Do covector fields correspond to homomorphisms of $ \mathscr C^\infty $-modules from the sheaf of vector fields to the sheaf of smooth functions?

Question

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The question

Let $ M $ be a smooth manifold. Let $ \mathscr C_M^\infty $ be the sheaf of $ \mathscr C^\infty $-functions on $ M $, and let $ \mathscr X $ be the $ \mathscr C_M^\infty $-module of vector fields on $ M $, i.e. $$ \mathscr X(U) = \Gamma(U,\mathrm TM) = \{\text{sections $ X\colon U\to \mathrm TM $}\} $$ for any $ U\subset M $ open, where of course $ \mathrm TM $ is the tangent bundle of $ M $.

The dual $ \mathscr C_M^\infty $-module $ \mathscr X^* $ of $ \mathscr X $ is defined by $$ \mathscr X^*(U) = \underline\hom_{\mathscr C_M^\infty}(\mathscr X,\mathscr C_M^\infty)(U) = \{\text{morphisms of $ \mathscr C_M^\infty $-modules $ \mathscr X{\restriction_U}\to \mathscr C_U^\infty $}\} $$ for any $ U\subset M $ open, where $ \mathscr X{\restriction_U} $ is the restriction of the sheaf $ \mathscr X $ to the open subsets of $ U $.

I then took $ U = M $ and I tried to show that $$ \mathscr X^*(M) \cong \mathscr X(M)^* $$ where $ \mathscr X(M)^* $ is the dual $ \mathscr C_M^\infty(M) $-module of the $ \mathscr C_M^\infty(M) $-module $ \mathscr X(M) $, but I didn't succeeded.

An obvious map $$ \mathscr X^*(M) \rightarrow \mathscr X(M)^* $$ is the one that takes $ \omega = (\omega_U)_{\text{$ U\subset M $ open}}\in \mathscr X^*(M) $ to $ \omega_M\in \mathscr X(M)^* $.

I thought that to define a map in the opposite direction $$ \mathscr X^*(M) \leftarrow \mathscr X(M)^* $$ one could start with an $ \omega\colon \mathscr X(M)\to \mathscr C_M^\infty(M) $ and define $$ \omega_U(X) = \omega(\tilde X) $$ for all $ X\in \mathscr X(U) $, where $ \tilde X $ is something like an "extension" of $ X\colon U\to \mathrm TM $ to all of $ M $. I think that such a $ \tilde X $ could be defined using partitions of unity, but I'm not sure about that so I'm asking here for help.

Answer 1

A smooth section $X$ of a vector bundle (e.g., a smooth vector field) over an open subset $U⊂M$ can be presented in the form $X=g Y$, where $g$ is a smooth function on $U$ and $Y$ is a smooth section over $U$ that extends to a smooth section $Z$ over $M$.

Using such a presentation, it is easy to define $ω_U$: $$ω_U(X) = ω_U(g Y) = g ω_U(Y) = g ω(Z).$$

To show the existence of a pair $(g,Y)$, use a smooth partition of unity to pick a smooth metric on the vector bundle and also pick a smooth function $h$ on $M$ that vanishes on $M∖U$ and is strictly positive elsewhere. Set $$g(x)=\exp(‖X(x)‖^2) h(x)^{-1}.$$ Now the section $Y=g^{-1}X$ is smooth and extends by zero to a smooth section $Z$ over $M$.