Certain Geometric Aspects of a Class of Almost Contact Structures on a Smooth Metric Manifold

The classication of Smooth Geometrical Manifolds still remains an open problem. The concept of almost contact Riemannian manifolds provides neat descriptions and distinctions between classes of odd and even dimensional manifolds and their geometries. We construct an almost contact structure which is related to almost contact 3-structure carried on a smooth Riemannian manifold (M, gM) of dimension (5n + 4) such that gcd(2, n) = 1. Starting with the almost contact metric manifolds (N4n+3, gN) endowed with structure tensors (ϕi, ξj , ηk) such that 1 ≤ i, j, k ≤ 3 of types (1, 1), (1, 0), (0, 1) respectively, we establish that there exists a structure (ϕ4, ξ4, η4) on (N4n+3 ⊗ Rd) ≈ M; gcd(4, d) = 1, d|2n + 1, constructed as linear combinations of the three structures on (N4n+3, gN) . We study some algebraic properties of the tensors of the constructed almost contact structure and further explore the Geometry of the two manifolds (N4n+3⊗Rd) ≈ M and N4n+3 via a !submersion F : (N4n+3 ⊗Rd) ↩→ (N4n+3) and the metrics gM respective gN between them. This provides new forms of Gauss-Weigarten's equations, Gauss-Codazzi equations and the Ricci equations incorporating the submersion other than the First and second Fundamental coecients only. Fundamentally, this research has revealed that the structure (ϕ4, ξ4, η4) is constructible and it is carried on the hidden compartment of the manifold M∼=(N4n+3 ⊗ Rd) (d|2n + 1) which is related to the manifold (N4n+3).