We consider minimal anti-invariant semiparallel submanifolds of generalized Sasakian space forms. We show that the submanifolds are totally geodesic under certain conditions. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 796 – 802 ¨ ITAK ˙ c TUB ⃝ doi: On semiparallel anti-invariant submanifolds of generalized Sasakian space forms 1 ¨ ¨ 1,∗, Fatma GURLER ¨ Cihan OZG UR , Cengizhan MURATHAN2 Department of Mathematics, Balıkesir University, C ¸ a˘ gı¸s, Balıkesir, Turkey 2 Department of Mathematics, Uluda˘ g University, Bursa, Turkey 1 Received: • Accepted: • Published Online: • Printed: Abstract: We consider minimal anti-invariant semiparallel submanifolds of generalized Sasakian space forms. We show that the submanifolds are totally geodesic under certain conditions. Key words: Semiparallel submanifold, generalized Sasakian space form, Laplacian of the second fundamental form, totally geodesic submanifold 1. Introduction Let (M, g) and (N, ge) be Riemannian manifolds and f : M → N an isometric immersion. Denote by σ and ∇ its second fundamental form and van der Waerden–Bortolotti connection, respectively. If ∇σ = 0 , then the submanifold M is said to have a parallel second fundamental form [6]. The act of R to the second fundamental form σ is defined by ( ) R(X, Y ) · σ (Z, W ) = R⊥ (X, Y )h(Z, W ) − σ(R(X, Y )Z, W ) − σ(Z, R(X, Y )W ) = (∇X ∇Y σ)(Z, W ) − (∇Y ∇X σ)(Z, W ), (1) where R is the curvature tensor of the van der Waerden–Bortolotti connection ∇. Semiparallel submanifolds were introduced by Deprez in [7]. If R·σ = 0, then f is called semiparallel. It is clear that if f has parallel second fundamental form, then it is semiparallel. Hence, a semiparallel submanifold can be considered as a natural generalization of a submanifold with a parallel second fundamental form. Semiparallel submanifolds have been studied by various authors; see, for example [3, 7, 8, 9, 13, 16] and the references therein. Recently, in [18], Yıldız et al. studied C -totally real pseudoparallel submanifolds of Sasakian
