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Can someone give me an intuitive explanation of the pullback bundle of a vector bundle in differential geometry? You can apply it to the tangent bundle as that's probably easier to visualise.

Am I right that the pullback bundle (where the vector bundle is the tangent bundle) of a map $u: M \to N$ is a subset of the tangent bundle $TN$?

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A vector bundle on $N$ is intuitively a family of vector spaces indexed by points of $N$. If we have a map $f: M \to N$ and a family of vector spaces indexed by points of $N$, we get a family of vector spaces indexed by points of $M$ by attaching to $p \in M$ the vector space attached to $f(p) \in N$. – countinghaus Nov 26 '12 at 15:21
(The reason this intuition isn't rigorous is that I need to say the family "varies smoothly" as $n$ varies in $N$, which is why you use the definition of vector bundle that you do (and why we need $f$ to be a smooth map to get a pullback bundle).) – countinghaus Nov 26 '12 at 15:22

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