Tight Spherical Embeddings (Updated Version)

TL;DR

This paper proves that all compact isoparametric hypersurfaces in spheres are tight and taut, meaning they have the minimal number of critical points for certain functions, based on foundational work in differential geometry.

Contribution

It establishes the tightness and tautness of all compact isoparametric hypersurfaces in spheres, extending classical results with updated notes and references.

Findings

All compact isoparametric hypersurfaces are tight.

Focal submanifolds of these hypersurfaces are also taut.

Results are based on M"{u}nzner's foundational work.

Abstract

This is an updated version of a paper 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 $M^n \subset S^{n+1} \subset {\bf R}^{n+2}$ is tight, i.e., every non-degenerate linear height function $\ell_p$, $p \in S^{n+1}$, has the minimum number of critical points on $M^n$ required by the Morse inequalities. Since $M^n$ lies in the sphere $S^{n+1}$, this implies that $M^n$ is also taut in $S^{n+1}$, i.e., every non-degenerate spherical distance function has the minimum number of critical points on $M^n$. A second result is that the focal submanifolds of isoparametric hypersurfaces in $S^{n+1}$ must also be taut. The proofs of these results are based on M\"{u}nzner's fundamental work on the structure of a family of isoparametric hypersurfaces in a sphere.