Ehresmann Connection of the tangential bundle & Chern classes

I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking.

The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the projection from $TE$ onto the Vertical Bundle. So $v \in \Omega^1(E,TE)$

One can define a Curvature form from this given by $\omega = dv + v\wedge v$, and have $\omega \in \Omega^2(E,TE)$.

Now, we can do this with the Tangent Bundle $TM\to M$ as well, and have for our Ehresmann connection $v\in \Omega^1(TM,TTM)$ and Curvature $\omega\in \Omega^2(TM,TTM)$.

According to http://en.wikipedia.org/wiki/Curvature_form , this curvature form $\omega$ should equal the Riemann Curvature Tensor $R$, $R(X,Y) = \omega(X,Y)$

But as I understand it, our Ehresmann Connection $v$ maps elements of $TTM$ back to $TTM$. And the Levi-Civita Connection is only interested in Elements of $TM$.

(1) how come the Curvature of the two agree? Aren't they something completely different?

Now I'm having the same problem with Chern classes. Chern Classes are given as Forms over $M$ ($\Omega(M,TM)$). But we can compute these Chern Classes via the Curvature form, which is a Form over $E$, $\Omega(E,TE)$ (or in our case $\Omega(TM,TTM)$, however, here I don't even understand how this can be in general).

(2) Again, aren't these two very different?

Solution 1:

I am not sure I completely understand your question but maybe this will help.

A differential form on the principal bundle is the pull back (under the bundle projection map) of a form on the manifold if and only if the form is invariant under the action of the Lie group and if it is zero whenever one of the vectors it is being evaluated on is vertical. This is easy to prove.

The curvature 2-form of a connection satisfies the second condition since by definition K(X,Y) = dw(hX,hY) where w is the connection 1-form on the principal bundle and hX and hY are the horizontal components of the vectors, X and Y. But it does not satisfy the first condition since its values change by that adjoint map under the action of the Lie group on the fiber.

So to get the curvature form down onto the manifold, one needs a local section of the principal bundle( a global section does not usually exist). The pull back of the curvature form under a local section is called a gauge field. If one has a Levi-Cevita connection then a local section is an orthonormal frame and the Lie group is SO(n).The connection 1-forms are seen to be skew symmetric because the Lie algebra of SO(n) is the skew symmetric matrices.

With Characteristic classes such as Chern classes both conditions are satisfied. Chern classes are the projections to the manifold of differential forms that are both invariant under the group action on the fiber and which evaluate to zero whenever one of the parameter vectors is vertical. They do not require a local section of the principal bundle but rather are globally defined on both the principal bundle and the manifold.