Manifolds with $4\frac{1}{2}$-Positive Curvature Operator of the Second Kind
Li, Xiaolong
Li, Xiaolong
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2022-08-04
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Article
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Curvature operator of the second kind,Nishikawa’s conjecture,Sphere theorem,Positive isotropic curvature
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Citation
Li, X. Manifolds with $4\frac{1}{2}$-Positive Curvature Operator of the Second Kind. J Geom Anal 32, 281 (2022). https://doi.org/10.1007/s12220-022-01033-8
Abstract
We show that a closed four-manifold with $4\frac{1}{2}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both ${\mathbb{CP}\mathbb{}}^2$and ${\mathbb {S}}^3 \times {\mathbb {S}}^1$have $4\frac{1}{2}$-nonnegative curvature operator of the second kind. In higher dimensions $n\ge 5$, we show that closed Riemannian manifolds with $4\frac{1}{2}$-positive curvature operator of the second kind are homeomorphic to spherical space forms. These results are proved by showing that $4\frac{1}{2}$-positive curvature operator of the second kind implies both positive isotropic curvature and positive Ricci curvature. Rigidity results for $4\frac{1}{2}$-nonnegative curvature operator of the second kind are also obtained.
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Springer Nature
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The Journal of Geometric Analysis
Volume 32
Volume 32
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1559-002X