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Let $f:M\to N$ be a smooth map between smooth manifolds.

Is there a natural way to give a smooth vector bundle structure to $\bigsqcup_{p\in M} \mathcal L(T_pM, T_{f(p)}N)$.

where $\mathcal L(V, W)$ denotes the space of all the linear maps between vector spaces $V$ and $W$.

I ask this because I want to make the following statement:

Let $f:M\to N$ be a smooth map. Then the map $p\mapsto df_p$ is smooth.

($df_p$ lives in $\bigsqcup_{p\in M} \mathcal L(T_pM, T_{f(p)}N)$).

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    $\begingroup$ Yes. First define the pullback bundle $f^*TN$ on $M$, then the bundle you want is $Hom(TM, f^*TN)$. $\endgroup$ – user99914 May 29 '15 at 16:19
  • $\begingroup$ @John Thanks. Do you know of a good reference where I can read about the pullback bundle. I cold not find it in Lee's Introduction to Smooth Manifolds (2nd Ediiton). Also, what is meant by Hom above? $\endgroup$ – caffeinemachine May 29 '15 at 16:21
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    $\begingroup$ This famous book by Hatcher math.cornell.edu/~hatcher/VBKT/VB.pdf covers some materials on vector bundle (the first chapter). For $Hom$ is just another construction . The point is that: if $E_1, E_2$ are two vector bundles on $M$, then $Hom(E_1, E_2)$ is another vector bundle on $M$ so that at each fiber it's $L((E_1)_x, (E_2)_x)$ (all linear map from $(E_1)_x$ to $(E_2)_x$. $\endgroup$ – user99914 May 29 '15 at 16:36
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    $\begingroup$ Consider a vector bundle $E$ on $N$ with description $\{U_{\alpha}, g_{\alpha\beta}\}$, where $g_{\alpha\beta}$ are the transition functions. Then $f^* E$ is the vector bundle on $M$ with description $\{f^{-1}(U_{\alpha}), g_{\alpha\beta} \circ f\}$. Further, If $F$ is another vector bundle on $N$, there is a vector bundle $Hom(E,F)$ whose fibre over any $n \in N$ is isomorphic to $Hom(E_n, F_n) = \mathcal{L}(E_n,F_n)$. $\endgroup$ – A.P. May 29 '15 at 16:40
  • $\begingroup$ @A.P. Thanks for the details! $\endgroup$ – caffeinemachine May 29 '15 at 16:58

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