On nonpositively curved compact Riemannian manifolds with degenerate Ricci tensor

Abstract:

In this article, we prove that if the Ricci tensor of a compact nonpositively curved manifold $M^n$ is nowhere negative definite, then it admits local flat factors. To be more precise, let $U\subseteq M$ be the open subset where the Ricci tensor has maximum rank $r$. Then for any connected component $U_a$ of $U$, its universal cover $\widetilde {U_a}$ is isometric to $\textbf {R}^{n\! -\! r} \! \times \! N^r_a$, where $N^r_a$ is a nonpositively curved manifold with negative Ricci curvature.

In particular, if $M^n$ is compact, nonpositively curved without Euclidean de Rham factor, and both the manifold and the metric are real analytic, then its Ricci tensor must be negative definite somewhere.