Frame bundle

In mathematics, a frame bundle is a principal fiber bundle ${\displaystyle F(E)}$ associated with any vector bundle ${\displaystyle E}$. The fiber of ${\displaystyle F(E)}$ over a point ${\displaystyle x}$ is the set of all ordered bases, or frames, for ${\displaystyle E_{x}}$. The general linear group acts naturally on ${\displaystyle F(E)}$ via a change of basis, giving the frame bundle the structure of a principal ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$-bundle (where k is the rank of ${\displaystyle E}$).

The frame bundle of a smooth manifold is the one associated with its tangent bundle. For this reason it is sometimes called the tangent frame bundle.