# Connection forms on manifolds in Euclidean space

Gold Member
This question comes from trying to generalize something that it easy to see for surfaces.

Start with an oriented surface smoothly embedded in Euclidean space. The embedding determines two mappings of the unit tangent circle bundle into Euclideam space.

Given a unit length tangent vector,e, at p the first map sends it to the parallel vector at the origin. The second map maps e to its 90 degree positively oriented tangential rotation.

These two maps, e and ie, determine a 1 form on the tangent circle bundle by the rule

w = <de,ie> where <,> is the Euclidean inner product and de is the differential of e.

It is standard and easy to see that w determines a Levi Civita connection on the surface. That is: w is invariant under rotations of the tangent circles and is normalized.

I tried to generalize this construction to higher dimensional manifolds. In this case one gets n-1 one forms like e but I had trouble showing invariance under rotation.

What is the correct generalization?