2023/02/13 by Jianquan Ge, Yi Zhou, Ge, Jianquan +1
Mathematics · Medicine · #53A07 #53C40 #53C42 #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2302.06105
openalex publication_date 2023/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by Bryant's research on austere subspaces and Cartan's isoparametric hypersurfaces with 3 distinct principal curvatures, we construct three families of austere submanifolds with flat normal bundle in unit spheres. From these examples we find three irreducible proper Dupin hypersurfaces with 5 distinct principal curvatures of different multiplicities. Thus, we give a negative answer to an open question raised by Thorbergsson in 2000 which is instructive for the local classification of proper Dupin hypersurfaces. Moreover, as an application, we obtain an upper bound estimate for the dimension of austere subspaces.