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).

On Subcritically Stein Fillable 5-manifolds

We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case where the fundamental group is finite cyclic, and we show that on the 5-sphere, the standard contact structure is the unique subcritically ?llable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

ON FINSLER MANIFOLDS WHOSE TANGENT BUNDLE HAS THE g-NATURAL METRIC

In this paper, we introduce a Riemannian metric [Formula: see text] and a family of framed f-structures on the slit tangent bundle [Formula: see text] of a Finsler manifold Fn = (M, F). Then we prove that there exists an almost contact structure on the tangent bundle, when this structure is restricted to the Finslerian indicatrix. We show that this structure is Sasakian if and only if Fn is of positive constant curvature 1. Finally, we prove that (i) Fn is a locally flat Riemannian manifold if and only if [Formula: see text], (ii) the Jacobi operator [Formula: see text] is zero or commuting if and only if (M, F) have the zero flag curvature.

Normal Variations of Invariant Hypersurfaces of Framed Manifolds

A hypersurface of a globally framed f-manifold (briefly, a framed manifold), does not in general possess a framed structure as one may see by considering the 4-sphere S4 in R5 or S5. For, a hypersurface so endowed carries an almost complex structure, or else, it admits a nonsingular differentiable vector field. Since an almost complex manifold may be considered as being globally framed, with no complementary frames, this situation is in marked contrast with the well known fact that a hypersurface (real codimension 1) of an almost complex manifold admits a framed structure, more specifically, an almost contact structure.