Isoparametric functions and exotic spheres
By Jianquan Ge, Zizhou Tang. 2010
74 citations recorded by the source corpus.
The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly transnormal function on any exotic 4-sphere if it exists. Furthermore, by projecting an $S^3$-invariant isoparametric function on $Sp(2)$, we construct a properly transnormal but not an isoparametric function on the Gromoll-Meyer sphere with two points as the focal varieties.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/5c126862e876d170628f58dfb0ff1d28455822dd
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Geometry of isoparametric hypersurfaces in Riemannian manifolds (2010, background): “ural result given by [Wa87] says that the focal varieties M± of a transnormal function are smooth submanifolds and each regular level hypersurface Mt is a tube over either of M±. In our previous work [GT09], we called an isoparametric (or transnormal) function proper if each component of M± has codi”
- Isoparametric hypersurfaces with four principal curvatures, II (2010, background): “It is noteworthy that in recent years there has been much effort to investigate isoparametric foliations on Riemannian manifolds other than the standard spheres, such as exotic spheres [19], [29], compact manifolds of positive scalar curvature [30], complex and quaternionic projective spaces [10], [”