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...

space. A common example of a principal bundle is the frame bundle F ( E ) {\displaystyle F(E)} of a vector bundle E {\displaystyle E} , which consists of...

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 various classical...

also defines a parallel transport on the frame bundle. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection...

{\displaystyle (M,g),\,} that is, an equivariant lift of the oriented orthonormal frame bundle F S O ( M ) → M {\displaystyle \mathrm {F} _{SO}(M)\to M} with respect...

tautological bundle G → G/H. A moving frame is a section of this bundle. It is moving in the sense that as the point of the base varies, the frame in the fibre...

frame 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...

Indeed, the vector bundles associated with the frame bundle are all sub-bundles of trivial bundles that extend to the ambient Euclidean space; a first...

linear connection on the frame bundle of a smooth manifold. Let π : P → M {\displaystyle \pi :P\to M} be a smooth principal G-bundle over a smooth manifold...

parallelizable. Chart (topology) Differentiable manifold Frame bundle Orthonormal frame bundle Principal bundle Connection (mathematics) G-structure Web (differential...

(orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle...

geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle F s ( M ) {\displaystyle F^{s}(M)} for some s...

only if the tangent bundle is trivial. By definition, a manifold M {\displaystyle M} is framed if and only if the tangent bundle T M {\displaystyle TM}...

orientation. In more concise terms, this says that the structure group of the frame bundle of E, which is the real general linear group GLn(R), can be reduced to...

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...

{F}}(TX)} is the frame bundle of the tangent bundle of the manifold X {\displaystyle X} , or more generally the frame bundle of a vector bundle over X {\displaystyle...

In symplectic geometry, the symplectic frame bundle of a given symplectic manifold ( M , ω ) {\displaystyle (M,\omega )\,} is the canonical principal S...

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. If for...

a lab frame, would be the particle detectors at the detection facility of a particle accelerator. Breit frame Center-of-mass frame Frame bundle Inertial...

choice of coordinate frame is a (local) section through that bundle. Coupled to the frame bundle is a second bundle, the spinor bundle. A section through...

In mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is locally a product space, but globally...

geometry of fiber bundles, the n vector fields { e a } a = 1 … n {\displaystyle \{e_{a}\}_{a=1\dots n}} define a section of the frame bundle i.e. a parallelization...

(n)} of the general linear group with the frame bundle of M. The resulting line bundle is known as the bundle of s-densities, and is denoted by | Λ | M...

frame field is associated with a unique coframe field, and vice versa; a coframe field is a set of four orthogonal sections of the cotangent bundle....

contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically...

{\displaystyle \theta } is the canonical vector-valued 1-form on the frame bundle, the torsion Θ {\displaystyle \Theta } of the connection form ω {\displaystyle...

structure group of the tangent bundle can be reduced in this way. Similar observations can be made for the frame bundle. Another way to define orientations...

{\displaystyle X_{K}\,} on the orthonormal frame bundle of its natural lift X ^ {\displaystyle {\hat {X}}\,} defined on the bundle of linear frames. Generalisations...

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...

vierbein is equivalent to a section of the frame bundle, and so defines a local trivialization of the frame bundle. To write down the equation we also need...