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    Manifolds with $4\frac{1}{2}$-Positive Curvature Operator of the Second Kind

    Date
    2022-08-04
    Author
    Li, Xiaolong
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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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    URI
    https://doi.org/10.1007/s12220-022-01033-8
    https://soar.wichita.edu/handle/10057/24866
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