# 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.

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.