Dtyjlui

I quote the construction given in Madsen's Calculus to Cohomology. This is more or less the construction explained here defining connection on dual bundle.

For a vector bundle $Omega^i(xi) := Omega^i(M) otimes_{Omega^0(M)} Omega^0(xi)$, where $Omega^0(xi)$ are smooth sections, $Omega^i(M)$ are sections of $i$ forms.

1. We define a non singular pairing
$$ Omega^i(xi) otimes Omega^j(xi^*) xrightarrow{wedge} Omega^{i+j}(xi otimes xi^*) rightarrow Omega^{i+j}(M)$$
given by
$$(w otimes s, tau otimes s^*) mapsto (w wedge tau) otimes langle s,s^* rangle $$
where $langle s,s^* rangle $ is given from the bundle map $xi otimes xi^* rightarrow underline{Bbb K}$ the trivial bundle.

2. We define the connection on $xi^*$ for all $s in Omega^0(xi),s^* in Omega^0(xi^*)$, we have
$$ d(s,s^*) = (nabla_{xi}(s), s^*) + (s, nabla_{xi^*} (s^*))$$

I have two questions:

a. Is it clear that this definition of $nabla_{xi^*}$ yields a
connection.

b. Why does this definition guarantee existence ?

Let's work with local frames. Suppose that ${ e_i }$ is a local frame for $xi$, and suppose that ${ e_j^star }$ is the dual frame for $xi^{star}$ (i.e. $langle e_i , e_j^star rangle = delta_{ij}$). And suppose that $nabla_{xi} e_i = sum_komega_{ik} e_k$ for suitable one-forms $omega_{ik}$. An arbitrary section of $xi^star$ can be written as $s^star = sum_j s^star_j e_j^star$, for suitable smooth functions $s_j$, and our aim is to determine $nabla_{xi^star} s$. We can calculate as follows:

begin{align} (e_i, nabla_{xi^star}s^star) &= d(e_i, s^star) - (nabla_xi e_i , s^star) \ &= sum_j dleft( s^star _j langle e_i, e_j^star rangle right) -sum_{jk} s^star_j omega_{ik}langle e_k, e_j^star rangle \ &=ds^star_i-sum_j omega_{ij}s^star_j. end{align}
This implies that
$$ nabla_{xi^star} s^star = sum_i( ds^star_i - sum_j omega_{ij} s^star_j )e_i^star. (star)$$

We still need to show that the $nabla_{xi^star}$ defined in $(star)$ is a connection. This follows from this calculation:
begin{align} nabla_{xi^star} (fs^star) &= sum_i( d(fs^star_i) - sum_j omega_{ij} fs^star_j) e^star_i \ &=(df) sum_is^star_i e^star_i + f sum_i ( ds^star_i - sum_j omega_{ij} s^star_j) e^star_i \ &= (df) s^star + fnabla_{xi^star} s^star end{align}

We also need to show that $d(s,s^*) = (nabla_{xi}(s), s^*) + (s, nabla_{xi^*} (s^*))$ holds for arbitrary $s$ when $nabla_{xi^star}$ is defined by $(star)$. Writing $s$ in the form $s = sum_i s_i e_i$ for suitable smooth functions $s_i$, this statement follows from the calculations:begin{align} d(s, s^star) &= dleft( sum_{ij} s_i s^star_jlangle e_i, e_j^star rangle right) = sum_i d(s_i s^star_i) =sum_i(ds_i) s_i^star + s_i (ds^star_i) \ (nabla_xi s, s^star) &= sum_{ij} (ds_i + sum_k omega_{ki}s_k) s^star _jlangle e_i, e_j^star rangle =sum_i (ds_i) s_i^star + sum_{ik} omega_{ki} s_k s_i^star \ (s, nabla_{xi^star} s^star ) &= sum_{ij} s_i (ds_j^star - sum_k omega_{jk} s_k^star ) langle e_i , e_j^star rangle =sum_j s_j (ds^star_j) - sum_{jk}omega_{jk} s_j s_k^star end{align}

Edit: As for coordinate independence, one way is to do the brute force calculation. If ${ f_i }$ is a different local frame, valid on a different patch, with $f_i = sum_{ij} g_{ij} e_j$ on the overlap between the two patches for suitable non-vanishing smooth functions $g_{ij}$, then the dual bases are related by $f_i^star = sum_{ij}(g^{-1})_{ji} e_j^star$. And if $nabla_xi f_i = sum_k zeta_{ik} f_k$ for suitable one-forms $zeta_{ik}$, then you can show that
$$ zeta_{ik} = sum_l dg_{il} (g^{-1})_{lk} + sum_{lm} g_{il} omega_{lm} (g^{-1})_{mk},$$
and from here, you can verify that definition $(star)$ is consistent whether you work using the ${ e_i }$ frame or the ${ f_i }$ frame.

But actually, I don't think we need to do all this work! Since $(star)$ is the unique $nabla_{xi^star}s^star$ satisfying $d(s,s^*) = (nabla_{xi}(s), s^*) + (s, nabla_{xi^*} (s^*))$ for all $s$ on the first patch, and since $(star)$-written-in-the-new-frame is the unique $nabla_{xi^star}s^star$ satisfying $d(s,s^*) = (nabla_{xi}(s), s^*) + (s, nabla_{xi^*} (s^*))$ for all $s$ on the second patch, the two definitions must be equal on the overlap!