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...

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...

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

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...

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...

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

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...

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

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...

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...

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...

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

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....

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...

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....

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

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...