex vector bundle on a complex manifold
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 map π : E → X
Holomorphic_vector_bundle
geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the higher order frame bundle F r ( M ) {\displaystyle F^{r}(M)} , for
Natural_bundle
the Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by
Iitaka_dimension
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
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)
1513 Ottoman nautical chart
search for potentially overlooked maps. Halil Edhem found a disregarded bundle of material containing an unusual parchment map. They showed the parchment to
Piri_Reis_map
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)
Principal fiber bundle
bundle is a fiber bundle where the fiber is the circle S 1 {\displaystyle S^{1}} . Oriented circle bundles are also known as principal U(1)-bundles,
Circle_bundle
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
Construct allowing differentiation of tangent vector fields of manifolds
ω on the frame bundle FM or GL(M) of a manifold M. In more detail, ω is a smooth map from the tangent bundle T(FM) of the frame bundle to the space of
Affine_connection
Manifold upon which it is possible to perform calculus
new charts is the tangent bundle for the charts Uα. The transition maps on this atlas are defined from the transition maps on the original manifold, and
Differentiable_manifold
Neighborhood of a submanifold
the normal bundle of S in M. Here S plays the role of the curve and M the role of the plane containing the curve. Consider the natural map i : N 0 → S
Tubular_neighborhood
Long exact sequence
sequence. Consider a fiber-oriented sphere bundle with total space E, base space M, fiber Sk and projection map π {\displaystyle \pi } : S k ↪ E ⟶ π M .
Gysin_homomorphism
Mathematics concept
vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle π : E → B
Vertical and horizontal bundles
Vertical_and_horizontal_bundles
data of a map f ∨ : B ∨ → A ∨ {\displaystyle f^{\vee }:B^{\vee }\to A^{\vee }} is the same as giving a family of degree zero line bundles on A {\displaystyle
Dual_abelian_variety
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
Differential geometry construct on fiber bundles
on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear
Ehresmann_connection
Concept in mathematics
a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or
Normal_bundle
double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πTM,M) of
Double_tangent_bundle
Assignment of a tensor continuously varying across a region of space
{\displaystyle T} of the tensor bundle V ⊗ p ⊗ ( V ∗ ) ⊗ q {\displaystyle V^{\otimes p}\otimes (V^{*})^{\otimes q}} , which maps every element of the manifold
Tensor_field
Homotopy theory in algebraic topology
the projection in a vector bundle, fiber bundle or fibration, where there need be no unique way of lifting. Assume all maps are continuous functions between
Homotopy_lifting_property
Constructs a fiber bundle from a base space, fiber and a set of transition functions
In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle with a structure group from a given base space, fiber
Fiber bundle construction theorem
Fiber_bundle_construction_theorem
Mathematical construct of fiber bundles
representation V of G such that the associated bundle map from the tangent bundle TM to the associated bundle P×G V is a bundle isomorphism. (In particular, V and
Solder_form
{\displaystyle f} is the identity map (i.e., the map which fixes every point of the torus) then the resulting torus bundle M ( f ) {\displaystyle M(f)} is
Torus_bundle
Fiber bundle Principal bundle Frame bundle Hopf bundle Associated bundle Vector bundle Tangent bundle Cotangent bundle Line bundle Jet bundle Sheaf (mathematics)
List of differential geometry topics
List_of_differential_geometry_topics
Topological construct
trivialized fiber bundles with fiber F {\displaystyle F} and structure group G {\displaystyle G} over the two hemispheres, then given a map f : S n − 1 →
Clutching_construction
Concept in economics
consumption bundles by order of preference. A graph of indifference curves for several utility levels of an individual consumer is called an indifference map. Points
Indifference_curve
Algebraic structure in linear algebra
with a continuous map π:E→X such that for every x in X, the fiber π−1(x) is a vector space. The case dim V = 1 is called a line bundle. For any vector space
Vector_space
Math/physics concept
connection is compatible with the structure of a G-bundle on E provided that the associated parallel transport maps always send one G-frame to another. Formally
Connection_form
Special type of principal bundle
\operatorname {SU} (2)} -bundles (or principal Sp ( 1 ) {\displaystyle \operatorname {Sp} (1)} -bundles) are special principal bundles with the second special
Principal_SU(2)-bundle
Diffeomorphism that has a hyperbolic structure on the tangent bundle
differentiable map f on M has a hyperbolic structure on the tangent bundle, then it is called an Anosov map. Examples include the Bernoulli map, and Arnold's
Anosov_diffeomorphism
Concept in differential topology
