On a Submanifold of a Submanifold of a Riemannian Manifold and the Gauss Map

The fundamental properties of frame bundles of a Submanifold of a Riemannian manifold are described by S. Kobayashi and K. Nomizu in [2]. Using the similar method, we will study frame bundles of a Submanifold of a Submanifold of a Riemannian manifold. The main purpose of this paper is to associate the Gauss (generalized) map to a submanifold of a Submanifold of Euclidean space. M. Obata [3] associates the Gauss map to a submanifold of a simply-connected complete N-space of constant curvature. We will study the relationship between the Gauss map in the sense of Obata and that of our sense in the forthcoming paper.