ORBITS OF s-REPRESENTATIONS WITH DEGENERATE GAUSS MAPPINGS

A submanifold is called tangentially degenerate if its Gauss mapping is degenerate. The investigation of tangentially degeneracy of submanifolds has long history. For example the classification of surfaces in R with degenerate Gauss mapping is equivalent to the classification of flat surfaces in R. As a result, that is one of planes, cylinders, cones or tangent developable surfaces. In this paper we shall investigate the Gauss mapping of a submanifold in the sphere, that is defined as a mapping to a Grassmannian manifold. The definition of the Gauss mapping, which here we deal with, will be given in Section 2. Ferus [4] obtained a remarkable result for tangentially degeneracy of submanifolds in the sphere. He showed that there exists a number, so-called the Ferus number, such that if the rank of the Gauss mapping is less than the Ferus number, then a submanifold must be a totally geodesic sphere. However, in general it is still unknown whether there exist submanifolds which satisfy the Ferus equality, that is, the equality of the Ferus inequality. In their papers [9, 10, 11], Ishikawa, Kimura and Miyaoka studied submanifolds with degenerate Gauss mappings in the sphere via a method of isoparametric hypersurfaces. They showed that Cartan hypersurfaces and some focal submanifolds of homogeneous isoparametric hypersurfaces are tangentially degenerate. Moreover, some of them satisfy the Ferus equality. A homogeneous isoparametric hypersurface in the sphere is obtained as an orbit of an s-representation of a compact symmetric pair of rank 2. Therefore we shall study submanifolds with degenerate Gauss mappings via a method of symmetric spaces. Our strategy is to investigate the space of relative nullity of the orbits. In fact, the index of relative nullity is equal to the rank of tangentially degeneracy. We will study the second fundamental form of the orbits of s-representations by restricted root systems, and determine their spaces of relative nullity. As a result, we will obtain that the space of relative nullity of the orbits through a long root, or a short root of restricted root system of type G2, is coincide with the root space