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Two Kinds of Curvature Operators in Dimension Four
Abstract
We prove an invariant identity relating the curvature operators of the first and second kind in dimension four. Under the natural identification S20≅Λ+⊗Λ-, the trace-free Ricci tensor contributes through a single commutator term. As a consequence, we relate the curvature operator of the second kind on products of unit self-dual and anti-self-dual two-forms to biorthogonal sectional curvature. We then classify closed oriented four-manifolds satisfying K⊥max≤ S/6, where K⊥max denotes the maximum biorthogonal sectional curvature and S is the scalar curvature. The classification includes the cases in which the scalar curvature vanishes.
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BibTeX
@misc{cao2026kinds,
title = {{Two Kinds of Curvature Operators in Dimension Four}},
author = {Xiaodong Cao and Xiaolong Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10887},
eprint = {2610.10887},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10887},
}
RIS
TY - UNPB
TI - Two Kinds of Curvature Operators in Dimension Four
AU - Xiaodong Cao
AU - Xiaolong Li
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10887
UR - https://arxiv.org/abs/2610.10887
AB - We prove an invariant identity relating the curvature operators of the first and second kind in dimension four. Under the natural identification S^2_0≅Λ^+⊗Λ^-, the trace-free Ricci tensor contributes through a single commutator term. As a consequence, we relate the curvature operator of the second kind on products of unit self-dual and anti-self-dual two-forms to biorthogonal sectional curvature. We then classify closed oriented four-manifolds satisfying K^⊥_max≤ S/6, where K^⊥_max denotes the maximum biorthogonal sectional curvature and S is the scalar curvature. The classification includes the cases in which the scalar curvature vanishes.
ER -