Some rigidity theorems on Finsler manifolds

Songting Yin; Songting Yin

doi:10.3934/math.2021184

AIMS Mathematics

2021, Volume 6, Issue 3: 3025-3036. doi: 10.3934/math.2021184

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Research article

Some rigidity theorems on Finsler manifolds

Songting Yin ^,

Department of Mathematics and Computer Science, Tongling University, Tongling, 244000 Anhui, China

Received: 30 October 2020 Accepted: 05 January 2021 Published: 11 January 2021
MSC : 53C24, 53C60

We prove that, for a Finsler manifold with the weighted Ricci curvature bounded below by a positive number, it is a Finsler sphere if and only if the diam attains its maximal value, if and only if the volume attains its maximal value, and if and only if the first closed eigenvalue of the Finsler-Laplacian attains its lower bound. These generalize some rigidity theorems in Riemannian geometry to the Finsler setting.
- Randers sphere,
- the maximum diam,
- the weighted Ricci curvature
Citation: Songting Yin. Some rigidity theorems on Finsler manifolds[J]. AIMS Mathematics, 2021, 6(3): 3025-3036. doi: 10.3934/math.2021184

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Abstract

We prove that, for a Finsler manifold with the weighted Ricci curvature bounded below by a positive number, it is a Finsler sphere if and only if the diam attains its maximal value, if and only if the volume attains its maximal value, and if and only if the first closed eigenvalue of the Finsler-Laplacian attains its lower bound. These generalize some rigidity theorems in Riemannian geometry to the Finsler setting.

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