$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be $d$-dimensional oriented Riemannian manifolds ($d \ge 2$). Let $f:\M \to \N$ be smooth.
Let $1 \le k \le d-1$ be fixed. Suppose pullback by $f$ locally preserves co-closedness of closed and co-closed forms. Does it follow that $f$ preserves co-closedness of all forms?
Stating it formally, we consider the following two properties $f$ can have: For every open $U \subseteq \N$,
- $\omega\in \Omega^k(U)$ is co-closed $\Rightarrow f^*\omega$ is co-closed.
$\,\,\,\,$ 2. $\omega\in \Omega^k(U)$ is closed and co-closed $\Rightarrow f^*\omega$ is co-closed.
Does property 2 implies property 1?
For $k=1$, maps $f$ which have property $2$ are called harmonic morphisms.
Locally, the space of closed and co-closed forms is infinite-dimensional. Thus property $2$ provides an "infinite-dimensional amount" of information about $f$, so maybe there is a chance the answer is positive.
*There are non-isometric maps which satisfy property $(1)$. If $d$ is even, than conformal maps satisfy it for $k=\frac{d}{2}$. (This follows immediately from the conformal invariance of the Hodge dual in half the dimension).
