Аннотация:where dV denotes the volume element of M2. The equality holds when and only when M2 is a sphere in E3. The result (A) was proved by Fenchel [5] in 1929, and (B) by Willmore [6] in 1968. The results (A) and (B) were generalized to closed curves and surfaces in higher dimensional e-Lelidean space by Borsuk [1] and Chen [3], respectively. In this paper we give some generalizations of (A) and (B). We consider a closed manifold Mn of dimension n and an immersion x: Mn > En+N of Mn into euclidean space of dimension n + N. Let B, (x) be the bundle of unit normal vectors of x (Mn) so that a point of B, (x) is a pair (p,e) where e is a unit normal vector at x(p). Then B,(x) is a bundle of (N -1)-dimensional spheres over Mn and is a (smooth) manifold of dimension n + N1. Let dV be the volume element of Mn. There is a differential form doof degree N 1 on B, (x) such that its restriction to a fibre is the volume element
