Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. 


Let $p \in M$, and let $\alpha\_p \in \bigwedge^k(T\_pM)^\*$.



> 
> Does there exist an open neighbourhood $U$ of $p$, and a closed and co-closed $k$-form $\omega \in \Omega^k(U)$ satisfying $\omega\_p=\alpha\_p$?
> 
> 
> 




---


This question is equivalent to the following question:



> 
> Do closed and co-closed frames for $\bigwedge^k(T^\*M)$ always exist locally?
> 
> 
> 


Indeed, if we can specify the value of a form in a point, we can take a basis for $\bigwedge^k(T\_pM)^\*$, and so obtain forms which form a frame at $p$. Since "being a frame" is an open condition, we have a local frame. On the other hand, suppose that local closed and co-closed frames exist. Then, by choosing a linear combination with constant coefficients of that frame, we can realize any given value at $p$. 


*Comment:* In general we cannot expect such a frame to be induced from coordinates. Indeed, when we specialize to even dimension $d$, and $k=\frac{d}{2}$, then, for a generic metric $g$, [there are no coordinate systems where even one wedge $\mathrm{d}x^{i\_1}\wedge\cdots\wedge\mathrm{d}x^{i\_n}
$ is co-closed](https://mathoverflow.net/a/301584/46290). 



Can we specify the value of harmonic forms at a point?
The answer to your question is 'yes', but it may take me a little while to look up the appropriate references, since I'm traveling now. 


The main point is that the (usually overdetermined) system of PDE defined by $\mathrm{d}\alpha = \mathrm{d}^\*\alpha = 0$ for $\alpha\in\Omega^k(M)$ is involutive and hence, if the metric is real-analytic, it's formally integrable, so one can specify the $0$-jet of a local solution arbitrarily. Meanwhile, because the elliptic complex $\mathrm{d}+\mathrm{d}^\*:\Omega^\*(M)\to\Omega^\*(M)$ satisfies Spencer's $\delta$-estimate, this local formal integrability carries over to the smooth case, which is what you are asking.


I'll get back home later in the week, and I'll be able to post appropritate references then.


