In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X × G {\displaystyle...

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...

bundle I-bundle Natural bundle Principal bundle Projective bundle Pullback bundle Quasifibration Universal bundle Vector bundle Wu–Yang dictionary Seifert...

principal bundle. If, in addition, a right action is given on the fibre of the principal bundle, we describe how to construct any associated bundle by...

theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused...

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...

geometry, principal SU ⁡ ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles (or principal Sp ⁡ ( 1 ) {\displaystyle \operatorname {Sp} (1)} -bundles) are...

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...

Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action. A covariant...

geometry, principal U ⁡ ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles (or principal SO ⁡ ( 2 ) {\displaystyle \operatorname {SO} (2)} -bundles) are...

n-manifold M, for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures includes...

holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of these cases, the holonomy...

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...

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 and a continuous...

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....

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...

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...

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....

geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered...

formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form...

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...

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...

{S} }\colon {\mathbf {S} }\to M\,} associated to the corresponding principal bundle π P : P → M {\displaystyle \pi _{\mathbf {P} }\colon {\mathbf {P} }\to...

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...

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...

{\displaystyle p} . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on...

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...

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...

associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" and whether...

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...