We define a (rank $k$) vector bundle to be a fibre bundle with fibre $\mathbb R^k$ such that the local trivializations are fibre-by-fibre vector space isomorphisms. I'm curious whether this linearity condition is enough to stop certain fibre bundles from being turned into vector bundles.

Question: Given a fibre bundle $E\xrightarrow{p}B$ with fibre homeomorphic to $\mathbb R^k$, can we find

• an open cover $(U_\alpha)$ of $B$ and
• local trivializations $\phi_\alpha\colon p^{-1}(U_\alpha)\to U_\alpha\times \mathbb R^k$

such that the local trivializations are fibre-by-fibre vector space isomorphisms?

I expect that the answer is 'No', but I'd be very interested to see a counterexample.

We define a (rank $k$) vector bundle to be a fibre bundle with fibre $\mathbb R^k$ such that the local trivializations are fibre-by-fibre vector space isomorphisms. I'm curious whether this linearity condition is enough to stop certain fibre bundles from being turned into vector bundles.

Question: Given a fibre bundle $E\xrightarrow{p}B$ with fibre homeomorphic to $\mathbb R^k$, can we find

• an open cover $(U_\alpha)$ of $B$ and
• local trivializations $\phi_\alpha\colon p^{-1}(U_\alpha)\to U_\alpha\times \mathbb R^k$

such that the local trivializations are fibre-by-fibre vector space isomorphisms?

I expect that the answer is 'No', but I'd be very interested to see a counterexample.

A counterexample might take the form of a fibre bundle which demonstrably has no global section, or it might be a fibre bundle which cannot admit $GL_k(\mathbb R)$ as a structure group.

I'd be particularly interested if the example had the affine group $Aff(\mathbb R^k)$ as a structure group.