Given a Riemannian metric $g$ on a smooth manifold $M$, one defines an
$L^2$-inner product on the space $\bigwedge^\ast(M)$ of differential
forms by


$$
 \langle \alpha, \beta \rangle\_g = \int\_M \alpha \wedge \ast\_g \beta,
$$


where $\ast\_g$ denotes the Hodge-star operator relative to $g$, and
$\alpha, \beta$ are forms of the same degree.


**Question**: Does every inner product on $\bigwedge^\ast(M)$ as a graded
vector space come from some metric $g$? How about inner products on
$k$-forms $\bigwedge^k(M)$ for a single $k$, especially $0 < k < \dim
M$?



Inner products on differential forms
Fixing a $k$ for simplicity, there are many inner products on $\bigwedge^k (M)$ (which I would usually denote $\Omega^k (M)$). Since $\bigwedge^k T^\* M$ is a vector bundle, there are, for example, Sobolev $H^s$ inner products on its space of smooth sections for any natural number $s$. See, for example, Palais, *Foundations of Global Non-Linear Analysis*.


