Contact metric structures on 3-dimensional manifolds

概要

A differentiable manifold M 2n+1 is said to have a contact structure or to be a contact manifold if there exists a 1-form η over M 2n+1 such that η ∧ (dη )n ̸= 0. The condition η ∧ (dη )n ̸= 0 means that a contact manifold is orientable. It is known that a smooth hypersurface satisfying some conditions has a contact structure. As a special case S 2n+1 is a contact manifold. When a contact form η is given on M 2n+1 , there exists a system (ξ , φ, g ) of a vector field ξ , a tensor field φ of type (1,1) and a Riemannian metric g , which called a contact metric structure.

On the other hand the notion of almost contact metric structures is a generalization of the notion of contact metric structures. An almost contact metric structure does not assume the condition η ∧ (dη )n ̸= 0. From the point of view of the Riemannian geometry of contact metric manifolds we consider K-contact structures.

This paper consists of three chapters. In Chapter 1 we mention the notion of an almost contact metric structure (φ, ξ , η , g ) on M 2n+1 and give its examples. Next we show that on an almost contact metric manifold M 2n+1 we can construct a useful orthonormal basis called φ-basis. And we explain that on the almost contact metric manifold R2n+1 the sectional curvature of a vector X orthogonal to ξ and φX is equal to −3. Finally we show that on the Heisenberg group HR identified with R3left translation preserves η and g is a left invariant metric.

Chapter 2 we mention the notion of a contact metric structure (φ, ξ , η , g ) and give its examples. Remark that for a contact form η , ξ is unique but g and φ are not necessarily unique. Next we show that in Hopf ’s mapping π : S 3 −→ S 2 the value of dπ (ξ ) is equal to 0. Moreower we mention the notion of K-contact structure. We consider the sectional curvature of K-contact manifold M 2n+1 . Finally we check that the almost contact metric structure on M 2n × R is not a contact metric structure.

It is known that every compact orientable 3-dimensional manifold has a contact structure. In Chapter 3 we consider 3-dimensional contact manifolds, especially S 3 , R 3 and T 3 . We give a typical contact form η on S 3 , R3 and T 3 respectively. Then we completely determine their contact metric structures. Next, we check that such contact metric structures are η -Einstein or not. If M 3 = S 3 , (φ, ξ , η , g ) is η -Einstein if and only if g is the standard metric. If M 3 = R3 , all (φ, ξ , η , g ) are η -Einstein. If M 3 = T 3 , one parameter family of (φ, ξ , η , g ) are η -Einstein. We check that such contact metric structures are Sasakian or not. If M 3 = S 3 , (φ, ξ , η , g ) is Sasakian if and only if g is the standard metric. If M 3 = R3 , all (φ, ξ , η , g ) are Sasakian. If M 3 = T 3 , all (φ, ξ , η , g ) are not Sasakian. We check that such contact metric structures are K-contact or not. If M 3 = S 3 , (φ, ξ , η , g ) is K-contact if and only if g is the standard metric. If M 3 = R3 , all (φ, ξ , η , g ) are K-contact. If M 3 = T 3 , all (φ, ξ , η , g ) are not K-contact.