In Lee et al. (2020) [21], the authors of the present article proved two optimal inequalities involving the Casorati curvatures ${\delta }_C(n-1)$ and $\widehat{{\delta }_C}(n-1)$ of n-dimensional Legendrian submanifolds in Sasakian space forms and identified the classes of those submanifolds for which the equality cases of both inequalities hold. The aim of this paper is to generalize these results to the case of generalized Casorati curvatures ${\delta }_C(r;n-1)$ and $\widehat{{\delta }_C}(r;n-1)$, which are fundamental extrinsic invariants of Riemannian submanifolds originally introduced by Decu et al. (2008) [14] as a natural generalization of ${\delta }_C(n-1)$ and $\widehat{{\delta }_C}(n-1)$, where r is any real number such that $0<r<n(n-1)$ or $r>n(n-1)$, respectively. We also provide examples of submanifolds that are ideal for any given r.

在李等人。(2020) [21]，本文作者证明了两个涉及卡索拉蒂曲率的最优不等式${\delta }_C(n-1)$ 和 $\widehat{{\delta }_C}(n-1)$在 Sasakian 空间形式中的n维勒让德里亚子流形，并确定了这些子流形的类，其中两个不等式的相等情况都成立。本文的目的是将这些结果推广到广义卡索拉蒂曲率的情况${\delta }_C(r;n-1)$ 和 $\widehat{{\delta }_C}(r;n-1)$，它们是最初由 Decu 等人引入的黎曼子流形的基本外在不变量。(2008) [14] 作为对${\delta }_C(n-1)$ 和 $\widehat{{\delta }_C}(n-1)$，其中r是任何实数，使得$0<r<n(n-1)$ 或者 $r>n(n-1)$， 分别。我们还提供了适用于任何给定r的子流形示例。