On DeTurck uniqueness theorems for Ricci tensor

TL;DR

This paper generalizes DeTurck and Koiso's uniqueness theorem for Ricci curvature, extending it to non-compact manifolds with nonnegative sectional curvature and finite total scalar curvature.

Contribution

It extends the DeTurck-Koiso theorem to broader classes of Riemannian manifolds, including non-compact cases with specific curvature conditions.

Findings

Unique determination of Levi-Civita connection by Ricci curvature on compact Einstein manifolds.

Extension of the uniqueness result to complete non-compact manifolds with nonnegative sectional curvature.

Applicability to manifolds with finite total scalar curvature.

Abstract

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.