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

F. Mahi; M. Belkhelfa

doi:10.15330/cmp.12.1.69-78

F. Mahi Department of Exact Sciences, Ecole Normale Supérieure de Mostaganem, Mostaganem 27000, Algeria
M. Belkhelfa Laboratoire de Physique Quantique de la Matière et Modélisations Mathématiques, Faculty of Exact Sciences, Mustapha Stambouli University, Mascara 29000, Algeria https://orcid.org/0000-0002-2570-6742

https://doi.org/10.15330/cmp.12.1.69-78

Keywords: $\mathcal{S}$-space form, Legendrian submanifold, normally flat submanifold, pseudo-parallel submanifold, Ricci generalized pseudo-parallel submanifold

Published online: 2020-06-12

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 $c\neq -3s$.