Some Notes on Submanifolds of an Euclidean Space with Conformal Gauss Map

Let （M,g） be an m-dimensional Riemannian manifold and i:M→Ean isometricimmersion of （M,g） into an n-dimensional Euclidean space E. Let VM be anopen set in which the immersion i:M→Eis given by x=x（y）, （h=1,…,n; α=1,…,m）. Here and in the sequel x（h=1,…,n） are rectangular coordinates of Eand y（α=1,…,m） are local coordinates of a generic point in V. The tangent planeiM=i（M）, P∈V, of iM can be considered after a suitable parallel displacementas a point Γ（P） of the Grassmann manifold G（m,n-m）.The mapping Γ:iM→G（m,n-m）, ipΓ（iP） =Γ（P） is called the Gauss map. The mapping Γ:M→G（m,n-m）,p（P） is called the Gauss map associated with the immersion i, and（M）=F（iM） the Gauss image of M.