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arXiv · 2610.07687

New Curvature Operator of the Second Kind Conditions for Rigidity

Xiaolong Li·Peter Petersen
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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 $C_{n}$, with $\frac{C_{n}}{n}\to\frac{5}{4}$, such that a complete Einstein manifold of dimension $n\geq5$ 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.

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Xiaolong Li, Peter Petersen. 2026-10-06. New Curvature Operator of the Second Kind Conditions for Rigidity. https://arxiv.org/abs/2610.07687

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