Frame Bundle - Orthonormal Frame Bundle

Orthonormal Frame Bundle

If a vector bundle E is equipped with a Riemannian bundle metric then each fiber E_x is not only a vector space but an inner product space. It is then possible to talk about the set of all of orthonormal frames for E_x. An orthonormal frame for E_x is an ordered orthonormal basis for E_x, or, equivalently, a linear isometry

where Rk is equipped with the standard Euclidean metric. The orthogonal group O(k) acts freely and transitively on the set of all orthonormal frames via right composition. In other words, the set of all orthonormal frames is a right O(k)-torsor.

The orthonormal frame bundle of E, denoted F_O(E), is the set of all orthonormal frames at each point x in the base space X. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank k Riemannian vector bundle E → X is a principal O(k)-bundle over X. Again, the construction works just as well in the smooth category.

If the vector bundle E is orientable then one can define the oriented orthonormal frame bundle of E, denoted F_SO(E), as the principal SO(k)-bundle of all positively-oriented orthonormal frames.

If M is an n-dimensional Riemannian manifold, then the orthonormal frame bundle of M, denoted F_OM or O(M), is the orthonormal frame bundle associated to the tangent bundle of M (which is equipped with a Riemannian metric by definition). If M is orientable, then one also has the oriented orthonormal frame bundle F_SOM.

Given a Riemannian vector bundle E, the orthonormal frame bundle is a principal O(k)-subbundle of the general linear frame bundle. In other words, the inclusion map

is principal bundle map. One says that F_O(E) is a reduction of the structure group of F_GL(E) from GL_k(R) to O(k).