tailieunhanh - Horizontally submersions of contact CR-submanifolds

In this paper, we discuss some geometric properties of almost contact metric submersions involving symplectic manifolds. We show that the structures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are related to (1, 2) - symplectic structures. For horizontally submersions of contact CR submanifolds of quasi-K-cosymplectic and quasiKenmotsu manifolds, we study the principal characteristics and prove that their total spaces are CR-product. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 436 – 453 ¨ ITAK ˙ c TUB ⃝ doi: Horizontally submersions of contact CR-submanifolds 1 Fortun´ e MASSAMBA1,∗, Tshikunguila TSHIKUNA-MATAMBA2 School of Mathematics, Statistics, and Computer Science, University of KwaZulu-Natal, Scottsville, South Africa 2 Department of Mathematical Sciences, College of Science, Engineering, and Technology, University of South Africa, Pretoria, South Africa Received: • Accepted: • Published Online: • Printed: Abstract: In this paper, we discuss some geometric properties of almost contact metric submersions involving symplectic manifolds. We show that the structures of quasi- K -cosymplectic and quasi-Kenmotsu manifolds are related to (1, 2) symplectic structures. For horizontally submersions of contact CR -submanifolds of quasi- K -cosymplectic and quasiKenmotsu manifolds, we study the principal characteristics and prove that their total spaces are CR -product. Curvature properties between curvatures of quasi- K -cosymplectic and quasi-Kenmotsu manifolds and the base spaces of such submersions are also established. We finally prove that, under a certain condition, the contact CR -submanifold of a quasi Kenmotsu manifold is locally a product of a totally geodesic leaf of an integrable horizontal distribution and a curve tangent to the normal distribution. Key words: CR -submanifold, almost Hermitian manifold, almost contact metric submersion, symplectic manifold, horizontal submersion 1. Introduction Riemannian submersions between Riemannian manifolds were initiated by O’Neill [14]. Almost contact metric submersions were developed by Chinea [8] and Watson [17]. The theory of almost contact metric submersions intertwines contact geometry with the almost Hermitian one. For instance, the base space of an almost contact metric submersion of type II , in the