AI & ChatGPT searches , social queries for PRINCIPAL BUNDLE
Search references for PRINCIPAL BUNDLE. Phrases containing PRINCIPAL BUNDLE
See searches and references containing PRINCIPAL BUNDLE!
Search references for PRINCIPAL BUNDLE. Phrases containing PRINCIPAL BUNDLE
See searches and references containing PRINCIPAL BUNDLE!PRINCIPAL 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
Concept in mathematics
transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal G-connection on a principal G-bundle P {\displaystyle
Connection_(principal_bundle)
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
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
Fiber bundle
transition from a bundle with fiber F {\displaystyle F} , on which G {\displaystyle G} acts, to the associated principal bundle (namely the bundle where the fiber
Associated_bundle
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)
Differential geometry construct on fiber bundles
Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action. A covariant
Ehresmann_connection
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
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
Holonomy
Principal bundle formulation of the Higgs field
characterized as a reduction of the structure group G {\displaystyle G} of a principal bundle P → X {\displaystyle P\to X} to its closed subgroup H {\displaystyle
Higgs_field_(classical)
Special type of principal bundle
geometry, principal SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles (or principal Sp ( 1 ) {\displaystyle \operatorname {Sp} (1)} -bundles) are
Principal_SU(2)-bundle
Special type of principal bundle
geometry, principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles (or principal SO ( 2 ) {\displaystyle \operatorname {SO} (2)} -bundles) are
Principal_U(1)-bundle
Structure group sub-bundle on a tangent frame bundle
structure group G {\displaystyle G} , is a principal G {\displaystyle G} -subbundle of the tangent frame bundle F M {\displaystyle {\text{F}}M} (or GL
G-structure_on_a_manifold
Math/physics concept
formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form
Connection_form
Algebraic geometry analog of a principal bundle in algebraic topology
In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski
Torsor_(algebraic_geometry)
Group of gauge symmetries in Yang–Mills theory
symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle P → X {\displaystyle P\to X} with a structure
Gauge_group_(mathematics)
Generalization of a fiber bundle
principal bundle is a fiber bundle endowed with a right group action with certain properties. One example of a principal bundle is the frame bundle.
Bundle_(mathematics)
Term in differential geometry
geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered
Curvature_form
Right inverse of a fiber bundle map
bundle of M {\displaystyle M} . Likewise, a 1-form on M {\displaystyle M} is a section of the cotangent bundle. Sections, particularly of principal bundles
Section_(fiber_bundle)
Concept in differential geometry
language of principal bundles. The collection of oriented orthonormal frames of a vector bundle form a frame bundle PSO(E), which is a principal bundle under
Spin_structure
Set on which a group acts freely and transitively
a base point). The principal homogeneous space concept is a special case of that of principal bundle: it means a principal bundle with base a single point
Principal_homogeneous_space
There is a composite bundle P → P / H → X {\displaystyle P\to P/H\to X} where P → P / H {\displaystyle P\to P/H} is a principal bundle with a structure group
Connection_(composite_bundle)
Function in mathematics
Lie groups. An Ehresmann connection is a connection in a fibre bundle or a principal bundle by specifying the allowed directions of motion of the field.
Connection_(mathematics)
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
Mathematical theories
non-zero. This can be used to find obstructions to trivializations of principal bundles. Because any map can be turned into a fibration, this construction
Obstruction_theory
Geometric structure
{S} }\colon {\mathbf {S} }\to M\,} associated to the corresponding principal bundle π P : P → M {\displaystyle \pi _{\mathbf {P} }\colon {\mathbf {P} }\to
Spinor_bundle
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
Partial differential equations whose solutions are instantons
of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the
Yang–Mills_equations
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
this bundle shows that the higher homotopy groups of spheres are not trivial in general. It also provides a basic example of a principal bundle, by identifying
Hopf_fibration
algebraic geometry, given a line bundle L on a smooth variety X, the bundle of n-th order principal parts of L is a vector bundle of rank ( n + dim ( X ) n )
Bundle_of_principal_parts
Gauge field loop operator
G} forming what's known as a fiber of the fiber bundle. These fiber bundles are called principal bundles. Locally the resulting space looks like R d × G
Wilson_loop
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
bundles associated to these principal bundles via the natural action of G on F k {\displaystyle \mathbb {F} ^{k}} are just the tautological bundles over
Stiefel_manifold
be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric. If M is a
Bundle_metric
_{q}} and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by Bun G ( X ) {\displaystyle \operatorname {Bun}
Moduli stack of principal bundles
Moduli_stack_of_principal_bundles
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)
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
