Affine immersion ofn-dimensional manifold intoR n+n(n+1)/2 and affine minimality

Abstract

Abstract. We formulate an affine theory of immersions of an n-dimensional manifold into the Euclidean space of dimension n + n(n + 1)/2 and give a characterization of critical immersions relative to the induced volume functional in terms of the affine shape operator. (1991): 53A15. Mathematics Subject Classification

A unimodularly invariant theory for immersions into the affine space

“…For the "extremal" cases p = 1 and p = 1 2 n(n + 1) in Weise's approach the Ricci tensor is symmetric. The first case is well known (see (i)) and the latter proved in [Sas95]. In both "extremal" cases Weise's normal coincides with our choice of a unimodular normal.…”

“…For the special case of n = p = 2 Theorem 6.3 has been proved in [DMVV94], and for the case n = 2, p = 3 in [DDVL94] for special deformations. This result in [DDVL94] was generalized in [Sas95] to arbitrary dimension n and p = n(n + 1)/2. (ii) In case of the normal of Weise the critical points of the area functional are in general not the immersions with vanishing mean curvature in the sense considered here (see [Kli51], where a different notion of mean curvature is used).…”

“…(iii) The family of hypersurfaces with vanishing unimodular mean curvature is known to be be a rich class. For the maximal codimension p = 1 2 n(n+1) the Veronese immersion is an example of an immersion with vanishing unimodular mean curvature as shown in [Sas95].…”