Isoparametric submanifolds

In this talk we consider two classes of submanifolds in Euclidean spaces that are characterized by simple local invariants. The first class consists of submanifolds with costant principal curvatures. These are by definition submanifolds $Mn$ such that for every parallel normal vector field $\xi(t)$ along a curve in $Mn$ the eigenvalues of the shape operator $A_{\xi(t)}$ are constant, see [St] and [0l2]. The second class of submanifolds we will consider consists of isoparametric submainfolds. These are by definition submanifolds $Mn$ such that the normal bundle is flat and the eigenvalues of the shape operator $A_\xi$ are constant for every locally defined parallel normal vector field $\xi$, see [Ha], [CW], [Te] and [PT 2]. It is of course clear that isoparametric submanifolds have constant principal curvatures