of the universal SO ( n ) {\displaystyle \operatorname {SO} (n)} principal bundle ESO ( n ) → BSO ( n ) {\displaystyle \operatorname {ESO} (n)\rightarrow
Classifying_space_for_SO(n)
Association of cohomology classes to principal bundles
associating to 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"
Characteristic_class
Type of differentiable manifold
{\displaystyle p} . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on
Parallelizable_manifold
Construct allowing differentiation of tangent vector fields of manifolds
either as a Cartan connection for the affine group or as a principal connection on the frame bundle. The main invariants of an affine connection are its torsion
Affine_connection
System of moving vectors in differential geometry
supplies a lifting of curves from the manifold to the total space of a principal bundle. Such curve lifting may sometimes be thought of as the parallel transport
Parallel_transport
Solitons in Euclidean spacetime
Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of
Instanton
mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle. Physically, such a monopole can be interpreted
Monopole_(mathematics)
Topics referred to by the same term
(affine bundle) Connection (composite bundle) Connection (fibred manifold) Connection (principal bundle), gives the derivative of a section of a principal bundle
Connection
Quotient of a weakly contractible space by a free action
has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle E G → B G {\displaystyle EG\to
Classifying_space
Generalization of affine connections
specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold
Cartan_connection
Physical theory with fields invariant under the action of local "gauge" Lie groups
Connections (gauge connection) define this principal bundle, yielding a covariant derivative ∇ in each associated vector bundle. If a local frame is chosen (a local
Gauge_theory
Topological construct
group. This is a bundle over N {\displaystyle N} with fibre Homeo ( F ) {\displaystyle \operatorname {Homeo} (F)} and is a principal bundle. Denote it by
Clutching_construction
forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections. A Lie-algebra-valued
Lie algebra–valued differential form
Lie_algebra–valued_differential_form
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
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
of the universal SU ( n ) {\displaystyle \operatorname {SU} (n)} principal bundle ESU ( n ) → BSU ( n ) {\displaystyle \operatorname {ESU} (n)\rightarrow
Classifying_space_for_SU(n)
N)\to \operatorname {Emb} (M,N)/\operatorname {Diff} (M)} is a principal fiber bundle with structure group Diff ( M ) {\displaystyle \operatorname {Diff}
Convenient_vector_space
mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal G {\displaystyle G} -bundle P {\displaystyle P} over a manifold M {\displaystyle M}
Atiyah_algebroid
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
Type of derivative in differential geometry
principal bundle. Now, if we're given a vector field Y over M (but not the principal bundle) but we also have a connection over the principal bundle,
Lie_derivative
Mathematical concept
form can also be characterized abstractly as the unique principal connection on the principal bundle G. Indeed, it is the unique g = TeG valued 1-form on
Maurer–Cartan_form
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
Mechanism that explains the generation of mass for gauge bosons
cannot be too much more massive than the vectors. Electromagnetic mass Higgs bundle Quantum triviality Weinberg angle Yang–Mills–Higgs equations Englert's co-author
Higgs_mechanism
Differential geometry technique
The most economical way to do this is to use a G-subbundle PM of the principal bundle of linear coframes LM, although this approach can lead to unnecessary
Cartan's_equivalence_method
Internal groupoid in the category of smooth manifolds
s^{-1}(x)} at a point x ∈ M {\displaystyle x\in M} is a principal G x {\displaystyle G_{x}} -bundle over the orbit O x {\displaystyle {\mathcal {O}}_{x}}
Lie_groupoid
Mathematical construct of fiber bundles
form on the frame bundle of a manifold. The reason for the name is that a solder form solders (or attaches) the abstract principal bundle to the manifold
Solder_form
Hypothetical particle with one magnetic pole
over a principal G-bundle over spacetime. G is the gauge group, and it acts on each fiber of the bundle separately. A connection on a G-bundle tells you
Magnetic_monopole
Topics referred to by the same term
(mathematics), a connection over a principal bundle G with a section (the Higgs field) of the associated adjoint bundle Monopole, the first term in a multipole
Monopole
Attempt to extend Yang–Mills theory to gravity
G} of a principal bundle P → X {\displaystyle P\to X} is reducible to a closed subgroup H {\displaystyle H} , i.e., there exists a principal subbundle
Gauge_gravitation_theory
Classical field theories on fiber bundles
subtlety. An associated vector bundle E → π M {\displaystyle E\xrightarrow {\pi } M} associated to the principal bundle P {\displaystyle P} through a representation
Covariant classical field theory
Covariant_classical_field_theory
Concept in differential geometry
of a 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
Exterior_covariant_derivative
Intrinsic geometric structures in mathematics
connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the
Riemannian connection on a surface
Riemannian_connection_on_a_surface
Natural moving frame in differential geometry of surfaces
structure of a principal bundle on M (the structure group for the bundle is O(p) × O(n − p).) This principal bundle embeds into the bundle of Euclidean