tangent bundle of M. Then identity bundle map idE: E →E is an E-valued one form on M. The tautological one-form is a unique one-form on the frame bundle of
Vector-valued differential form
Vector-valued_differential_form
Type of topological space
dimension. Real projective space has a natural line bundle over it, called the tautological bundle. More precisely, this is called the tautological subbundle
Real_projective_space
Type of fiber bundle
In differential geometry, an area of mathematics, an affine bundle is a fiber bundle whose typical fiber is an affine space and whose trivialization morphisms
Affine_bundle
In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution to a Cartesian equation of the form: X 2 + a X Y + b Y 2 = P (
Conic_bundle
Exceptional functor
fiber bundles, where they yield maps that have the opposite of the usual variance. They are thus called wrong way maps, umkehr maps, Gysin maps, as they
Shriek_map
Concept in differential geometry
differentiable principal bundle or vector bundle with a connection. Let G be a Lie group and P → M be a principal G-bundle on a smooth manifold M. Suppose
Exterior_covariant_derivative
Tool to classify manifolds within a homotopy type in dim > 4
from the stable tangent bundle of M {\displaystyle M} to some bundle ξ {\displaystyle \xi } over X {\displaystyle X} . Two such maps are equivalent if there
Surgery_exact_sequence
Concept in geometric topology
has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold M to X matching the fundamental
Normal_invariant
Infinite-dimensional group in topology
geometrically as bundle gerbes since any K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} -bundle can be realized as the homotopy fiber of a map giving a homotopy
String_group
Generalization of vector bundles
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
Special type of principal bundle
\operatorname {U} (1)} -bundles (or principal SO ( 2 ) {\displaystyle \operatorname {SO} (2)} -bundles) are special principal bundles with the first unitary
Principal_U(1)-bundle
a vector-bundle map f : E → F {\displaystyle f:E\to F} over a variety X (that is, a scheme X-morphism between the total spaces of the bundles), the degeneracy
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Exact homotopy case
together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to
Classifying_space_for_U(n)
Concept in mathematics
mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To
Tensor_bundle
Characteristic classes of vector bundles
vector bundle to a classifying space (an infinite Grassmannian in this case). For any complex vector bundle V over a manifold M, there exists a map f from
Chern_class
Set of charts that describes a manifold
the transition maps between charts of an atlas preserve a local trivialization, then the atlas defines the structure of a fibre bundle. Smooth atlas Smooth
Atlas_(topology)
Continuous deformation between two continuous functions
^{n}-\{0\}\to S^{n-1}} is a fiber bundle with fiber R > 0 {\displaystyle \mathbb {R} _{>0}} . Every vector bundle is a fiber bundle with a fiber homotopy equivalent
Homotopy
Composite bundles Y → Σ → X {\displaystyle Y\to \Sigma \to X} play a prominent role in gauge theory with symmetry breaking, e.g., gauge gravitation theory
Connection_(composite_bundle)
Concept in differential geometry
\langle \cdot ,\cdot \rangle :E\times E\to M\times \mathbb {R} } , and a bundle map ρ : E → T M {\displaystyle \rho :E\to TM} (called anchor) subject to the
Courant_algebroid
Set of applications and supporting libraries
common release schedule. The bundle is composed of over 200 applications. Examples of prominent applications in the bundle include the file manager Dolphin
KDE_Gear
2021 video game
including new maps, game modes and seasonal events, while the game continues to monetize via the battle pass system and cosmetic bundles featured in the
Call_of_Duty:_Vanguard
Fixed-point theorem for smooth manifolds
needed relates to the elliptic complex of vector bundles E j {\displaystyle E_{j}} , namely a bundle map φ j : f − 1 ( E j ) → E j {\displaystyle \varphi
Atiyah–Bott fixed-point theorem
Atiyah–Bott_fixed-point_theorem
Scheme in algebraic geometry
normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry. The normal cone CXY or
Normal cone (algebraic geometry)
Normal_cone_(algebraic_geometry)
{\displaystyle R(V,K)=R(V,K_{V})\,} of sections of powers of the canonical bundle K. Its nth graded component (for n ≥ 0 {\displaystyle n\geq 0} ) is: R n
Canonical_ring
Kind of complex manifold
{O}}_{X}^{*})\to H^{2}(X,\mathbb {Z} )} is the first Chern class map, sending an isomorphism class of a line bundle to its associated first Chern class. It turns out
Complex_torus
Association of cohomology classes to principal bundles
each principal bundle of a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" and
Characteristic_class
Generalization of affine connections
manifold with a smooth map π: P → M which looks locally like the trivial bundle M × H → M. The frame bundle of M is a principal GL(n)-bundle, while if M is a
Cartan_connection
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