Open Access
January 2014 Geometry of isoparametric hypersurfaces in Riemannian manifolds
Jianquan Ge, Zizhou Tang
Asian J. Math. 18(1): 117-126 (January 2014).


In our previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exists at least one minimal isoparametric hypersurface. In this paper, we show such a minimal isoparametric hypersurface is also unique in the family if the ambient manifold has positive Ricci curvature. Moreover, we give a proof of Theorem D claimed by Q.M.Wang (without proof) which asserts that the focal submanifolds of an isoparametric function on a complete Riemannian manifold are minimal. Further, we study isoparametric hypersurfaces with constant principal curvatures in general Riemannian manifolds. It turns out that in this case the focal submanifolds have the same properties as those in the standard sphere, i.e., the shape operator with respect to any normal direction has common constant principal curvatures. Some necessary conditions involving Ricci curvature and scalar curvature are also derived.


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Jianquan Ge. Zizhou Tang. "Geometry of isoparametric hypersurfaces in Riemannian manifolds." Asian J. Math. 18 (1) 117 - 126, January 2014.


Published: January 2014
First available in Project Euclid: 27 August 2014

zbMATH: 1292.53040
MathSciNet: MR3215342

Primary: 53C20.

Keywords: constant mean curvature , focal submanifold , Isoparametric function

Rights: Copyright © 2014 International Press of Boston

Vol.18 • No. 1 • January 2014
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