Bundle-valued conjugate-linear Hodge star
= Bundle-valued conjugate-linear Hodge star
{title2=$*_E:A^{p,q}(E)\to A^{n-p,n-q}(E^*)$}
For inner products linear in the first argument, define $*_E$ by contraction in $\psi\wedge *_E\eta=(\psi,\eta)dV$. It is conjugate-linear and takes $(p,q)$-forms with values in $E$ to $(n-p,n-q)$-forms with values in $E^*$. With induced dual metrics, $*_{E^*}*_E=(-1)^{p+q}$.