Isoparametric families on projective spaces (Q1107852)

Let \(\Sigma\) be a transnormal system on the complex or quaternionic projective space \(FP^ m\) which contains a hypersurface M. Then the number of distinct principal curvatures of M is 2, 3, 5 or 7, and M may have nonconstant principal curvatures. One of the key ingredients to obtain these results is the number k of nonhorizontal eigenspaces \(T\bar M\), where \(\bar M=\pi^{-1}(M)\) and \(\pi: S^ n\to FP^ m\) is the Hopf fibration. In fact, k is constant on \(\bar M\) and M has constant principal curvatures if and only if \(k=2\). One of the interesting facts is the existence of non-isometric hypersurfaces of \(CP^ m\) whose corresponding isoparametric hypersurfaces are isometric in \(S^{2m+1}\).