  In this article a relation between curvature functionals for surfaces in the
Euclidean space and area functionals in relative differential geometry will be
given. Relative differential geometry can be described as the geometry of
surfaces in the affine space, endowed with a distinguished "relative normal
vector field" which generalises the notion of unit normal vector field N from
Euclidean differential geometry. A concise review of relative differential
geometry will be presented. The main result, to which the title of this article
refers, will be given in the third section. Here we consider, for a function
$f$ of two variables, relative normal vector fields of the form
$f(H,K)\,N-\grad_{\II}(f(H,K))$ for non-degenerate surfaces in the Euclidean
three-dimensional space. A comparison of the variation of the curvature
functional $\int f(H,K)\,\dd\Omega$ with the relative area functional obtained
from the above relative normal vector field, results in a distinguishing
property for the one-parameter family of relative normal vector fields which
was introduced by F.\ Manhart, and which is obtained by choosing
$f(H,K)=|K|^{\alpha}$ (where we will assume that $\alpha\neq 1$). More
precisely, the following will be shown in theorem~6: ``{The curvature
functionals $(\ast)$ for which the critical points coincide with the
relative-minimal surfaces with respect to the relative normal vector field
$(\dagger)$, are essentially those obtained from Manhart's family}." In the
fourth section, we give a characterisation of the sphere by means of relations
between the support function and the curvatures. In the last section, we
combine the previously described results and arrive at a variational
characterisation of the sphere.

Differential Geometry