On surfaces immersed in Euclidean space R4

Let M be a closed oriented surface immersed in R-4. Associated it one has the generalized Gauss map from M into the Grassmann manifold G(4,2). This note will be concerned with the geometry of the generalized Gauss map by using the moving frame theory and the quaternion interpretation of Plucker coordinates. As one of consequences, we get the celebrated theorem of Chern and Spanier, Ho. man and Osserman, who proved it by quite different methods. At last, we give an explicit construction of a series of immersions of S-2 in R-4 with any given normal Euler number.