Topology and Curvature of Isoparametric Families in Spheres

Chao Qian; Zizhou Tang; Wenjiao Yan

doi:10.1007/s40304-021-00259-2

Topology and Curvature of Isoparametric Families in Spheres

Chao Qian, Zizhou Tang^*, Wenjiao Yan

^*此作品的通讯作者

数学学院

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3 引用（Scopus）

摘要

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

源语言	英语
页（从-至）	439-475
页数	37
期刊	Communications in Mathematics and Statistics
卷	11
期	2
DOI	https://doi.org/10.1007/s40304-021-00259-2
出版状态	已出版 - 6月 2023

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10.1007/s40304-021-00259-2

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Qian, C., Tang, Z., & Yan, W. (2023). Topology and Curvature of Isoparametric Families in Spheres. Communications in Mathematics and Statistics, 11(2), 439-475. https://doi.org/10.1007/s40304-021-00259-2

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abstract = "An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.",

keywords = "Diffeomorphism, Focal submanifold, Homeomorphism, Homotopy equivalent, Isoparametric hypersurface, Lusternik–Schnirelmann category, Parallelizability, Ricci curvature, Sectional curvature",

author = "Chao Qian and Zizhou Tang and Wenjiao Yan",

note = "Publisher Copyright: {\textcopyright} 2022, School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2023",

month = jun,

doi = "10.1007/s40304-021-00259-2",

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journal = "Communications in Mathematics and Statistics",

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AU - Qian, Chao

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AU - Yan, Wenjiao

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N2 - An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

AB - An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

KW - Diffeomorphism

KW - Focal submanifold

KW - Homeomorphism

KW - Homotopy equivalent

KW - Isoparametric hypersurface

KW - Lusternik–Schnirelmann category

KW - Parallelizability

KW - Ricci curvature

KW - Sectional curvature

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Topology and Curvature of Isoparametric Families in Spheres

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