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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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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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    Wow, thanks! I'll check these material out.2011-10-20