PreprintarXiv licencearXiv
Tachibana's Theorem Revisited
Abstract
We prove Tachibana-type rigidity theorems for closed manifolds under weakened partial-sum conditions on the eigenvalues of curvature operators. Specifically, an Einstein manifold of dimension n≥4 is locally symmetric if its curvature operator is (2(n−1))/3-nonnegative. The corresponding thresholds are (2(n+1))/3 for the primitive curvature operator of a Kähler-Einstein manifold of complex dimension n≥2, and (2(n+2))/3 for the curvature operator of a quaternionic-Kähler manifold of real dimension 4n≥8.
Related
BibTeX
@misc{li2026tachibanas,
title = {{Tachibana's Theorem Revisited}},
author = {Xiaolong Li and Peter Petersen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36226},
eprint = {2609.36226},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36226},
}
RIS
TY - UNPB
TI - Tachibana's Theorem Revisited
AU - Xiaolong Li
AU - Peter Petersen
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36226
UR - https://arxiv.org/abs/2609.36226
AB - We prove Tachibana-type rigidity theorems for closed manifolds under weakened partial-sum conditions on the eigenvalues of curvature operators. Specifically, an Einstein manifold of dimension n≥4 is locally symmetric if its curvature operator is (2(n−1))/3-nonnegative. The corresponding thresholds are (2(n+1))/3 for the primitive curvature operator of a Kähler-Einstein manifold of complex dimension n≥2, and (2(n+2))/3 for the curvature operator of a quaternionic-Kähler manifold of real dimension 4n≥8.
ER -