Global Geometrical Approach to Contact Manifolds

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In this project we studied the following researches.1. Study of CR twistor space over a 5-dim contact metric manifold was developed. In analogy of 4-dim manifold it is shown that the CR twistor space admits an almost CR structure and it is verified that this almost CR structure is integrable under the curvature conditions on a given base contact metric 5-manifold, that the anti-self-dual Weyl conformal tensor vanishes and also the scalar curvature s=-4.2. 4-dimensional geometry can be applied to the contact subbundle of contact metric manifolds. By Tachibana's theorem and also by N.Tanaka's systematic theory on CR geometry harmonic k-forms over a compact Sasakian (2n+1)-manifold take values in the contact subbundle, when k<n+1. So the self-duality in Sasakian contact structure was defined like 4-dim manifold theory. Remark that Sasakian contact structure turns out to be nothing but a normal strongly pseudo convex CR structure, a main subject in CR geometry.3. Study of Legendrian surfaces minimally immersed in a Sasakian contact 5-manifold was proceeded in terms of Hopf differential, the cubic differential and also in terms of the second variation.