Darboux_frame
Modern theory of gravitation that combines supersymmetry and general relativity
a Spin(3,1) principal bundle over it. This principal bundle represents the local Lorentz symmetry. In addition, we have a vector bundle T over the manifold
Supergravity
category over the category of S-schemes, where an object over T is a principal G-bundle P → T {\displaystyle P\to T} together with equivariant map P → X {\displaystyle
Quotient_stack
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
Correspondsnce between Higgs bundles and fundamental group representations
vector bundle with flat connection as follows. The universal cover X ^ {\displaystyle {\hat {X}}} of X {\displaystyle X} is a principal bundle over X
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
Topologically nontrivial gauge transformations
topological space M, a topological group G and a principal G-bundle over M, a global section of that principal bundle is a gauge fixing and the process of replacing
Large_gauge_transformation
Formulation to quantize gauge field theories in physics
right-invariant local fields on the principal gauge bundle, and different local sections through a portion of the gauge bundle, related by passive transformations
BRST_quantization
Manifold upon which it is possible to perform calculus
as a section of the frame bundle F(M), a GL(n, R) principal bundle made up of the set of all frames over M. The frame bundle is useful because tensor fields
Differentiable_manifold
Generalization of an ordered basis of a vector space
of the pullback of the tautological bundle to M. Intrinsically a moving frame can be defined on a principal bundle P over a manifold. In this case, a moving
Moving_frame
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
Concept in differential topology
a smooth vector bundle of rank k over M and let π : F(E) → M be the (associated) frame bundle of E, which is a principal GLk(R) bundle over M. The pullback
Vector-valued differential form
Vector-valued_differential_form
to the tangent bundle being trivial. Partition of unity PL-map Poincaré lemma Principal bundle – A principal bundle is a fiber bundle P → B {\displaystyle
Glossary of differential geometry and topology
Glossary_of_differential_geometry_and_topology
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
Limit of spheres in algebraic topology
contractible and hence appears as the total space of multiple universal principal bundles. With the usual definition S n = { x ∈ R n + 1 | ‖ x ‖ 2 = 1 } {\displaystyle
Infinite-dimensional_sphere
Introduction and Reference on Differential Geometry
discussion of curvature representation of characteristic classes of principal bundles (Chern–Weil theory), it covers Euler classes, Chern classes, and Pontryagin
Foundations of Differential Geometry
Foundations_of_Differential_Geometry
Branch of mathematics studying (smooth) functions of manifolds
associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted"
Geometric_topology
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
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
forms a G L ( n ) {\displaystyle GL(n)} principal bundle over M {\displaystyle M} , which is called the coframe bundle. Frame fields in general relativity
Coframe
Algebraic topology theory
{\displaystyle G} on X {\displaystyle X} and the principal bundle E G → B G {\displaystyle EG\to BG} . The bundle X → E G × G X → B G {\displaystyle X\to EG\times
Equivariant_cohomology
Operation on fibered manifolds
of the quotient bundle C = J1P/G → M, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle VP/G → M whose typical
Connection_(fibred_manifold)
algebra bundle is a vector bundle. Examples include the tensor-algebra bundle, exterior bundle, and symmetric bundle associated to a given vector bundle, as
Algebra_bundle
Construct in mathematics
principal H {\displaystyle H} -bundles on U {\displaystyle U} with isomorphism as morphisms (thus the category is a groupoid). As principal bundles glue
Gerbe
formula comes from the fact that it applies to the moduli stack of principal bundles on a curve over a finite field (in some instances indirectly, via
Behrend's_trace_formula
Mathematical behavior near singularities
geometry, an analogous role is played by parallel transport. In a principal bundle B {\displaystyle B} over a smooth manifold M {\displaystyle M} , a
Monodromy
Conflict of interest when one person acts on another's behalf
The principal–agent problem (often abbreviated agency problem) refers to the conflict in interests and priorities that arises when one person or entity
Principal–agent_problem
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
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
Expression that may be integrated over a region
U(1) principal bundle on which both electromagnetism and general gauge theories may be described. The connection form for the principal bundle is the
Differential_form
Concept in number theory
local-global principles, and adelic descriptions of divisors, line bundles, and principal bundles on algebraic curves. Let K {\displaystyle K} be a global field
Adele_ring
Object in differential geometry
characterization of torsion, applies to the frame bundle FM of the manifold M. This principal bundle is equipped with a connection form ω, a gl(n)-valued
Torsion_tensor
Relativistic quantum mechanical wave equation
μ a {\displaystyle A_{\mu }^{a}} , formally the connection on the principal bundle, which necessarily transforms in the adjoint representation of the
Dirac_equation
PRINCIPAL BUNDLE
PRINCIPAL BUNDLE
PRINCIPAL BUNDLE
PRINCIPAL BUNDLE
PRINCIPAL BUNDLE
PRINCIPAL BUNDLE
PRINCIPAL BUNDLE
PRINCIPAL BUNDLE
PRINCIPAL BUNDLE