2025/10/29 by Cecil, Thomas E., Ryan, Patrick J.
#53B25 #53C40 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.25611
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 Mn ⊂ Sn+1 ⊂ \bf Rn+2 is tight, i.e., every non-degenerate linear height function ℓp, 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 fundamental work on the structure of a family of isoparametric hypersurfaces in a sphere.