Géométrie des sous-variétés dans les espaces de formes complexes et S-variétés.

Abstract

Geometry of submanifolds in Complex space forms and S-manifolds This work investigate the pseudo-parallel submanifolds of (2n+s)-dimensional Smanifolds of constant φ-sectional curvature c (in short, S-space forms Ḿ). These spaces generalize the Sasakian space forms (i.e. Sasakian manifolds of constant φ-sectional curvature). We are first interested in the second order parallel symmetric tensor on an S-manifold, we establish a formula of this tensor for s≥1. Thus, we prove the non-existence of parallel hypersurfaces of an S-space form, with c≠s. We apply these results to show the nonexistence of semi-parallel hypersurfaces of this space. In addition, we obtain the same negative results concerning the pseudo-parallel hypersurfaces in a Sasakian space form, and pseudo-parallel hypersurfaces that meet certain conditions in the S-space form Ḿ(-3s) of dimension 2n+2+s. On the other hand, the Legendrian pseudo-parallel submanifolds are examined, the necessary conditions of such submanifolds are given to be semi-parallel, totally geodesic or minimal. We also studied another type of pseudo-parallel for these submanifolds, namely Ricci generalized pseudo-parallel. Finally, we consider an invariant, pseudo-parallel and Ricci generalized pseudo-parallel submanifolds of an S-manifold. It is shown that these submanifolds are totally geodesic under certain conditions.