What is an example of a pair of finite dimensional $C^{\infty}$ manifolds $E$ and $M$, and a smooth function $\pi:E\rightarrow M$ such that $\pi^{-1}(p)$ has a vector space structure for each $p\in M\ $ (all of them with same dimension), but it is not a vector bundle?
An example of a triple $(E,\pi,M)$ which is not a vector bundle
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differential-geometry
vector-bundles
fiber-bundles