  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 which are a critical point of the area functional of the
second fundamental form under various constraints.
