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...

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, 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...

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...

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

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

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

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...

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...

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...

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....

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...

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,...

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...

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

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...

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...

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...

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...

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...

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...

N)\to \operatorname {Emb} (M,N)/\operatorname {Diff} (M)} is a principal fiber bundle with structure group Diff ⁡ ( M ) {\displaystyle \operatorname {Diff}...

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...