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lavinia

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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?