Frame
A set of linearly independent vectors taken in a definite order and placed at a common origin. Any three non-parallel vectors not lying in one plane can serve as a frame for the vectors in space. If the vectors building the frame are mutually orthogonal, then the frame is called orthogonal; if in this case the length of the vectors is equal to one, the frame is called orthonormal.
Comments
Usually a frame is called a basis (of vectors in space). In this sense, the word "frame" is also used in physics (frame of reference, cf. Reference system). For Frénet frame see Frénet trihedron.
A framing of an -dimensional differentiable manifold
is a vector bundle isomorphism of its tangent bundle
with the trivial bundle
(so that
is parallelizable). Using the standard basis
of
such an isomorphism defines a frame field: it assigns to every
a frame, or basis, of the tangent space at that point.
The frame bundle over a manifold is the principal fibre bundle with structure group
whose fibre over
is the collection of all bases (frames) of
, the tangent space at that point.
A -frame
in
is an ordered set of
linearly independent vectors. Let
denote the set of all
-frames in
. Let
be the subgroup of
leaving a fixed frame
invariant. Then
. Thus,
has a real-analytic structure. It is called the Stiefel manifold of
-frames in
-space.
References
[a1] | N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951) |
Frame. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frame&oldid=16777