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October, 2017 Hypersurfaces of Randers Spaces with Constant Mean Curvature
Jintang Li
Taiwanese J. Math. 21(5): 979-996 (October, 2017). DOI: 10.11650/tjm/7945


Let $(\overline{M}^{n+1}, \overline{F})$ be a complete simply connected Randers space with $\overline{F}(x,Y) = \overline{a}(x,Y)+ \overline{b}(x,Y)$, where $\overline{a}(x,Y) = \sqrt{\overline{a}_{ij}(x) Y^i Y^j}$ and $\overline{b}(x,Y) = \overline{b}_i(x) Y^i$ are a Riemannian metric and a $1$-form on the smooth $(n+1)$-dimensional manifold $\overline{M}$ respectively. Assume the $1$-form $\overline{b}$ is parallel with respect to $\overline{a}$ and the sectional curvature $\overline{K}_{\overline{M}}$ of $\overline{M}$ with respect to $\overline{a}$ satisfies $\delta(n) \leq \overline{K}_{\overline{M}} \leq 1$. In this paper, we study the compact hypersurface $(M,F)$ of the Randers space $(\overline{M}^{n+1}, \overline{F})$ with constant mean curvature $|H|$ and prove that if the norm square $S$ of the second fundamental form of $(M,F)$ with respect to the Finsler metric $\overline{F}$ satisfies a certain inequality, then $S = n|H|^2$ and $M$ is the unit sphere or equality holds. In that case, we describe all $M$ that satisfy this equality, which generalizes the result of [8] from the Riemannian case to the Randers space.


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Jintang Li. "Hypersurfaces of Randers Spaces with Constant Mean Curvature." Taiwanese J. Math. 21 (5) 979 - 996, October, 2017.


Received: 24 August 2016; Revised: 27 November 2016; Accepted: 3 January 2017; Published: October, 2017
First available in Project Euclid: 1 August 2017

zbMATH: 06871355
MathSciNet: MR3707880
Digital Object Identifier: 10.11650/tjm/7945

Primary: 53C60
Secondary: 53C40

Keywords: Finsler manifolds , hypersurfaces , Randers spaces

Rights: Copyright © 2017 The Mathematical Society of the Republic of China


Vol.21 • No. 5 • October, 2017
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