Differential geometry of submanifolds with planar normal sections

Summary Submanifolds of E^m with (pointwise) planar normal sections were studied in [3, 4]. In the present paper, we will prove that the first normal spaces of such submanifolds are always parallel in the normal bundle. Moreover, we will prove that a spherical submanifold has parallel second fundamental form if and only if it has pointwise planar normal sections. Some applications of these fundamental results will also be given.