On uniqueness theorems for Ricci tensor. (English) Zbl 1373.53045

The authors generalize the well-known theorem of DeTurck-Koiso asserting that, on a compact non-negatively curved Einstein manifold, the Ricci tensor uniquely determines the Levi-Civita connection. The generalization replaces the Einstein property with the condition that the Ricci tensor is bounded above by the metric. In addition, the authors extend this result to complete non-compact manifolds with finite total scalar curvature.


53C20 Global Riemannian geometry, including pinching
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