AFFINE SURFACES IN ℝ⁵

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

“…Put Vx~XN = EF~jXp and V~o = 7i;G. By definition, we have This is seen to be the same as the condition considered in [2]; refer to Theorem 1.2 of the paper. The affine normal space also coincides with that given in [2]. This can be considered to be also an element of End(End(N'), End(T)), which we denote by A ± to clarify the difference from A.…”

“…With this definition, the pair {h ~ } and {Ho} also satisfies (1) and (2), that is, {h"} and {Ha} are the inverse of each other.…”

“…As for surfaces in R 4, Nomizu and Vrancken ( [5]) gave a new formulation showing that there is also a canonical way of introducing an equiaffine structure. This is also the case of surfaces in R 5 as shown in [2]. The first aim of this paper is generalizing this study to the case of n-dimensional submanifolds in R ~°+p, where p = n(n + 1)/2, by transferring the projective theory of n-dimensional submanifolds in the projective space P~+" to the affine case, which was treated in [6].…”