Connection forms on manifolds in Euclidean space

[lavinia]

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?

 

[jvicens]

Thank you for sharing your thoughts and questions regarding the generalization of Levi Civita connection on higher dimensional manifolds. I am always interested in exploring new ideas and finding solutions to challenging problems. After reading your post, I did some research and would like to share my insights with you.

Firstly, let me clarify that the concept of Levi Civita connection is closely related to the Riemannian geometry, which deals with manifolds equipped with a metric tensor. In your post, you mentioned that the embedding in Euclidean space determines two mappings of the unit tangent circle bundle into Euclidean space. This is indeed true for surfaces, but for higher dimensional manifolds, we need to consider the unit tangent sphere bundle instead.

The unit tangent sphere bundle is a submanifold of the tangent bundle, which consists of all unit length tangent vectors at every point on the manifold. Similar to the surface case, we can define two mappings on the unit tangent sphere bundle, one sending a unit tangent vector to its parallel vector at the origin, and the other sending it to its 90 degree positively oriented tangential rotation.

Now, the key to generalizing the Levi Civita connection is to consider the tangent sphere bundle as a principal bundle with the structure group being the orthogonal group O(n). This means that at every point on the manifold, we can rotate the tangent sphere by an element of O(n) and still obtain the same sphere. In other words, the sphere is invariant under rotations by O(n).

Using this idea, we can define a connection on the tangent sphere bundle by considering the infinitesimal rotations, which are elements of the Lie algebra of O(n). This is a set of n x n skew-symmetric matrices, denoted by so(n). Now, we can define a 1-form on the tangent sphere bundle, similar to the one you mentioned in your post, but instead using the Lie algebra elements <de,ie>, where de is the differential of the unit tangent vector and ie is the infinitesimal rotation.

This 1-form will be invariant under infinitesimal rotations, and by using the structure of the tangent sphere bundle as a principal bundle, we can show that it is also invariant under rotations by O(n). Therefore, this 1-form determines a Levi Civita connection on the manifold, just like in the surface case.

I hope this helps you in your generalization process. If you have any further questions