# Differential of a smooth map in terms of sheaves of deriviations

$$\newcommand{\C}{\mathscr{C}^\infty}\newcommand{\blank}{{-}}\newcommand{\from}{\colon}\newcommand{\after}{\circ}\newcommand{\Der}{\mathrm{Der}}$$Recall that for a smooth manifold $$M$$, there is an equivalence of categories between vector bundles on $$M$$ and locally free sheaves of $$\C_M$$-modules, given by mapping a vector bundle $$E \to M$$ to the sheaf of sections $$\Gamma(\blank, E)$$. If $$p \from E \to M$$ and $$p' \from E' \to N$$ are two vector bundles and $$F \from M \to N$$ is a smooth map, then a map of vector bundles $$E \to E'$$ covering $$F$$ is by definition a smooth map $$\Phi \from E \to E'$$ such that $$p' \after \Phi = F \after p$$ which is linear on the fibers. In terms of sheaves, this corresponds to a morphism $$\Gamma(\blank, E) \to F^*\Gamma(\blank,E') := \C_M \otimes_{F^{-1}\C_N} F^{-1} \Gamma(\blank, E')$$.

In particular, the tangent bundle $$T_M$$ corresponds to the sheaf of derivations of the sheaf of algebras $$\C_M$$. If $$F \from M \to N$$ is a smooth map of manifolds, then it is well-known that the differential of $$M$$ defines a map of vector bundles $$DF \from T_M \to T_N$$ covering $$F$$. Can this morphism be described in terms of the morphism of sheaves $$\Der(\C_M) \to F^*\Der(\C_N)$$?

a priori no, because $$TM,TN$$ (as sheaves) are sheaves on different spaces (live in different categories somehow) so there is no morphism between them, only after pulling back via $$F$$.
However, if $$M \hookrightarrow N$$ is a closed immersion, you have a map $$F^*\Omega_N=\Omega_N|_M \twoheadrightarrow \Omega_M$$ of differential forms. That map you can dualise to get a map $$TM \rightarrow TN|_M$$ which is often an embedding (for instance if everything is smooth). Note this works because you have two sheaves on the same space (namely $$M$$). This construction comes from algebraic geometry, hope this still helps!
• I wrote $\mathrm{Der}(\mathscr{C}_M^\infty) \to F^*\mathrm{Der}(\mathscr{C}_N^\infty)$. These are both sheaves on $M$. Nov 19, 2022 at 16:01
• yes that's true - maybe I misunderstood you, I thought you were looking for a map of sheaves between $M,N$ in analogy of $DF$. Nov 19, 2022 at 16:14
• the only classical map involving a derivation would be the normal bundle $TM/TN$ see also here Nov 19, 2022 at 16:22