I'd just like to check whether my visualization for a way to get a non-flat connection is correct. The definition I am using for a connection is, for a fiber bundle $\rho:E \to B$, a smooth assignment to each $e \in E$ a subspace of $T_eE$ transverse to the fiber containing $e$.

For simplicity in visualization, I'll choose $\mathbb{R} \times \mathbb{R}$ as the connection, as a fiber bundle over the first factor. The flat connection would then just be the assignment of the tangent space parallel to the first factor, so $(x,y)$ gets associated the tangent space that looks like $\mathbb{R} \times y$.

So my idea for creating a non-flat connection is to take some diffeomorphism $\phi$ of $\mathbb{R} \times \mathbb{R}$ that always fixes the horizontal subspaces, but maps fibers that used to be straight lines into wavy lines (that are still never horizontal)

Here's a picture in paint, where the black curves are fibers and the red lines are a few 1-dim subspaces of the tangent spaces at points in the fibers enter image description here

So then if we straighten the fibers by the diffiomorphism $\phi^{-1}$, and similarly push forward the tangent spaces in the picture by $d\phi^{-1}$, it seems like this would be non-flat connection.

Is this correct? Is this a good way to visualize what a non-flat connection looks like - that is, can we usually (in a local product neighbohood) view the connection as some horizontal foliation, through which the fibers move transversely, but not necessarily orthogonally?

I have other related questions, but I suppose it's better to just to clarify this and then ask those as separate questions afterwards.

  • $\begingroup$ Thanks for a good, thought stimulating question! $\endgroup$ – Yuri Vyatkin May 18 '14 at 7:42

You give a definition of connection but there is no mention of how you define flatness, so I'm afraid you misunderstand it.

"Wavy" does not mean "non-flat". In differential geometry flatness is the synonym of vanishing curvature. A more synthetic way of seeing flatness is as being locally "equivalent" to a flat model (in our case this is the Euclidean space with the standard Euclidean metric and the standard Euclidean connection).

Now, your notion of connection is so-called Eheresmann connection. It has a corresponding notion of curvature, and there is also a notion of integrabitily of distibutions which is related to the curvature by the following

Fact. If $E\to M$ is a fiber bundle, then a connection is flat if and only if the corresponding horizontal distribution $HE \subset TE$ is integrable.

On the other hand, a 1-dimensional distribution is always integrable as we know from the theory of ODE.

This means, that in you example $\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ all connections will be flat, no matter how wavy your fibers look like :-)

To prepare this answer I have used Chapter 5. Curvature on Bundles from Ch.Wendl's "Lecture Notes on Bundles and Connections". There (p.119) you can find a better picture that helps to visualize the curvature of a connection in this approach:

enter image description here

  • 1
    $\begingroup$ Yep, I was definitely misunderstanding (or at least forgetting) some things :) Thanks. $\endgroup$ – Carl May 18 '14 at 7:21

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