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 $\omega\in \Omega^p(M)$ and $X\in T_pM$, do we have $$*(\nabla_X \omega)=\nabla_X (*\omega)\quad?$$

  • $\begingroup$ @abx since we're plugging a vector field $X$ into the covariant derivative, $\nabla_X\omega \in \Omega^p$ as long as $\omega \in \Omega^p$? The degrees certainly line up here. $\endgroup$ – Rohil Prasad yesterday
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    $\begingroup$ Anyways, this derivation is a straightforward exercise. Use the fact that the two tensors are equal if and only if their inner products with the forms of the corresponding degrees are equal. Then use the fact that the volume form is covariantly constant with respect to the Levi-Civita derivative along with the definition of the Hodge star. $\endgroup$ – Rohil Prasad yesterday
  • $\begingroup$ @Rohil Prasad: you are right of course. Comment deleted. $\endgroup$ – abx yesterday

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