Classification of Casorati ideal Legendrian submanifolds in Sasakian space forms II

Abstract

In Lee et al. (2020) [21], the authors of the present article proved two optimal inequalities involving the Casorati curvatures δC(n−1) and δCˆ(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 δC(r;n−1) and δCˆ(r;n−1), which are fundamental extrinsic invariants of Riemannian submanifolds originally introduced by Decu et al. (2008) [14] as a natural generalization of δC(n−1) and δCˆ(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.