Theorems of existence and non-existence of conformal Killing forms

On an $n$-dimensional compact, orientable, connected Riemannian manifold, we consider the curvature operator acting on the space of covariant traceless symmetric $2$-tensors. We prove that, if the curvature operator is negative, the manifold admits no nonzero conformal Killing $p$-forms for $p=1,2,\dots,n-1$. On the other hand, we prove that the dimension of the vector space of conformal Killing $p$-forms on an $n$-dimensional compact simply-connected conformally flat Riemannian manifold $(M, g)$ is not zero.