TAILIEUCHUNG - Extended cross product in a 3-dimensional almost contact metric manifold with applications to curve theory
In this work, we define a new cross product in 3-dimensional almost contact metric manifold and we study the theory of curves using this new cross product in this manifold. Besides, in the works of Baikousis, Blair and Cho et al. | Turk J Math 36 (2012) , 305 – 318. ¨ ITAK ˙ c TUB doi: Extended cross product in a 3-dimensional almost contact metric manifold with applications to curve theory C ¸ etin Camcı Abstract In this work, we define a new cross product in 3-dimensional almost contact metric manifold and we study the theory of curves using this new cross product in this manifold. Besides, in the works of Baikousis, Blair [1] and Cho et al. [4], we observe that some theorems are incomplete and excessively generalized are thus their alternative proofs presented. Key Words: Sasakian Manifold, Legendre curve, cross product 1. Introduction n Let M be a (2n+1)-dimensional differentiable manifold. If there exist a 1-form η , such that ηΛ (dη) = 0 on M , then (M, η) is called a contact manifold and η a contact 1-form [2]. A unique vector field ξ is called Reeb vector field (or characteristic vector field) where η(ξ) = 1 and dη(ξ, .) = 0 [2]. In a contact manifold, the contact distribution is defined by D = {X ∈ χ(M ) : η(X) = 0} . A (2n + 1)-dimensional differentiable manifold M is called an almost contact manifold if there is an almost contact structure (φ, ξ, η) consisting of a tensor field φ type (1, 1), a vector field ξ , and a 1-form η satisfying φ2 = −I + η ⊗ ξ, and (one of) η(ξ) = 1, φξ = 0, η ◦ φ = 0. () If the induced almost complex structure J on the product manifold M 2n+1 × R defined by J d X, f dt d = ϕX − fξ, η(X) dt is integrable then the structure (ϕ, ξ, η) is said to be normal, where X is tangent to M , t is the coordinate of R and f is a smooth function on M 2n+1 × R [2]. M becomes an almost contact metric manifold with an almost contact metric structure (φ, ξ, η, g), if g(φX, φY ) = g(X, Y ) − η(X)η(Y ), 2000 AMS Mathematics Subject Classification: Primary 53C15; Secondary 53C25. 305 CAMCI or equivalently, g(X, φY ) = −g(φX, Y ) and g (X, ξ) = η(X) for all X, Y ∈ T M , where g is a Riemannian metric tensor of M [2]. For a 3-dimensional .
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