Fiber Bundles

Core idea

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
• Fiber Of A Map Preimage Fiber
• 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 Chart Coordinate Chart
• Smooth Embedding
• Smooth Immersion
• Smooth Manifold
• Smooth Map
• Smooth Submersion
• Tangent Bundle
• Tangent Space At A Point
• Vector Field
• Wedge Product Of Differential Forms

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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 [M,BG]
• 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