New Curvature Operator of the Second Kind Conditions for Rigidity
Abstract
We establish Tachibana-type rigidity theorems with the aid of the curvature operator of the second kind (COSK). A sharp estimate that combines the symmetric and commutator actions on Weyl tensors leads to a constant Cn, with Cn/n→5/4, such that a complete Einstein manifold of dimension n≥5 with Cn-nonnegative COSK has constant nonnegative sectional curvature. We also explore shifted curvature operators of the second kind leading to similar rigidity results, offer an elegant Ricci-corrected Bochner identity for harmonic forms, and classify the complete simply connected locally symmetric Einstein manifolds in the resulting shifted cone. Explicit spaces establish the sharpness of the tensor estimates, and complete lists of the eigenvalues of the COSKs for symmetric spaces identify which of them are borderline cases.
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BibTeX
@misc{li2026curvature,
title = {{New Curvature Operator of the Second Kind Conditions for Rigidity}},
author = {Xiaolong Li and Peter Petersen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07687},
eprint = {2610.07687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07687},
}
RIS
TY - UNPB
TI - New Curvature Operator of the Second Kind Conditions for Rigidity
AU - Xiaolong Li
AU - Peter Petersen
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07687
UR - https://arxiv.org/abs/2610.07687
AB - We establish Tachibana-type rigidity theorems with the aid of the curvature operator of the second kind (COSK). A sharp estimate that combines the symmetric and commutator actions on Weyl tensors leads to a constant C_n, with C_n/n→5/4, such that a complete Einstein manifold of dimension n≥5 with C_n-nonnegative COSK has constant nonnegative sectional curvature. We also explore shifted curvature operators of the second kind leading to similar rigidity results, offer an elegant Ricci-corrected Bochner identity for harmonic forms, and classify the complete simply connected locally symmetric Einstein manifolds in the resulting shifted cone. Explicit spaces establish the sharpness of the tensor estimates, and complete lists of the eigenvalues of the COSKs for symmetric spaces identify which of them are borderline cases.
ER -