Geodesic
On Uniqueness Theoremsfor Ricci Tensor
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 55 (2016) no. 1, pp. 47-52.

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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.
Classification : 53C20
Keywords: Uniqueness theorem for Ricci tensor; compact and complete Riemannian manifolds; vanishing theorem 
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Khripunova, Marina B.; Stepanov, Sergey E.; Tsyganok, Irina I.; Mikeš, Josef. On Uniqueness Theoremsfor Ricci Tensor. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 55 (2016) no. 1, pp. 47-52. http://geodesic.mathdoc.fr/item/AUPO_2016__55_1_a6/