I would like to see how to derive the connection 1 form on the bundle of oriented orthonormal frames for a manifold embedded in Euclidean space.
It seems that the geometry resides in a collection of 1 forms on the frame bundle the arise naturally from the embedding. Given a frame (e1,...,en) over a point in the manifold, map it to e1. This is a smooth map of the frame bundle into Euclidean space. The 1 forms are then defined as
wi = de1.ei 1=2,...n
The action of the rotation group on the frame bundle leaves these forms invariant because it preserves the Euclidean inner product. That is
dL(e1).L(ei) = Ld(e1).L(ei) = de1.ei
It is tempting to think that these 1 forms are the components of a connection 1 form but I do not not see how to formalize it.