Vector bundle

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Definition and first consequences

Continuous Surjection Finite-dimensional Open neighborhood Natural number

27 related topics

Smooth vector bundles

Diffeomorphisms Smoothness Cp Infinitely differentiable Real analytic

20 related topics

Sections and locally free sheaves

Sections Vector field Zero element of the vector space Pointwise Scalar multiplication

18 related topics

Operations on vector bundles

Operations Dual vector bundle Dual vector space Dual space Functorial

17 related topics

Topics to explore

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Overview

Mathematics
Topological
Family Family of sets
Vector spaces Vector space
Space Space (mathematics)
Topological space
Smooth (or differentiable) manifolds Manifold
Algebraic variety
For all
Fiber bundles Fiber bundle
Tangent bundles Tangent bundle
Tangent space
Hairy ball theorem
Parallelizable Parallelizable manifold
Real Real number
Complex numbers Complex number
Complex vector bundles Complex vector bundle
Category of topological spaces

Definition and first consequences

Continuous Continuous function
Surjection
Finite-dimensional Hamel dimension
Open neighborhood
Natural number
Homeomorphism
Vectors Euclidean vector
Linear Linear map
Isomorphism
Local trivialization
Dimension
Function Function (mathematics)
Locally constant
Connected component Locally connected space
Line bundle
Cartesian product
Composite function
GL ( k ) {\displaystyle {\text{GL}}(k)} -valued General linear group
Transition functions Transition map
Set Set (mathematics)
Čech cocycle
Action Group action
Associated Associated bundle
Fibre bundle construction theorem
Topological subspace
Restriction Restriction of a map
Möbius band

Vector bundle morphisms

Morphism
Induced Induced map
Category Category (mathematics)
Bundle map
Inverse Inverse element
Identity map Identity function
Commutes Commutative diagram
Abelian Abelian category
Kernel Kernel (category theory)
Pullback bundle

Sections and locally free sheaves

Sections Section (fiber bundle)
Vector field
Zero element of the vector space Zero vector
Pointwise
Scalar multiplication
Sheaf Sheaf (mathematics)
Module Module (mathematics)
Ring Ring (mathematics)
Real-valued functions Real-valued function
Locally free Locally free sheaf
Tuple
Equivalent Equivalence of categories
Finitely generated Finitely generated module
Sheaves of OX-modules Sheaf of modules
Kernels Kernel (algebra)
Cokernels Cokernel
If and only if
Linearly independent

Operations on vector bundles

Additional structures and generalizations

Vector bundle metric Metric (vector bundle)
Positive definite Definite bilinear form
Euclidean space
Complex structure Linear complex structure
Complex-linear Complex linear structure
Reduction of the structure group of a bundle
Topological fields Topological field
Banach space
Banach bundle
Banach manifolds Banach manifold
Sphere bundles Sphere bundle

Smooth vector bundles

K-theory

Hausdorff Hausdorff space
Abelian group
Isomorphism classes Isomorphism class
Whitney sum
Relation Relation (mathematics)
Exact sequence
Topological K-theory
KO-theory
Compact supports Compact support
Periodicity theorem Bott periodicity
Raoul Bott
Algebraic geometry
Coherent sheaves Coherent sheaf
Scheme Scheme (mathematics)
Equivalence relation
Smooth Smooth morphism

General notions

Topology and differential geometry

Connection Connection (vector bundle)
Gauge theory Gauge theory (mathematics)

Algebraic and analytic geometry

Picard group

Advanced semantic analysis

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Map overview Semantic statistics

Nodes158

Edges157

Triples53

Avg. degree1.99

Density0.012658

Components1

How this topic connects Entity context

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Vector bundle

Top relations

related to Additional structures and generalizations · 8

Vector bundle → Banach, Euclidean, For, If, More, Specifically, Usually, Vector

related to K-theory · 8

Vector bundle → Hausdorff, K-theory, KO-theory, Raoul Bott, S2X, The, The K-theory, Whitney

related to Smooth vector bundles · 7

Vector bundle → Cp, Cω-bundles, Depending, In, That, The, TM

related to External links · 6

Vector bundle → EMS Press, Encyclopedia, Mathematics, MathOverflowWhy, Vector, Why

related to Operations on vector bundles · 5

Vector bundle → Ex, For, Formally, Most, The

related to Sections and locally free sheaves · 5

Vector bundle → As, Given, Let, The, With

related to Subbundles · 5

Vector bundle → For, Given, In, Möbius, One

is a · 3

Vector bundle → Cr vector bundle, tangent bundle, topological construction that makes precise the idea of a family of vector spaces parameterized by another space X

related to Topology and differential geometry · 2

Vector bundle → Connection, Gauge

related to Algebraic and analytic geometry · 1

Vector bundle → Algebraic

Important terminology Word statistics

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Important terminology

vector bundle bundles space displaystyle real smooth fiber trivial spaces structure example also continuous tangent category manifold general called every

Entity relationships Subject–Predicate–Object triples

Subject	Predicate	Object	Confidence	Src
Vector bundle	is a	topological construction that makes precise the idea of a family of vector spaces parameterized by another space X	0.90	text
Vector bundle	is a	tangent bundle	0.90	text
Vector bundle	is a	Cr vector bundle	0.90	text
Vector bundle	related to Additional structures and generalizations	Vector	0.60	section
Vector bundle	related to Additional structures and generalizations	For	0.60	section
Vector bundle	related to Additional structures and generalizations	Usually	0.60	section
Vector bundle	related to Additional structures and generalizations	Euclidean	0.60	section
Vector bundle	related to Additional structures and generalizations	More	0.60	section
Vector bundle	related to Additional structures and generalizations	If	0.60	section
Vector bundle	related to Additional structures and generalizations	Banach	0.60	section
Vector bundle	related to Additional structures and generalizations	Specifically	0.60	section
Vector bundle	related to Algebraic and analytic geometry	Algebraic	0.60	section

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Min side: 3

Research this topic

Explore this topic

Definition and first consequences

Smooth vector bundles

Sections and locally free sheaves

Operations on vector bundles

Key facts & relationships

Topics to explore

Overview

Definition and first consequences

Vector bundle morphisms

Sections and locally free sheaves

Operations on vector bundles

Additional structures and generalizations

Smooth vector bundles

K-theory

General notions

Topology and differential geometry

Algebraic and analytic geometry

Advanced semantic analysis

Map overview Semantic statistics

Vector bundle

How this topic connects Entity context

Vector bundle

Top relations

Important terminology Word statistics

Important terminology

Entity relationships Subject–Predicate–Object triples

Related concept clusters Concept neighborhoods

Connections between topic areas Semantic bridges