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PreprintCC BY 4.0arXiv

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 Rick≥0 for some 1≤ k≤ 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≤ k≤ n−2 is sharp: in every dimension n≥4, nonnegative Ricci curvature alone does not imply the constant rank property in either setting.

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Cite

BibTeXRIS
BibTeX
@misc{lai2026constant,
  title = {{Constant Rank Theorems under Intermediate Ricci Curvature}},
  author = {Mijia Lai and Guoqiang Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17691},
  eprint = {2610.17691},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17691},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Constant Rank Theorems under Intermediate Ricci Curvature
AU  - Mijia Lai
AU  - Guoqiang Wu
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.17691
UR  - https://arxiv.org/abs/2610.17691
AB  - 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≥0 for some 1≤ k≤ 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≤ k≤ n−2 is sharp: in every dimension n≥4, nonnegative Ricci curvature alone does not imply the constant rank property in either setting.
ER  -