Let $V$ be a finite-dimensional vector space with euclidean product, and let $U$ be a subspace. Now let $P$ be the projection of $V$ onto $U$, and let $\omega$ be any alternating multilinear $k$-form. Let $\omega_t$ denote $P^\ast \omega$, the tangential part of $\omega$.
Then $\omega_n = \omega - \omega_t$ is called the normal part of $\omega$. It is now claimed that $\star(\omega_n) = (\star\omega)_t$.
But I do not believe that. I think it holds if $\omega$ is in the direct sum of $\Lambda^k U$ and $\Lambda^k (U^\perp)$, but this direct sum is generally not $\Lambda^k(V)$. Can you give me a proof?