Let $g,h$ be two Riemannian metric on $M$ related by $g=h+\phi$ where $\phi$ is a symmetric $(0,2)$-tensor and $g$ transforms $h$-orthonormal frame $\{e_i\}$ to $g$-orthonormal frame $\{a_i e_i\}$ for $a_i\in \Bbb R$. Then
How $\star_g$ related to $\star_h$?
Update: Somewhere I read that if A is a $n\times n$-matrix, and if $e_i\in T_pM$, then $$\star(Ae_1\wedge\cdots\wedge Ae_k)=\det(A)\star(e_1\wedge\cdots\wedge e_k),\quad n=\dim M, k\leq n$$ Is this equivalent to my question?