Paper · Oct 13, 2026

Constant Rank Theorems under Intermediate Ricci Curvature

Mijia Lai, Guoqiang Wu

math.DGarXiv:2610.17691
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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 -intermediate Ricci curvature satisfies for some , 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 is sharp: in every dimension , nonnegative Ricci curvature alone does not imply the constant rank property in either setting.

Ricci solitons

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  1. Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature

    Matheus Horácio, Wenqi Li, Jianyu Ou et al. PDF

    Topics Ricci solitons · Same author Guoqiang Wu · Shared terms “cover splits”, “universal cover”, “Ricci tensor”

  2. Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

    Fengjiang Li, Wenqi Li, Jianyu Ou et al. PDF

    Topics Ricci solitons · Same author Guoqiang Wu · Shared terms “universal cover”, “gradient Ricci”

  3. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al. PDF

    Topics Ricci solitons · Same author Guoqiang Wu · Shared terms “Ricci tensor”, “gradient Ricci”

  4. Rigidity of shrinking gradient ricci soliton with constant scalar curvature

    Fengjiang Li, Yuanyuan Qu and Guoqiang Wu PDF

    Topics Ricci solitons · Same author Guoqiang Wu · Shared terms “gradient Ricci”

  5. A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu PDF

    Topics Ricci solitons · Same author Guoqiang Wu · Shared terms “gradient Ricci”

  6. Some rigidity results on shrinking gradient Ricci soliton

    Jianyu Ou, Yuanyuan Qu and Guoqiang Wu PDF

    Topics Ricci solitons · Same author Guoqiang Wu · Shared terms “gradient Ricci”

  7. Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid

    Wenqi Li, Jianyu Ou, Guoqiang Wu et al. PDF

    Topics Ricci solitons · Same author Guoqiang Wu · Shared terms “gradient Ricci”

  8. Rigidity of Five-Dimensional shrinking gradient Ricci solitons

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al. PDF

    Topics Ricci solitons · Same author Guoqiang Wu · Shared terms “gradient Ricci”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.