# dg.differential geometry – Reference for non-parallel harmonic \$k\$-forms

I want to get some deep understanding on closed orientable Riemannian manifolds admitting $$k$$-forms ($$kgeq 2$$) $$omega$$ that satisfices the following conditions:
$$nabla omeganeq 0,quad Deltaomega=0.$$
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.