Let M be an «-dimensional manifold immersed in a euclidean m-space E m . Let S and a be the length of second fundamental form and the length of mean curvature vector and let p be the scalar curvature of M. Then p = n 2 a 2 -S 2 . From Proposition 2.2 of [2], p satisfies the following pinching property: Let F be a field and let H t (M; F) be the Zth homology group of M over F. We define a topological invariant $(M) by The purpose of this note is to announce the following results. The detailed

more »

... ofs will appear elsewhere. THEOREM 1. Let M be an n-dimensional closed manifold immersed in a euclidean m-space E m t Then (1) f M S n dV>y(}(M), where (2) in"l 2 c n /2, if n is even, 7 " ) «<" +1 >/ 2 c" +1 c m _ 1 /2c m (m -\) l '\ if n is odd, and c n is the area of unit n-spheres. The equality sign holds only when M is imbedded as a hypersphere of a linear (n + l)-subspace of E m . AMS (MOS) subject classifications (1970). Primary 53B25, 53C40. Key words and phrases. Length of second fundamental form, total mean curvature, scalar curvature, homology groups.