Constant Rank Theorems under Intermediate Ricci Curvature

Mijia Lai; Guoqiang Wu

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arXiv · 2610.17691

Constant Rank Theorems under Intermediate Ricci Curvature

Mijia Lai·Guoqiang Wu

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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 $1\leq k\leq n-2$, 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 $1\leq k\leq n-2$ is sharp: in every dimension $n\geq4$, nonnegative Ricci curvature alone does not imply the constant rank property in either setting.

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BibTeX RIS

Mijia Lai, Guoqiang Wu. 2026-10-13. Constant Rank Theorems under Intermediate Ricci Curvature. https://arxiv.org/abs/2610.17691

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