摘要
Let M be a closed oriented surface immersed in R4 . 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 Plcker coordinates. As one of consequences,we get the celebrated theorem of Chern and Spanier,Hoffman and Osserman,who proved it by quite different methods. At last,we give an explicit construction of a series of immersions of S2 in R4 with any given normal Euler number.
Let M be a closed oriented surface immersed in R4 . 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 Plcker coordinates. As one of consequences,we get the celebrated theorem of Chern and Spanier,Hoffman and Osserman,who proved it by quite different methods. At last,we give an explicit construction of a series of immersions of S2 in R4 with any given normal Euler number.
基金
supported by National Natural Science Foundation of China(Grant Nos. 10531090 and 10229101)
the Chang Jiang Scholars Program
