CR-Submanifolds of Generalized -Space Forms

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. Dedicated to my sister and my parents for their endless support, kind and sacrifices. 1. Introduction In 1978, Bejancu introduced and studied CR-submanifolds of a K？hler manifold [1, 2]. Since then, 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]. Recently Falcitelli and Pastore [7] introduced generalized globally framed -space forms. Globally framed -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. In this paper, we study CR-submanifolds of generalized -space forms. 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]. In the class of -structures introduced in 1963 by Yano [9], the so-called -structures with complemented frames, also called globally framed -structures or -structures with parallelizable kernel (briefly .-structures) [10–13] are particulary interesting. An .-manifold is a manifold on which an -structure is defined, that is a -tensor field satisfying , of rank , such that the subbundle is parallelizable. Then, there exists a global frame , , for the subbundle , with dual 1-form , satisfying , from which , follow. An .-structure on a manifold is said to be normal if the tensor field vanishes, denoting the Nijenhuis torsion of . It is known that one can consider a Riemannian metric on associated with an .-structure , such that , for any , and the structure is then called a metric .-structure. Therefore, splits as complementary orthogonal sum of its subbundles and . We denote their respective differentiable distributions by and . Let denote the 2-form on defined by , for any . Several subclasses have been studied from different points of view [10, 11, 14–16], also dropping the