Legendrian normally flat submanifols of $\mathcal{S}$-space forms

Abstract

In the present study, we consider a Legendrian normally flat submanifold $M$ of $(2n+s)$-dimensional $\mathcal{S}$-space form $\widetilde{M}^{2n+s}(c)$ of constant $\varphi$-sectional curvature $c$. We have shown that if $M$ is pseudo-parallel then $M$ is semi-parallel or totally geodesic. We also prove that if $M$ is Ricci generalized pseudo-parallel, then either it is minimal or $L=\frac{1}{n-1}$, when $c\neq -3s$.