  Given a family $(F,h) \to X \times S$ of Hermite-Einstein bundles on a
compact K\"ahler manifold $(X,g)$ we consider the higher direct image sheaves
$R^q p_* \mathcal{O}(F)$ on $S$, where $p: X \times S \to S$ is the projection.
On the complement of an analytic subset these sheaves are locally free and
carry a natural metric, induced by the $L_2$ inner product of harmonic forms on
the fibers. We compute the curvature of this metric which has a simpler form
for families with fixed determinant and families of endomorphism bundles.
Furthermore, we discuss the metric for moduli spaces of stable vector bundles.
