Abstract

The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called `mean curvature of the second fundamental form.' Several new characteristic properties of (hyper)spheres in which the mean curvature of the second fundamental form occurs are given. In particular, it is shown that the spheres are the only ovaloids that are a critical point of the area functional of the second fundamental form under various constraints.