Hi everybody. I think I get a problem with the definitions of the connections 1-form of a vector bundle.

Let's consider the sphere $S^2$ with its tangent bundle as a vector bundle. Let's take a tangent vector field $A$ regular on the sphere and construct using local patches these connections 1-forms:

$\omega^{\alpha}_{\beta}=$ $\delta^{{\alpha},{\beta}}\sum_j A_jdx^j$,

where $\delta^{{\alpha},{\beta}}$ is the Kronecker delta. I supposed that the vector field is regular and defined in the whole sphere, so the connection 1-forms do vanish in a certain point, because of the hairy ball theorem. Is it a problem? Why? I don't find in the definitions that the connections 1- form can't be zero...

Anyway from these connections we can construct the curvature 2-form and the first Chern number integrating that curvature. But the 2-form to integrate in ordero obtain the first Chern number here is essentially an exact form ($\Omega$=$dA$) and so the integration through the compact surface is zero.

But the first Chern number of these vector bundle should not be zero...