On an Inequality of Mean Curvature

where < , > denotes the scalar product in E, cn the area of the unit rc-sphere, and dV the volume element of M. The equality sign of (1) holds when and only when M" is imbedded as a hypersphere in an («+ l)-dimensional subspace of E (Chen [3], [4]; see also Chen [1] and Willmore [6], [7]). It is very interesting to know whether the inequality (1) can be improved for some special submanifolds in E. Recently, Willmore [8] proved that if M = SxS, m — 3 and x(M) is generated by carrying a small circle around a closed curve such that the centre moves along the curve and the plane of the circle is in the plane normal to the curve at each point, then the inequality (1) for n = 2 can be improved as follows: