SEMI-INVARIANT $\xi ^{\bot}$-SUBMANIFOLDS OF GENERALIZED QUASI-SASAKIAN MANIFOLDS

Abstract

A structure on an almost contact metric manifold is defined as a generalization of well-known cases: Sasakian, quasi-Sasakian, Kenmotsu and cosymplectic. Then we consider a semi-invariant ξ ⊥ -submanifold of a manifold endowed with such a structure and two topics are studied: the integrability of distributions defined by this submanifold and characterizations for the totally umbilical case. In particular we recover results of Kenmotsu [8], Eum [6] and Papaghiuc [12].

Skew semi-invariant submanifolds of generalized quasi-Sasakian manifolds

“…Since every generalized quasi-Sasakian manifold is normal (see [8], Theorem 7), the proof is obvious.…”

“…Form equation (18), it follows that∇ X ξ = ϕ 2 X = −X + η(X)ξ. Using (19), (8) and η(X) = 0 in (6), we get…”

“…Moreover, some related topics were studied by V. V. Goldberg and R. Rosca [16][17][18][19][20]. In 2012, C. Calin et al [8] have studied the semi-invariant ξ ⊥ -submanifold of a generalized quasi-Sasakian manifold. Later on, A. Bejancu defined and studied a semiinvariant submanifold of a locally product manifold [4].…”