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

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

““There is Lowell, who’s striving Parnassus to climb

With a whole bale of isms tied together with rhyme,

He might get on alone, spite of brambles and boulders,

But he can’t with that *bundle* he has on his shoulders,

The top of the hill he will ne’er come nigh reaching

Till he learns the distinction ‘twixt singing and preaching;”

—James Russell Lowell (1819–1891)

“But angels come to lead frail minds to rest

In chaste desires, on heavenly beauty bound.

You *frame* my thoughts, and fashion me within;

You stop my tongue, and teach my heart to speak;”

—Edmund Spenser (1552?–1599)