CR-Submanifolds of Generalized -Space Forms

Mahmood Jaafari Matehkolaee

doi:10.1155/2013/654780

Abstract

We study sectional curvature, Ricci tensor, and scalar curvature of submanifolds of generalized -space forms. Then we give an upper bound for foliate -horizontal (and vertical) CR-submanifold of a generalized -space form and an upper bound for minimal -horizontal (and vertical) CR-submanifold of a generalized -space form. Finally, we give the same results for special cases of generalized -space forms such as -space forms, generalized Sasakian space forms, Sasakian space forms, Kenmotsu space forms, cosymplectic space forms, and almost -manifolds.

Highlights

In 1978, Bejancu introduced and studied CR-submanifolds of a Kahler manifold [1, 2]
Many papers appeared on this topic with ambient manifold such as Sasakian space form [3], cosymplectic space form [4], and Kenmotsu space form [5, 6]
The theory of a submanifold of a Sasaki manifold was investigated from two different points of view: one is the case where submanifolds are tangent to the structure vector and the other is the case where those are normal to the structure vector [8]

Summary

Introduction

In 1978, Bejancu introduced and studied CR-submanifolds of a Kahler manifold [1, 2]. Since many papers appeared on this topic with ambient manifold such as Sasakian space form [3], cosymplectic space form [4], and Kenmotsu space form [5, 6]. Framed f-manifolds are studied from the point of view of the curvature and are introduced and the interrelation with generalized Sasakian and generalized complex space forms is pointed out. We study CR-submanifolds of generalized f-space forms. An f.p.k.-structure on a manifold M2n+s is said to be normal if the tensor field N = [φ, φ] + 2dηi ⊗ ξi vanishes, [φ, φ] denoting the Nijenhuis torsion of φ. The purpose of the present paper is to study Ricci tensor, sectional curvature, and scalar curvature of submanifolds of a generalized f.p.k.-space form.

Preliminaries

Sectional Curvature of Submanifolds

The Ricci Tensor and Scalar Curvature of a Submanifold