By using T. Oprea's optimization methods on submanifolds, we give another proof of the inequalities relating the normalized $\delta-$Casorati curvature $\hat{ \delta}_c(n-1)$ for submanifolds in real space forms. Also, inequalities relating the normalized $\delta-$Casorati curvature $\delta_C(n-1)$ for submanifolds in real space forms are obtained. Besides, we characterize a kind of Casorati ideal hypersurface of Euclidean 4-space. We also show that this kind of Casorati ideal hypersurface is rigid.
Categories: math.DG