A condition for isoparametric hypersurfaces of \(S^ n\) to be homogeneous (Q1059292)

An isoparametric hypersurface M with g distinct principal curvatures in the sphere is ''algebraically homogeneous'' in the sense that the second fundamental forms at any two points coincide up to linear isometries of the corresponding tangent spaces. The author shows that M is homogeneous if and only if a similar algebraic homogeneity holds for all covariant derivatives up to order g-2 of the second fundamental form.