Main Article Content
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).
Chinea D, Gonzalez C. A classication of almost contact metric manifolds. Ann. Mat. Pura Appl. 1990;(4)156:15-36.
Gray A, Hervella LM. The sixteen classes of almost Hermitian manifolds and their linear imvarients. Ann. Mat. Pura Appl. 1980;123:35-58.
Kuo YY. On Almost contact 3-structure. Tohoku Math. Journ. 1970;22:325-332.
Udriste C. Structures presque coquaternioniennes. Bull. Math. Roumanie RS. 1969;13(61):487- 507.
Boyer C, Galicki K. 3-Sasakian manifolds. Surveys in differential geometry: Essays on Einstein manifolds. Surv. Dier. Geom., VI, Int. Press, Boston, MA. 1999;123-184.
Cappelletti Montano B, De Nicola A. 3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem. J. Geom. Phys. 2007;57:2509-2520.
Kuo YY, Tachibana S. On the distribution appeared in contact 3-structure. Taita J. of Math. 1970;2:17-24.
Tachibama S, Yu WN. On a Riemannian space admitting more than one Sasakian structure.
Tohoku Math. J. 1970;22:536-540.
Yano K, Eum SS, Ki UH. On almost contact ane 3-structure. Kodai Math. Sem. Rep. 1973;25.
Sasaki S. Spherical space forms with normal contact metric 3-structure. J. Diff. Geom. 1972;6:307-315.
Ishihara S, Konishi M. Fibred Riemanian Spaces with Sasakian 3-structure. Differential Geometry in Honour of Yano K, Tokyo. 1972;179-194.
Ishihara S, Konishi M. On f-three-structure. Hokkaido Math. J. 1972;1:127-135.
Ishihara S, Konishi M. Fibred Riemannian space with triple of Killing vectors. Kddai Math. Sem. Rep. 1973;25.
Tanno S. Killing vectors on contact Riemannian manifolds and bering related to the Hopf brations. Tohoku Math. J. 1971;23:313-334.
Yano K, Ishihara S, Konishi M. Normality of almost contact 3-structure Tohoku Math. Journ. 1973;25:167-175.
Blaga AM. An isoparametric function on almost k-contact manifolds. An. St. Univ. Ovidius Constanta. 2009;17(1):1522.
Mihai A, Rosca R. On vertical skew symmetric almost contact 3-structures. J. Geom. 2005;82:146155.