The **frame bundle** of *E*, denoted by F(*E*) or F_{GL}(*E*), is the disjoint union of all the *F*_{x}:

Each point in F(*E*) is a pair (*x*, *p*) where *x* is a point in *X* and *p* is a frame at *x*. There is a natural projection *π* : F(*E*) → *X* which sends (*x*, *p*) to *x*. The group GL_{k}(**R**) acts on F(*E*) on the right as above. This action is clearly free and the orbits are just the fibers of *π*.

The frame bundle F(*E*) can be given a natural topology and bundle structure determined by that of *E*. Let (*U*_{i}, *φ*_{i}) be a local trivialization of *E*. Then for each *x* ∈ *U*_{i} one has a linear isomorphism *φ*_{i,x} : *E*_{x} → **R***k*. This data determines a bijection

given by

With these bijections, each *π*−1(*U*_{i}) can be given the topology of *U*_{i} × GL_{k}(**R**). The topology on F(*E*) is the final topology coinduced by the inclusion maps *π*−1(*U*_{i}) → F(*E*).

With all of the above data the frame bundle F(*E*) becomes a principal fiber bundle over *X* with structure group GL_{k}(**R**) and local trivializations ({*U*_{i}}, {*ψ*_{i}}). One can check that the transition functions of F(*E*) are the same as those of *E*.

The above all works in the smooth category as well: if *E* is a smooth vector bundle over a smooth manifold *M* then the frame bundle of *E* can be given the structure of a smooth principal bundle over *M*.

Read more about Frame Bundle: Associated Vector Bundles, Tangent Frame Bundle, Orthonormal Frame Bundle, *G*-structures

### Other articles related to "frame bundle, bundle, frame, bundles":

**Frame Bundle**

... See also Connection (principal

**bundle**) Let M be a surface embedded in E3 ... An ordered basis or

**frame**v, w in the tangent space is said to be oriented if det(v, w) is positive ... The tangent

**bundle**of M consists of pairs (p, v) in M x E3 such that v lies in the tangent plane to M at p ...

... See also Connection form and Connection (principal

**bundle**) The theory of connections according to Élie Cartan, and later Charles Ehresmann, revolves around a principal

**bundle**E the exterior ... All "natural" vector

**bundles**associated with the manifold M, such as the tangent

**bundle**, the cotangent

**bundle**or the exterior

**bundles**, can be constructed from the

**frame bundle**using the representation ... as a way of lifting vector fields on M to vector fields on the

**frame bundle**E invariant under the action of the structure group K ...

... Cartan's approach was rephrased in the modern language of principal

**bundles**by Ehresmann, after which the subject rapidly took its current form following contributions by Chern, Ambrose and ... operators acting on functions on the manifold, to differential operators on the

**frame bundle**in the case of an embedded surface, the lift is very simply described ... 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 order differential operator can always be applied to a section of a ...

**Frame Bundle**-

*G*-structures

... natural to consider a subbundle of the full

**frame bundle**of M which is adapted to the given structure ... it is natural to consider the orthonormal

**frame bundle**of M ... The orthonormal

**frame bundle**is just a reduction of the structure group of FGL(M) to the orthogonal group O(n) ...

### Famous quotes containing the words bundle and/or frame:

“We styled ourselves the Knights of the Umbrella and the *Bundle*; for, wherever we went ... the umbrella and the *bundle* went with us; for we wished to be ready to digress at any moment. We made it our home nowhere in particular, but everywhere where our umbrella and *bundle* were.”

—Henry David Thoreau (1817–1862)

“Predictions of the future are never anything but projections of present automatic processes and procedures, that is, of occurrences that are likely to come to pass if men do not act and if nothing unexpected happens; every action, for better or worse, and every accident necessarily destroys the whole pattern in whose *frame* the prediction moves and where it finds its evidence.”

—Hannah Arendt (1906–1975)