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revised How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base
I have added a reference
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comment How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base
Dear anonymous I have found this note by Scott Morrison addressing the fact pointed out by Mark Grant: arxiv.org/pdf/math/0105161v1.pdf. I hope it helps.
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answered How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base