A New Differentiable Sphere Theorem and Its Applications

Year 2024, Volume: 17 Issue: 2, 388 - 393

Vladimir Rovenski , Sergey Stepanov

https://doi.org/10.36890/iejg.1529961

Abstract

In this paper, we use the Lichnerowicz Laplacian to prove new results: the sphere theorem and the integral inequality for Einstein's infinitesimal deformations, which allow us to characterize spherical space forms. Our version of the sphere theorem states that a closed connected Riemannian manifold $(M, g)$ of even dimension $n>3$ is diffeomorphic to a Euclidean sphere or a real projective space if the inequality $Ric_{\rm max}(x) < n K_{\rm min}(x) g$ is true at each point $x\in M$, where $Ric_{\rm max}(x)$ is the maximum of the Ricci curvature, and $K_{\rm min}(x)$ is the minimum of the sectional curvature of $(M, g)$ at $x$. Since this inequality implies positive sectional curvature; therefore, our result partially answers Hopf's old open question.

Keywords

Lichnerowicz Laplacian, differentiable sphere theorem, Einstein's infinitesimal deformation, spherical space form, curvature operator of the second kind

Ethical Statement

Not required

Supporting Institution

No

Thanks

Not required

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