090929q

=2009 September 29th - questions=

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1 Harold W
Two questions about the forgetful functor from smooth (C-infinity) vector bundles to smooth fiber bundles:

1) Suppose that $$E\to B$$ is a smooth vector bundle (E,B smooth manifolds). Does $$E\to B$$ as a smooth fiber bundle uniquely determine $$E\to B$$ as a vector bundle, up to isomorphism?

In other words, if $$E_1\to B_1$$, $$E_2\to B_2$$ are vector bundles which are isomorphic as smooth fiber bundles (via some fiber-preserving diffeomorphism, not necessarily linear on fibers), are they necessarily isomorphic as smooth vector bundles (via some fiber-preserving diffeomorphism, linear on fibers)?

2) Suppose that $$E\to B$$ is a smooth fiber bundle with fibers diffeomorphic to $$\mathbb{R}^n$$. Can it be given the linear structure of a vector bundle?

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