Section
Fiber Bundles
Differential geometry of fiber bundles, principal bundles, and connections.
Core idea
Bundle foundations
Start with a bundle and its projection to a base. Local product charts make it a fiber bundle; smooth charts give a smooth fiber bundle. Specifying group-valued transition data gives a G-bundle with a chosen structure group.
- Fiber of a map
- Typical fiber
- Local trivialization
- Bundle atlas
- Transition function
- Vector bundle
- Principal G-bundle
- Topological principal bundle
- Associated bundle
Definitions
- Closed Differential Form
- Cotangent Bundle
- De Rham Cohomology Group
- Diffeomorphism
- Differential K Form
- Differential Of A Lie Group Homomorphism Lie Algebra Homomorphism
- Differential Pushforward Of A Smooth Map
- Exact Differential Form
- Exponential Map Lie Group Exponential
- Exterior Derivative
- Coordinate transition map
- Fiber coordinates
- G-torsor
- Interior Product Contraction X
- Left Invariant Vector Field
- Left Translation L G
- Lie Algebra Of A Lie Group
- Lie Bracket
- Lie Derivative Of A Differential Form
- Lie Group
- Lie Group Homomorphism
- Lie Subgroup
- Pullback Of Covectors
- Pullback Of Differential Forms
- Regular Value
- Right Invariant Vector Field
- Right Translation R G
- Smooth Atlas
- Smooth compatibility of charts and atlases
- Smooth Chart Coordinate Chart
- Smooth Embedding
- Smooth Immersion
- Smooth Manifold
- Smooth Map
- Smooth structure
- Maximal smooth atlas
- Smooth G-valued Čech 1-cocycle
- Smooth Submersion
- Tangent Bundle
- Tangent-bundle cocycle from coordinate changes
- Tangent Space At A Point
- Vector Field
- Wedge Product Of Differential Forms
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Vector bundles, sections, and bundle operations
- Compactly supported section
- Category of vector bundles over a manifold
- Kernel and image bundles of a constant-rank morphism
- Line bundle
- Module of smooth sections
- Nowhere-vanishing section
- Quotient vector bundle
- Section of a fiber bundle
- Section of a vector bundle
- Sheaf of smooth sections
- Short exact sequence of vector bundles
- Sphere bundle
- Splitting theorem for vector bundles
- Support of a section
- Vector subbundle
- Vector bundles and locally free sheaves
- Zero section
- Nonabelian Čech H1 and principal bundles
Connections and curvature
- Chern connection
- Connection on a Hom bundle
- Connection on exterior and symmetric powers
- Dual connection
- Hermitian connection
- Pullback connection on a vector bundle
- Tensor product connection
- Vector-bundle-valued differential form
Reductions and geometric structures
- Connection compatible with a reduction
- Curvature of the Chern connection
- G-structure
- Intrinsic torsion of a G-structure
- Quaternionic vector bundle
- Spin connection
- Spin structure
Characteristic classes
- Characteristic number
- Chern number
- Chern roots
- Chern-Gauss-Bonnet theorem
- Chern-Weil homomorphism
- Euler class as pullback of the Thom class
- Euler class obstruction to a nowhere-zero section
- Invariant polynomial on a Lie algebra
- Pontryagin number
- Splitting principle
- Stiefel-Whitney class
- Thom class
- Thom isomorphism theorem
- Universal characteristic class
- Whitney sum formula
Serre-Swan duality
- Reconstructing a vector bundle from a projective module
- Section module is finitely generated projective
- Serre–Swan theorem
- Vector bundle from an idempotent matrix
Gauge actions and quotients
- Based gauge group
- Coulomb gauge
- Gauge theory
- Gauge-fixing condition
- Irreducible connection
- Moduli stack of connections
- Reducible connection
- Slice theorem for the gauge action
- Sobolev completion of connections and gauge transformations
- Stabilizer of a connection
- Wilson loop
Analytic and complex gauge theory
- Covariant Hodge Laplacian
- Donaldson-Uhlenbeck-Yau theorem
- Formal adjoint of the covariant exterior derivative
- Hermitian Yang-Mills connection
- L2 inner product on bundle-valued forms
Yang-Mills theory and instantons
- Anti-self-dual moduli space
- ASD deformation complex
- Bogomolny monopole equation
- Instanton number
- Regular ASD connection
- Self-dual and anti-self-dual connection
- Uhlenbeck compactness theorem
- Uhlenbeck removable singularity theorem
- Yang-Mills energy identity
- Yang-Mills instanton
- Yang-Mills moduli space
Flat connections and moduli
- Atiyah-Bott symplectic form
- Character variety
- Curvature as a moment map
- Deformation complex of a flat connection
- Holonomy correspondence for flat connections
