Classifications of Isoparametric Hypersurfaces in Randers Space Forms
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Classifications of Isoparametric Hypersurfaces in Randers Space Forms Qun HE
Pei Long DONG
School of Mathematical Sciences, Tongji University, Shanghai 200092, P. R. China E-mail : [email protected] [email protected]
Song Ting YIN1) Department of Mathematics and Computer Science, Tongling University, Tongling 244000, P. R. China and Key Laboratory of Applied Mathematics (Putian University), Fujian Province University, Putian 351100, P. R. China E-mail : [email protected] Abstract In this paper, we give the complete classifications of isoparametric hypersurfaces in Randers space forms. By studying the principal curvatures of anisotropic submanifolds in a Randers space (N, F ) with the navigation data (h, W ), we find that a Randers space form (N, F, dµBH ) and the corresponding Riemannian space (N, h) have the same isoparametric hypersurfaces, but in general, their isoparametric functions are different. We give a necessary and sufficient condition for an isoparametric function of (N, h) to be isoparametric on (N, F, dµBH ), from which we get some examples of isoparametric functions. Keywords manifold
Isoparametric hypersurface, Randers space form, principal curvature, anisotropic sub-
MR(2010) Subject Classification
1
53C60, 53C40, 53B25
Introduction
In Riemannian geometry, the study of isoparametric hypersurfaces has a long history. Since 1938, Cartan had begun to study the isoparametric hypersurfaces in real space forms with constant sectional curvature c systematically. The classification of isoparametric hypersurfaces in space forms is a classical geometric problem with a history of almost one hundred years. Isoparametric hypersurfaces in Euclidean and hyperbolic spaces were classified in 1930’s [3, 21, 24]. For the classification of isoparametric hypersurfaces in a unit sphere, which is the most difficult case, there are many important results (as like [14, 25], etc.) and it was recently completely solved in [10]. Received August 6, 2019, accepted March 24, 2020 Supported by NNSFC (Grant Nos. 11471246 and 11971253), AHNSF (Grant No. 1608085MA03) and KLAMFJPU (Grant No. SX201805) 1) Corresponding author
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He Q. et al.
In Finsler geometry, the concept of isoparametric hypersurfaces has been introduced in [16]. Let (N, F, dμ) be an n-dimensional Finsler manifold with volume form dμ. A function f on (N, F, dμ) is said to be isoparametric if there are a ˜(t) and ˜b(t) such that F (∇f ) = a ˜(f ), (1.1) Δf = ˜b(f ), where ∇f and Δf denote the nonlinear gradient and Laplacian of f with respect to dμ, respectively (see Sections 2.1 and 2.3 for details). Studying and classifying isoparametric hypersurfaces in Finsler space forms are interesting problems naturally generalized from Riemannian geometry. In [16], the authors studied isoparametric hypersurfaces in Finsler space forms, and obtained the Cartan type formula and some classifications on the number of distinct principal curvatures or their multiplicitie
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