In this paper we give a criterion for a deformation of a hermitian vector bundle to be Ricci-flat. As an application we show that on a K\"ahler manifold, every deformation of a vector bundle can be made Ricci-flat whereas on some Hopf manifolds, the non-existence of a Ricci-flat deformation is related to non-trivial vector bundles on the universal cover ${ \mathbb C}^n\setminus\{ 0\}$. On a surface with $b_1(X)\not=0$ filtrable Ricci-rigid vector bundles prove to be very special. We apply this to Inoue and Hopf surfaces.
Categories: math.AG