- Moduli space of flat connections
- Representation variety
- Tangent space to flat-connection moduli
- Local system
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Uncategorized
- Adjoint action of a Lie group
- Adjoint bundle
- Adjoint representation of a Lie algebra
- Ambrose–Singer curvature span
- Ambrose–Singer holonomy theorem
- Associated bundle
- Associated connection theorem
- Associated vector bundle
- Atiyah algebroid of a principal bundle
- Atiyah sequence
- Basic differential form on a principal bundle
- Bundle atlas
- Bundle isomorphism
- Bundle map
- Bundle metric
- Bundle morphism
- Bundle of connections
- Bundle of orbits
- Cartan connection
- Cartan
- Cartan
- Characteristic class
- Chern character via Chern–Weil theory
- Chern class via Chern–Weil theory
- Chern–Simons form
- Gauge transformation behavior of Chern–Simons forms
- Chern–Weil form
- Chern–Weil theorem
- Classification of principal G-bundles by homotopy classes of maps into BG
- Classifying map of a principal bundle
- Classifying space BG
- Clutching function
- Coadjoint action of a Lie group
- Cocycle condition for transition functions
- Complex vector bundle
- Conjugation action of a Lie group on itself
- Connection 1-form on a principal bundle
- Connection on a vector bundle
- Adjoint bundle Ad(P)
- Adjoint Lie algebra bundle ad(P)
- Associated bundle from a principal bundle and a left G-space
- Atiyah algebroid TP/G and its anchor
- Change of connection formula for Chern Weil characteristic forms
- Construction: Connection on Fr(E) induced by a vector bundle connection (and conversely)
- Curvature of an induced associated connection via a representation
- Construction: Frame bundle Fr(E) of a vector bundle E
- Gauge transformation of local bundle data
- Holonomy element from parallel transport around a loop
- Horizontal lift of curves and uniqueness
- Induced connection on an associated bundle via horizontals
- Induced covariant derivative on sections of an associated vector bundle
- Induced map on associated bundles
- Construction: local trivialization from a local section
- Parallel transport map along a curve
- Product principal bundle (fiber product over the base)
- Construction: pullback principal bundle
- Construction: quotient manifold P/G for a free proper action
- Reduction of structure group via H-valued transition functions
- Construction: Splitting of the Atiyah sequence from a principal connection
- Transition functions from local sections
- Convention: Ad(P) uses the conjugation action on G
- Convention: associated bundles use a left action on the fiber
- Convention: fundamental vector field uses the right action
- Convention: local curvature is F = dA + A wedge A
- Convention: manifolds are smooth, Hausdorff, and second countable
- Convention: principal bundles use a right G-action on P
- Theorem: A trivial principal bundle admits a global section
- Chern–Weil classes are independent of the connection
- Gauge equivalence classes of connections form an orbit space
- Every principal bundle admits a connection
- Every vector bundle admits a connection
- Isomorphic principal bundles have the same Chern–Weil classes
- Nontrivial principal bundle with no global section
- Covariant derivative of a section
- Covariant exterior derivative on ad(P)-valued forms
- Curvature
- Curvature 2-form in a frame
- Curvature 2-form of a principal connection
- Curvature of a vector bundle connection
- Differential of a Lie group homomorphism
- Differential of a smooth map
- Dirac monopole connection on the Hopf bundle
- Direct sum vector bundle (Whitney sum)
- Dual vector bundle
- Ehresmann connection
- Equivalence of cocycles
- Theorem: Principal connections are equivalent to splittings of the Atiyah sequence
- Equivalent bundle atlases
- Equivariant cohomology (Cartan model)
- Equivariant local trivialization
- Equivariant map
- Equivariant map associated to a section of an associated bundle
- Euler class via Chern–Weil theory
- Reducing a GL(n)-structure to O(n) using a bundle metric
- Existence of partitions of unity on paracompact manifolds
- Theorem: Existence of principal connections on smooth manifolds
- Theorem: Existence and uniqueness of horizontal lifts of curves
- Exponential map
- Extension of structure group
- Exterior covariant derivative
- Exterior power bundle
- Fiber of a map
- Fiber-preserving map
- Fibered manifold
- Flat connection on a trivial bundle
- Flat principal connection
- Flat vector bundle connection
