Affine immersion of \(n\)-dimensional manifold into \(\mathbb{R}^{n+n(n+1)/2}\) and affine minimality
In classical affine differential geometry, one studies hypersurfaces of the equiaffine space \(\mathbb{R}^{n+ 1}\), equipped with its natural equiaffine structure, and one induces a canonical equiaffine structure on the hypersurface by introducing a uniquely determined affine normal. Recently this procedure has been extended to surfaces in \(\mathbb{R}^4\) and \(\mathbb{R}^5\) by several authors. Moreover, in all cases the equivalence of ``critical volume'' and ``vanishing affine mean curvature'' has been established. There is an early attempt by Weise to deal with all possible dimensions \(n\) and codimensions at most \(n(n+ 1)/2\). His results, dating from late thirties, are interesting, but difficult to understand. Also, in his approach there is no equivalence of critical volume and vanishing affine mean curvature, and no equiaffine structure is mentioned (in fact, following Weise's approach for surfaces in \(\mathbb{R}^4\), neither two objectives hold, but for surfaces in \(\mathbb{R}^5\), they do). In the paper under review, the author deals with \(n\)-dimensional submanifolds in \(\mathbb{R}^{n+ n(n+ 1)/2}\) whose first osculating space has its maximal dimension \(n(n+ 1)/2\). He reconstructs Weise's affine normal space in a more understandable way and proves that indeed, for this special codimension, the induced volume form is parallel with respect to the induced connection and that the volume is critical for all deformations with compact support if and only if the trace of all shape operators is zero.
scientific article; zbMATH DE number 806292