Constant Rank Theorems under Intermediate Ricci Curvature

Abstract

We prove two constant rank theorems for the Ricci tensor in two geometric settings: manifolds with harmonic curvature and gradient Ricci solitons. In both cases, if the $k$-intermediate Ricci curvature satisfies $\Ric_k\geq0$ for some $1\le k\le n-2$, then the Ricci tensor has constant rank and its nullity distribution is parallel. In the complete case, the universal cover splits isometrically along the Ricci-null directions. We also show that the range $1\le k\le n-2$ is sharp: in every dimension $n\ge 4$, nonnegative Ricci curvature alone does not imply the constant rank property in either setting.