Say M is an n-dimensional product manifold of the form $\mathbb{R}_{+}$ $\times$ N where N is an n-1 dimensional manifold and the metric on M is of the form g$_{M}$=dr$^{2}$+r$^{2}$g$_{N}$. I can write an arbitrary k-form $\omega$ on M as $\omega$=dr $\wedge$ $\alpha$ + $\beta$, where $\alpha$ and $\beta$ are forms on N potentially parameterized by r.
My question is how can I relate the hodge star of $\omega$ with respect to the warped product metric to the hodge stars of the forms $\alpha$ and $\beta$ with respect to the metric g$_{N}$? I have been attempting to do this in coordinates but this has been a huge mess and I don't feel such an approach should be necessary since such a result would be independant of the coordinates on N anyway.