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Geometry of isoparametric hypersurfaces in Riemannian manifolds. (English) Zbl 1292.53040

Summary: 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.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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