# dg.differential geometry – covariant derivative commute with Hodge star

Suppose $$(M,g)$$ is a Riemannian manifold. Let $$nabla$$ be the covariant derivative associated with the Levi-Civita connection. Suppose $$*$$ is the Hodge operator with respect to the metric $$g$$ on $$Omega^p(M)$$. If $$omegain Omega^p(M)$$ and $$Xin T_pM$$, do we have
$$*(nabla_X omega)=nabla_X (*omega)quad?$$