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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...

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Connection 1-forms do not transform in the same way as ordinary 1-forms, so the local expressions you have written do not patch up to a well-defined connection. Otherwise you could just set all the connection 1-forms to zero and get a flat connection on any vector bundle. –  Johannes Nordström Sep 20 '12 at 9:33
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Thank you very much. I'm still not able to check that my definitions don't patch, but I'll try... Anyway I wanted to ask a related question that is: what do we need to talk about "Chern numbers"?

I'm a bit confused. Somewhere I read that I need a principal G-bundle. In that case I could consider (having base manyfold $S^2$) the frame bundle with the group $GL(2,R)$ acting on it?

Somewhere else (this is wikipedia) I found that the object to be considered is an Hermitian complex vector bundle. In this case I could consider the vector bundle of complexified tangent spaces of the sphere?

Are these cases both good and well defined? The Chern number can differ?

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This is not the appropriate forum. Try math.stackexchange.com –  Chris Gerig Sep 21 '12 at 17:58
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I'm not sure that I agree that this needs to shunted off to stackexchange. However, it is bad form to ask a question in this way. If you want to ask another question, you should ask a new question, and it should be more precisely formulated than this. –  Donu Arapura Sep 21 '12 at 18:25
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ok... maybe I'm not in the right forum for asking more clarifications. If somebody wants to help, I've posted where Gerig suggested: math.stackexchange.com/questions/200657/… –  ShortEdge Sep 22 '12 at 10:57
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