$\newcommand{\al}{\alpha}$
$\newcommand{\euc}{\mathcal{e}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\Det}{\operatorname{Det}}$


Let $M,N$ be smooth $n$-dimensional Riemannian manifolds (perhaps with smooth boundary), and let $\, f:M \to N$ be a smooth **immersion**. Let $\Omega^k(M,f^\*TN)$ be the space of $f^\*TN$-valued $k$-forms. 


Let $d:\Omega^k(M,f^\*TN) \to \Omega^{k+1}(M,f^\*TN)$ be the covariant exterior derivative associated with the pullback connection of the Levi-Civita connection on $N$ (via $f$), and let $\delta$ be its adjoint.


Denote $\delta\_{k}:\Omega^k(M,f^\*TN) \to \Omega^{k-1}(M,f^\*TN)$.


**Question:** Is $\ker \delta\_1$ is infinite dimensional?


Since $\delta=\star d \star$ (up to sign), $\dim(\ker \delta\_1)=\dim(\ker d\_{n-1})$. So, this is a question about $d$.




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**Edit:**


In the special case where $N$ is flat, $\delta\_{k} \circ \delta\_{k+1}=0$, so $\operatorname{Image}(\delta\_{k+1}) \subseteq \ker(\delta\_k)$.


In particulr, $\operatorname{Image}(\delta\_{2}) \subseteq \ker(\delta\_1)$.


Is $\operatorname{Image}(\delta\_{2})$ is infinite-dimensional? Can we at least something when $N=\mathbb{R}^n$? or even when $M=N=\mathbb{R}^n$?


Let's see what happens when $M=N=\mathbb{R}^n,f=\operatorname{Id}$:


$$ d\_{n-1}(\sigma)(e\_1,...,e\_n)=\sum\_{j=1}^{d} (-1)^{j+1} \nabla^{T\mathbb{R}^n}\_{e\_j} \big( \sigma(e\_1,...,\hat{e\_j},...,e\_d) \big)$$


Write $\sigma=(-1)^{j+1}a\_j^idx^1 \wedge \dots \wedge \hat{dx^j} \wedge \dots \wedge dx^d \otimes e\_i, a\_j^i \in C^{\infty}(\mathbb{R}^n)$. 


Then $ \sigma(e\_1,...,\hat{e\_j},...,e\_d)=(-1)^{j+1}a\_j^i e\_i $, so 


$$ \nabla^{T\mathbb{R}^n}\_{e\_j} \big( \sigma(e\_1,...,\hat{e\_j},...,e\_d) \big)= \nabla^{T\mathbb{R}^n}\_{e\_j}((-1)^{j+1}a\_j^i e\_i)=(-1)^{j+1}\frac{\partial a\_j^i}{\partial x\_j}e\_i.$$


Thus, $$\sigma \in \ker d\_{n-1} \iff \frac{\partial a\_j^i}{\partial x\_j}e\_i=0 $$ (in the last term there is a double summation, on $i,j$).


Since $e\_i$ are independent, this is equivalent to
$$\sum\_{j=1}^d \frac{\partial a\_j^i}{\partial x\_j}=0 \, \text{ for all } \, i=1,...,d $$


Denoting $\bar a^i=(a\_1^i,...,a\_d^i)$, we get that $\operatorname{div} (\bar a^i)=0$.


I guess it shouldn't be too hard to see now that $\dim(\ker d\_{n-1}) = \infty$. (By taking the $a\_j^i$ to be constants, one immediately gets $\dim(\ker d\_{n-1}) \ge n^2$. Since the condition on the divergence do not touch $\frac{\partial a\_j^i}{\partial x\_k}$ for $k \neq j$, it seems plausible that the dimension is indeed infinite. Perhaps someone can come with a slick argument to show this.)


The next case we should try is $M=N=\mathbb{R}^d$, and $f$ an arbitrary mapping....




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**Comment:**


I know that $\operatorname{Cof}(df) \in \ker \delta\_1$, where $\operatorname{Cof}(df)$ is the corresponding *cofactor map* of $df$:
 $$ \Cof df= (-1)^{d-1} \star\_{f^\*TN}^{n-1} (\wedge^{n-1} df) \star\_{TM}^1. $$ 


Since $f$ is an immersion, $\operatorname{Cof}(df) \neq 0$, so $\dim (\ker \delta\_1 ) \ge 1$.


For my purposes, It would suffice to know that $\ker \delta\_1$ always contains elements which are linearly independent of $\Cof df$.



Is the kernel of the coderivative infinite-dimensional?
Suppose that the vector field $v$ is a non-vanishing covariantly constant on $N$, then its pullback $f^\*v$ is a constant section of $f^\*TN \to M$ with respect to the pulled back connection. Then, for any $(n-1)$-form $\alpha$ and $(n-2)$-form $\beta$ on $M$, we have $d\_{n-2} (\beta \otimes f^\*v) = (d\beta) \otimes f^\*v$ and $d\_{n-1} (\alpha \otimes f^\*v) = (d\alpha) \otimes f^\*v$, where I used $d$ to denote the usual de Rham exterior derivative. Then clearly, $\mathrm{im}(d) \otimes f^\*v \subseteq \ker d\_{n-1}$ is an infinite dimensional subspace.


On the other hand, suppose that there are no non-vanishing constant sections of $f^\*TN \to M$. We might still be able to find some differential operator $q \colon \Gamma(E) \to \Gamma(\Lambda^{n-1}T^\*M \otimes\_M f^\*TN)$ whose image lands inside $\ker d\_{n-1}$, that is $d\_{n-1} \circ q = 0$. The trick is to first dualize. As you noted, $d\_{n-1}^\* = d\_0$, up to conjugation by invertible operators. Then any "compatibility operator" $q^\*$, such that $q^\* \circ d\_0 = 0$, gives us a candidate $q = (q^\*)^\*$, because $d\_{n-1} \circ q = d\_0^\* \circ (q^\*)^\* = (q^\* \circ d\_0)^\*$.


The non-existence of non-trivial constant sections of $f^\*TN \to M$ means that there exist integrability conditions for the equation $d\_0 v = 0$. In particular, after some massaging, we can come up with an operator $C\colon \Gamma(T^\*M \otimes\_M f^\*TN) \to \Gamma(f^\*TN)$ such that $C\circ d\_0 = \mathrm{id}$. As a consequence, we have the identity $(\mathrm{id} - d\_0 \circ C) \circ d\_0 = d\_0 - d\_0 \circ (C\circ d\_0) = 0$. Hence we have found a compatibility operator $q^\* = \mathrm{id} - d\_0 \circ C$, from which $q = \mathrm{id} - C^\* \circ d\_{n-1}$. In conclusion, we have $\mathrm{im}(q) \subseteq \ker d\_{n-1}$.


It remains to check that $\mathrm{im}(q)$ is indeed infinite dimensional. I don't have a slick argument for that, at the moment. But I'm sure that in individual cases it should not be hard to establish, at the very least by looking at the effect of $q$ on jets of higher and higher order at some fixed point $x\in M$.


