A nonisoparametric hypersurface with constant principal curvatures
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- by Alberto Rodríguez-Vázquez PDF
- Proc. Amer. Math. Soc. 147 (2019), 5417-5420 Request permission
Abstract:
In this note we construct an explicit example of a (compact) conformally flat Riemannian manifold which admits a totally geodesic foliation of codimension one with no isoparametric leaves. This answers negatively the question: Is every hypersurface with constant principal curvatures isoparametric?References
- Élie Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl. 17 (1938), no. 1, 177–191 (French). MR 1553310, DOI 10.1007/BF02410700
- José Carlos Díaz-Ramos and Miguel Domínguez-Vázquez, Isoparametric hypersurfaces in Damek-Ricci spaces, Adv. Math. 239 (2013), 1–17. MR 3045139, DOI 10.1016/j.aim.2013.02.010
- J. C. Díaz-Ramos, M. Domínguez-Vázquez, and V. Sanmartín-López, Submanifold geometry in symmetric spaces of noncompact type, São Paulo J. Math. Sci. (2019), 1–36, doi:10.1007/s40863-019-00119-6.
- Jianquan Ge, Zizhou Tang, and Wenjiao Yan, A filtration for isoparametric hypersurfaces in Riemannian manifolds, J. Math. Soc. Japan 67 (2015), no. 3, 1179–1212. MR 3376584, DOI 10.2969/jmsj/06731179
- Alfred Gray, Tubes, 2nd ed., Progress in Mathematics, vol. 221, Birkhäuser Verlag, Basel, 2004. With a preface by Vicente Miquel. MR 2024928, DOI 10.1007/978-3-0348-7966-8
- Hitoshi Takagi, Conformally flat Riemannian manifolds admitting a transitive group of isometries. II, Tohoku Math. J. (2) 27 (1975), no. 3, 445–451. MR 442853, DOI 10.2748/tmj/1203529254
Additional Information
- Alberto Rodríguez-Vázquez
- Affiliation: Department of Mathematics, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
- Email: a.rodriguez@usc.es
- Received by editor(s): March 19, 2019
- Published electronically: June 10, 2019
- Additional Notes: The author was supported by the project MTM2016-75897-P (AEI/FEDER, Spain) and an FPU fellowship from Ministerio de Ciencia, Innovación y Universidades.
- Communicated by: Jiaping Wang
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 5417-5420
- MSC (2010): Primary 53C40; Secondary 53B25, 53C12
- DOI: https://doi.org/10.1090/proc/14654
- MathSciNet review: 4021099