- Frame bundle of a manifold
- Frame bundle of a rank-n vector bundle
- Free action
- Gauge group
- Gauge transform of a local connection form
- Gauge transformation
- Good cover
- Hermitian metric
- Holonomy algebra
- Holonomy group
- Holonomy reduction principle
- Holonomy representation
- Homotopy class
- Hopf fibration as a principal U(1)-bundle
- Horizontal differential form on a principal bundle
- Horizontal distribution
- Horizontal lift of a curve
- Horizontal lift of a tangent vector
- Horizontal lift of a vector field
- Horizontal subbundle
- Integrable horizontal distribution
- Integrality of Chern classes
- Interior product
- Invariant differential form
- Invariant function
- Jet bundle (first jets of sections)
- Left Maurer–Cartan form
- Left translation
- Leibniz rule for a connection
- Lemma: Chern–Weil forms are basic
- Covariant exterior derivative preserves tensoriality
- Difference of two principal connections is tensorial
- Lemma: local curvature transforms by conjugation
- Local gauge transformation law for a connection
- Maurer–Cartan equation for the left Maurer–Cartan form
- Levi–Civita connection as a principal O(n)-connection
- Lie algebra of a Lie group
- Lie-algebra-valued k-form
- Lie derivative
- Local connection 1-form
- Local curvature 2-form
- Local curvature formula
- Local frame of a vector bundle
- Local gauge transformation
- Local trivialization
- Maurer–Cartan equation
- Moment map
- Naturality of Chern–Weil classes under pullback
- Open cover
- Orbit map
- Orbit of a group action
- Orientation of a real vector bundle
- Oriented frame
- Orthonormal frame bundle
- Orthonormal frame bundle
- Paracompact manifold
- Paracompact topological space
- Parallel section along a curve
- Parallel transport for an Ehresmann connection
- Theorem: Parallel transport defines a G-equivariant map between fibers
- Partition of unity subordinate to an open cover
- Pontryagin class via Chern–Weil theory
- Principal action
- Principal bundle automorphism
- Principal bundle isomorphism
- Principal bundle morphism
- Principal bundle over S1 from a clutching function
- Principal bundle transition function
- Principal connection
- Principal G-bundle
- Principal H-subbundle
- Principal homogeneous space (G-torsor)
- Proper action
- Flatness implies holonomy depends only on homotopy class of loops
- Flatness implies path-independence on simply connected domains
- Gauge group action on connections by pullback
- Leibniz rule for induced connections on associated bundles
- Parallel transport respects concatenation of paths
- Pullback
- Pullback bundle
- Theorem: Pullback of a principal bundle is a principal bundle
- Theorem: Pullback of a principal connection is a principal connection
- Pure gauge connection on a trivial bundle
- Quotient manifold (for a free proper action)
- Quotient space of an action (orbit space)
- Rank of a vector bundle
- Theorem: Reduction by cocycle (H-reduction iff H-valued transition functions exist)
- Reduction of structure group
- Representation of a Lie algebra
- Representation of a Lie group
- Reproduction property
- Restricted holonomy group
- Right Maurer–Cartan form
- Right principal action
- Right translation
- Section of Ad(P)
- Smooth action of a Lie group on a manifold
- Smooth chart
- Smooth fiber bundle
- Solder form on the frame bundle
- Special orthonormal frame bundle
- Special unitary frame bundle
- Splitting of the Atiyah sequence
- Stabilizer (isotropy subgroup)
- Symmetric power bundle
- The tangent bundle of the 2-sphere is nontrivial
- Tensor product vector bundle
- TFAE: Flat principal bundles (principal G-bundle with connection)
- TFAE: Metric-compatible connections on a metric vector bundle
- Equivalent descriptions of a principal connection
- Equivalent conditions for reduction of structure group
- Tensorial forms and ad(P)-valued forms
- Equivalent conditions for triviality of a principal bundle
- Connections on vector bundles via frame bundles
- Torsion 2-form
- Transgression form
- Transgression theorem (Chern–Simons)
- Transition function
- Transition matrix of a local frame
- Trivial fiber bundle
- Theorem: Global section implies a principal bundle is trivial
- Trivial principal bundle
- Trivial vector bundle
- Typical fiber
- Unitary frame bundle
- Universal principal bundle EG→BG
- Vector bundle
- Vector bundle morphism
- Vertical subbundle
- Vertical tangent space
- Vertical vector field
- Yang–Mills connection
- Yang–Mills equation
- Yang–Mills functional