A New Differentiable Sphere Theorem and Its Applications

Yıl 2024, Cilt: 17 Sayı: 2, 388 - 393

Vladimir Rovenski , Sergey Stepanov

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

Öz

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.

Anahtar Kelimeler

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

Etik Beyan

Not required

Destekleyen Kurum

No

Teşekkür

Not required

Kaynakça

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