English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

In this article a relation between curvature functionals for surfaces in theEuclidean space and area functionals in relative differential geometry will begiven. Relative differential geometry can be described as the geometry ofsurfaces in the affine space, endowed with a distinguished "relative normalvector field" which generalises the notion of unit normal vector field N fromEuclidean differential geometry. A concise review of relative differentialgeometry will be presented. The main result, to which the title of this articlerefers, 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 Euclideanthree-dimensional space. A comparison of the variation of the curvaturefunctional $\int f(H,K)\,\dd\Omega$ with the relative area functional obtainedfrom the above relative normal vector field, results in a distinguishingproperty for the one-parameter family of relative normal vector fields whichwas introduced by F.\ Manhart, and which is obtained by choosing$f(H,K)=|K|^{\alpha}$ (where we will assume that $\alpha\neq 1$). Moreprecisely, the following will be shown in theorem~6: ``{The curvaturefunctionals $(\ast)$ for which the critical points coincide with therelative-minimal surfaces with respect to the relative normal vector field$(\dagger)$, are essentially those obtained from Manhart's family}." In thefourth section, we give a characterisation of the sphere by means of relationsbetween the support function and the curvatures. In the last section, wecombine the previously described results and arrive at a variationalcharacterisation of the sphere.

A Characterisation of Manhart's Relative Normal Vector Fields