  In Riemannian geometry the prescribed Ricci curvature problem is as follows:
given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on
$M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the
following celebrated result: the Ricci curvature uniquely determines the
Levi-Civita connection on any compact Einstein manifold with non-negative
section curvature. In the present paper we generalize the result of DeTurck and
Koiso for a Riemannian manifold with non-negative section curvature. In
addition, we extended our result to complete non-compact Riemannian manifolds
with nonnegative sectional curvature and with finite total scalar curvature.
