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.

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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 $ n $- dimensional differentiable manifold $ M $ is a vector bundle isomorphism of its tangent bundle $ TM $ with the trivial bundle $ M \times \mathbf R ^ {n} $( so that $ M $ is parallelizable). Using the standard basis $ ( e _ {1} \dots e _ {n} ) $ of $ \mathbf R ^ {n} $ such an isomorphism defines a frame field: it assigns to every $ x \in M $ a frame, or basis, of the tangent space at that point.

The frame bundle over a manifold $ M $ is the principal fibre bundle with structure group $ \mathop{\rm GL} _ {n} ( \mathbf R ) $ whose fibre over $ x \in M $ is the collection of all bases (frames) of $ T _ {x} M $, the tangent space at that point.

A $ k $- frame $ v ^ {k} $ in $ \mathbf R ^ {n} $ is an ordered set of $ k $ linearly independent vectors. Let $ V _ {n,k} $ denote the set of all $ k $- frames in $ \mathbf R ^ {n} $. Let $ G ( k) $ be the subgroup of $ \mathop{\rm GL} _ {n} ( \mathbf R ) $ leaving a fixed frame $ v _ {0} ^ {k} $ invariant. Then $ V _ {n,k} = \mathop{\rm GL} _ {n} ( \mathbf R ) / G ( k) $. Thus, $ V _ {n,k} $ has a real-analytic structure. It is called the Stiefel manifold of $ k $- frames in $ n $- space.

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