SUBMANIFOLDS WITH CONSTANT PRINCIPAL CURVATURES AND NORMAL HOLONOMY GROUPS

A submanifold has by definition constant principal curvatures if the eigenvalues of the shape operators A_ξ are constant for any parallel normal field ξ along any curve. It is shown that a submanifold of Euclidean space has constant principal curvatures if and only if it is an isoparametric or a focal manifold of an isoparametric submanifold. Furthermore a "homogeneous slice theorem" is proved which says that the fibres of the projection of an isoparametric submanifold onto a full focal manifold are homogeneous isoparametric. To this end it is shown that any two points of an isoparametric submanifold can be connected by a piecewise differentiable curve whose pieces are tangent to one of the simultaneous eigenspaces E_i, i ∈ I, of the shape operators provided that the corresponding reflections generate the Weyl group of the isoparametric submanifold.