Tight Spherical Embeddings (Updated Version)

Authors

Thomas E. Cecil, College of the Holy CrossFollow
Patrick J. Ryan, McMaster UniversityFollow

Document Type

Article

Publication Date

10-29-2025

Department

Mathematics

Abstract

This is an updated version of the paper [14] which appeared in the proceedings of the 1979 Berlin Colloquium on Global Differential Geometry. This paper contains the original exposition together with some notes by the authors made in 2025 (as indicated in the text) that give references to descriptions of progress made in the field since the time of the original version of the paper. The main result of this paper is that every compact isoparametric hypersurface Mⁿ ⊂ Sⁿ⁺¹ ⊂ Rⁿ⁺² is tight, i.e., every non-degenerate linear height function ℓ_p, p ∈ Sⁿ⁺¹, has the minimum number of critical points on Mⁿ required by the Morse inequalities. Since Mⁿ lies in the sphere Sⁿ⁺¹, this implies that Mⁿ is also taut in Sⁿ⁺¹, i.e., every non-degenerate spherical distance function has the minimum number of critical points on Mⁿ. A second result is that the focal submanifolds of isoparametric hypersurfaces in Sⁿ⁺¹ must also be taut. The proofs of these results are based on Münzner’s [23]–[24] fundamental work on the structure of a family of isoparametric hypersurfaces in a sphere.

Comments

Mathematics Subject Classification numbers: 53B25, 53C40, 53C42

DOI

http://arxiv.org/abs/2510.25611

Repository Citation

Cecil, Thomas E. and Ryan, Patrick J., "Tight Spherical Embeddings (Updated Version)" (2025). Mathematics and Computer Science Department Faculty Scholarship. 26.
https://crossworks.holycross.edu/math_fac_scholarship/26