$\delta^{\sharp}(2,2)$-Ideal Centroaffine Hypersurfaces of Dimension 4

Ideal submanifolds have been studied from various aspects since Chen invented $\delta$-invariants in early 1990s (see [12] for a survey). In centroaffine differential geometry, Chen's invariants denoted by $\delta^{\sharp}$ are used to determine an optimal bound for the squared norm of the Tchebychev vector field of a hypersurface. We point out that a hypersurface attaining this bound is said to be an ideal centroaffine hypersurface. In this paper, we deal with $\delta^{\sharp}(2,2)$-ideal centroaffine hypersurfaces in $\mathbb{R}^{5}$ and in particularly, we focus on $4$-dimensional $\delta^{\sharp}(2,2)$-ideal centroaffine hypersurfaces of type $1$.