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arXiv · 2112.08465

Manifolds with nonnegative curvature operator of the second kind

Xiaolong Li
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Abstract

We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is diffeomorphic to a spherical space form, improving a recent result of Cao-Gursky-Tran assuming two-positivity. The second one states that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either diffeomorphic to a spherical space form, or flat, or isometric to a quotient of a compact irreducible symmetric space. This settles the nonnegativity part of Nishikawa's conjecture under a weaker assumption.

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Xiaolong Li. 2021-12-15. Manifolds with nonnegative curvature operator of the second kind. https://doi.org/10.1142/s0219199723500037

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