Some Theorems for Hypersurface of Randers Spaces
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Some Theorems for Hypersurface of Randers Spaces Jin Tang LI1) School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China E-mail : [email protected]
Jian Feng ZHANG Department of Mathematics, Lishui University, Lishui 323000, P. R. China E-mail : [email protected] Abstract In this paper, we consider the hypersurfaces of Randers space with constant flag curvature. n+1 n+1 , F ) be a Randers–Minkowski space. If (M n , F ) is a hypersurface of (M , F ) with (1) Let (M n+1 , F ) be a constant flag curvature K = 1, then we can prove that M is Riemannian. (2) Let (M n+1 Randers space with constant flag curvature. Assume (M, F ) is a compact hypersurface of (M ,F) with constant mean curvature |H|. Then a pinching theorem is established, which generalizes the result of [Proc. Amer. Math. Soc., 120, 1223–1229 (1994)] from the Riemannian case to the Randers space. Keywords
Randers space, hypersurfaces, constant mean curvature, constant flag curvature
MR(2010) Subject Classification
1
53C60, 53C40
Introduction
The Riemannian hypersurfaces are important in modern differential geometry. There has been a long history for the study of Riemannian hypersurfaces. Finsler manifold is a differentiable manifold with Finsler metric. Finsler metric is just Riemannian metric without the quadratic restriction. Recent studies on Finsler manifolds have taken on a new look and Finsler manifolds can also be applied to biology and physics, etc. In these researches, people find that there is a quite important metric constructed from a Riemannian metric α and a 1-form β on a smooth manifold M . We call this metric a Randers metric which was firstly studied by the physician Randers and be applied in studying the physics, biology and navigation problems, etc. In [7], Shen studied the projectively flat Randers metrics and has classified projectively flat Randers metrics with constant flag curvature. In [3], Bao, Robles and Shen have completed classification of strongly convex Randers metrics with constant flag curvature. As far as we know, on researches for Finsler manifolds, scholars studied mainly the local properties of Finsler manifolds in local coordinate bases. But there are very few global rigidity results on Finsler manifolds. This motivates us to start considering some properties for Finsler hypersurfaces of Finsler manifolds. In this paper, we consider the hypersurfaces of Randers space with constant flag curvature. Received November 13, 2019, accepted April 8, 2020 Supported by the National Natural Science Foundation of China (Grant No. 11871405) 1) Corresponding author
Li J. T. and Zhang J. F.
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For the hypersurfaces of Minkowski space, Professor Zhongmin Shen has given a problem as follows: n+1
Problem 1.1 Let (M , F ) be a Minkowski space and F the induced Finsler metric on the indicatrix S = F −1 (1). If F is of constant flag curvature K = 1, is F Euclidean? In §4, we consider the hypersurfaces of Randers–Minkowsk
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