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This concept appears in Bott&Tu's GTM82. A flat vector bundle is one who has a particular trivialization with locally constant transition functions. Then my question is whether every vector bundle over a manifold admits such a trivialization.

btw: Are the tags correct?

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up vote 9 down vote accepted

No. Using the Chern-Weil perspective on characteristic classes (see here), you can prove that all the rational Pontryagin classes of a flat vector bundle have to vanish. Thus all you need are vector bundles with non-vanishing rational Pontryagin classes, of which there are many.

A very nice source for this perspective on characteristic classes and flat bundles is Morita's book "Geometry of characteristic classes".

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I tried to put a link to wikipedia, but it seems to have been deleted... –  Adam Smith Oct 20 '11 at 0:51
    
Thanks t.b.!!!! –  Adam Smith Oct 20 '11 at 0:54
    
I fixed it. Wikipedia has this habit of using non-standard characters in links, and that is not very well supported by this site. Here it was the dash. Replacing it by the the dash on the keyboard usually works, otherwise you have to resort to percent encoding but that is somewhat painful. –  t.b. Oct 20 '11 at 0:56
    
@Adam Smith a very minor point : there is no "The" in the title of Morita's book. –  Olivier Bégassat Oct 20 '11 at 0:58
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In particular, the tangent bundle of $S^2$ can't be given a flat structure. –  Ryan Budney Oct 20 '11 at 4:05
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