Frame Bundle

The frame bundle on a Riemannian manifold  is a principal bundle. Over every point , the Riemannian metric determines the set of orthonormal frames, i.e., the possible choices for an orthonormal basis for the tangent space . The collection of orthonormal frames is the frame bundle.

The choice of an orthonormal frame at a point reflects a choice of coordinates, up to first order. Roughly speaking, the frame bundle reflects the ambiguity of choosing coordinates in Riemannian geometry. Consequently the frame bundle can be used to show that equations are well-defined, independent of coordinates, without any explicit reference to coordinates. A local bundle section of the frame bundle gives a moving frame, which can be used to calculate the classical tensors of differential geometry such as curvature.

An orthogonal matrix acts on an orthonormal basis to give another orthonormal basis. Consequently, the frame bundle on a -dimensional manifold admits an action by the orthogonal group , which makes the frame bundle a principal O(n)-bundle. Restricting attention to special kinds of frames, such as those which give a unitary basis, corresponds to frame bundle reduction.