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Let $(M,g)$ be a Riemannian manifold and let $E \to M$ be a real vector bundle over $M$. Let $d_A = d+A$ be a covariant derivative on $E$. It is an $\mathbb R$-linear map $d_A \colon \Gamma(E) \to \Gamma(T^*M \otimes E)$. Then the formal adjoint map $d_A^*$ acts from $\Gamma(T^*M \otimes E^*)$ to $\Gamma(E^*)$. But how to find explicitely this map? I know the answer but I don't know how to obtain it. If we had $\Gamma(T^*M \otimes E^*) \cong \Gamma(T^*M \otimes E)$ and $\Gamma(E^*) \cong \Gamma(E)$ and if we had scalar products defined on these two spaces then I could obtain the answer starting from the formula $\langle d_A s, \omega \rangle = \langle s, d_A^* \omega \rangle$, where $s \in \Gamma(E)$, $\omega \in \Gamma(T^*M \otimes E)$ and $\langle \cdot , \cdot \rangle$ are corresponding scalar products, but it's not so clear for me what to do in the case when we have no scalar products in these spaces. I think that we need to give some sense to $\langle \cdot, \cdot \rangle$. Given $(s,v) \in \Gamma(E) \times\Gamma(E^*)$ we can define $\bigl(v(s)\bigr)_x = v_x(s_x)$ to be a $C^\infty$-function on $M$ so we can define $\langle s, v \rangle = \langle v(s) \rangle_{L_2(M)}$. On the other hand, given $(\omega \otimes s, \eta \otimes v) \in \Gamma(T^*M \otimes E) \times \Gamma(T^* M \otimes E^*)$ (locally) we can define $\langle \omega \otimes s, \eta \otimes v\rangle = \int_M v(s) \omega \wedge *\eta$ and using these two expressions for $\langle \cdot , \cdot \rangle$ we can find formal adjoints as in the case of scalar product spaces. But are these considerations correct or there is some other way to find formal adjoint to $d_A$?

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What you say makes sense, I just think you need to be a little bit careful with the "duality pairings". You need duality pairings $\Gamma(E) \times\Gamma(E^*) \rightarrow \mathbb{R}$ and $\Gamma(T^*M \otimes E) \times \Gamma(T^* M \otimes E^*) \rightarrow \mathbb{R}$ indeed.

I think what people consider the "canonical" duality pairings to be in this context is: $$ \begin{array}{ccc} \Gamma(E) \times\Gamma(E^*) & \rightarrow &\mathbb{R} \\ (s, v) &\mapsto & \int_M v(s) \, dvol_g \end{array}$$ and $$ \begin{array}{ccc} \Gamma(T^*M \otimes E) \times \Gamma(T^* M \otimes E^*) & \rightarrow &\mathbb{R} \\ \langle \omega \otimes s, \eta \otimes v\rangle &\mapsto & \int_M \langle \omega, \eta\rangle_g \,v(s) \, dvol_g \end{array}$$ where here $\langle \omega, \eta\rangle_g$ is just (pointwise) the inner product on $T^*M$ derived from the inner product $g$ on $TM$.

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