Lecture 9: Moving Frames in the Nonhomogenous Case: Frame Bundles

LetM be an n-dimensional Riemannian manifold. ThenM is a differentiable manifold with a smoothly varying inner product 〈, 〉x on each tangent space TxM . An orthonormal frame at the point x ∈ M is an orthonormal basis {e1, . . . , en} of the tangent space TxM . The set of orthonormal frames at each point is isomorphic to the Lie group O(n), and the set of orthonormal frames on M forms a principal bundle over M with fiber O(n), called the frame bundle of M and denoted F(M). An orthonormal frame on the open set U ⊂ M is a choice of an orthonormal frame {e1(x), . . . , en(x)} at each point x ∈ U such that each ei is a smooth local section of the tangent bundle TM ; any such frame is a smooth section of the frame bundle F(M).