$\delta^{\sharp}(2,2)$-Ideal Centroaffine Hypersurfaces of Dimension $5$

Abstract

The notion of an ideal submanifold was introduced by Chen at the end of the last century. A survey of recent results in this area can be found in his book [9]. Recently, in [10], an optimal collection of Chen's inequalities was obtained for Lagrangian submanifolds in complex space forms. As shown in [2], these inequalities have an immediate counterpart in centroaffine differential geometry. Centroaffine hypersurfaces realising the equality in one of these inequalities are called ideal centroaffine hypersurfaces. So far, most results in this area have only been related with $3$- and $4$-dimensional $\delta^{\sharp}(2)$-ideal centroaffine hypersurfaces. The purpose of this paper is to classify $\delta^{\sharp}(2,2)$-ideal hypersurfaces of dimension~$5$ in centroaffine differential geometry.