I want to get some deep understanding on *closed* *orientable* Riemannian manifolds admitting $k$-forms ($k\geq 2$) $\omega$ that satisfices the following conditions:
$$\nabla \omega\neq 0,\quad \Delta\omega=0.$$
where $\Delta=(\delta +d)^2$ is the Laplace-Beltrami operator (Hodge Laplacian). What other useful properties can these forms have? Any reference would be greatly appreciated.



Reference for non-parallel harmonic $k$-forms
There are too many of these for them to have any particularly interesting structure. For example, consider any metric $g$ on the $3$-torus $\mathbb{T}^3$. By the Hodge theorem, the space of $g$-harmonic $2$-forms has dimension $3$. However, if, say, all of them were $g$-parallel, then their duals would be a basis of $g$-parallel $1$-forms, which would imply that the metric $g$ is flat. Thus, any non-flat metric $g$ on the $3$-torus will have a $g$-harmonic $2$-form that is not $g$-parallel.


In fact, as long as $H^2\_{\mathrm{deRham}}(M^3,\mathbb{R})\not=0$, where $M$ is an orientable compact $3$-manifold, the generic metric $g$ on $M$ will have a $g$-harmonic $2$-form that is not $g$-parallel, since having a nonzero $g$-parallel $2$-form would force $g$ to be locally a product metric, and the generic metric is not locally a product.


Unfortunately, I think that there not much 'deep' to say about such $2$-forms in general.


