η-PARALLEL CONTACT 3-MANIFOLDS

Abstract

In this paper, we give a classification of contact 3-manifolds whose Ricci tensors are ·-parallel. It is well-known that for a 3-dimensional Riemannian manifold its curvature tensor R is expressed only in terms of the Ricci tensor S, the metric tensor g and the scalar curvature r. Hence, in studying the 3-dimensional Riemannian geometry, we see at once that the condition of local symmetry (rR = 0) is equivalent to the Ricci-parallel condition (rS = 0). Recently Boeckx and the present author (7) proved that a locally symmetric contact Riemannian manifold is either Sasakian and of constant curvature 1 or locally isometric to the unit tangent sphere bundle (with its standard con- tact metric structure) of a Euclidean space. (In the 3-dimensional case, we may also refer to (5).) This result says that the local symmetry is rather a strong condition in contact Riemannian geometry and hence, it is natural to consider a weaker condition, that is ·-parallel. Let M = (M;·,g,',») be a contact Riemannian manifold. Then the contact form · determines the con- tact distribution D which is given by the kernel of ·. We say that the Ricci tensor S is ·-parallel if S satisfies g((rXS)Y,Z) = 0 for any X,Y,Z 2 D. In this paper, we shall study 3-dimensional contact Riemannian manifolds whose Ricci-tensors are ·-parallel. On the other hand, given a contact manifold M = (M;·) one may raise a following natural question (cf. (16)): Which metric is most proper among Rie- mannian metrics associated with ·? One method of finding the nice Riemann- ian metrics is to study the criticality in the variational sense. In particular, in (2) and (16) the authors showed that M satisfies r»h = 2h' if and only if it has the critical metric of the Dirichlet energy functional E(g) = R M kL»gk 2 dM defined on the set of all Riemannian metrics associated with the given contact form ·, where h = 1 L»' and L» is the Lie derivative with respect to ». (Chern- Hamilton (10) first studied critical metrics of the Dirichlet energy functional in