Geometry of Gauss Mapping

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First we investigated submanifolds M with degenerate Gauss mapping in spheres S^n. The Gauss map of M to real Grassmannian is constant if and only if M is totally geodesic in S^n, so the rank of the Gauss map measures the degree of how shape of M is near to the totally geodesic one. On the other hand, each leaf of the foliation given by the kernel of the differential of the Gauss map. So essential problem is that for a submanifold M in S^n foliated by great spheres, find the condition of which along each leaf the Gauss map is constant. In this research, we give general method to construct submanifolds foliated by great spheres in S^n by using the canonical sphere bundle over real Grassmanniam. Moreover, for either a circle bundle over complex submanifolds in complex quadrics or the twistor space over quaternionic symmetric spaces, we showed that along each fiber of the sphere bundles over submanifolds, the Gauss map is constant, and their twisted normal cones are special Lagrangian in complex Euclidean space.