On the existence of conformally flat contact metric manifolds

Abstract

By a $contact$ $manifold$ we mean a (2n + 1)-dimensional $C^\infty$ manifold M together with a global 1-form $\eta$ such that $\eta \land (d\eta)^n \ne O$. Given a contact form $\eta$, it is well known that there exists a unique vector field \xi, called the $characteristic vector field$, such that $d\eta(\xi, X) = O$ and normalized by \eta(\xi) = 1. We denote by D the $contact subbundle$ defined by the subspaces ${X \in T_mM: \eta(X) = O}$. A Riemannian metric $g$ is an $associated metric$ for a contact form \eta, if there exists a tensor fieid \phi of type (1,1) such that $\phi^2 = - I + \eta \otimes \xi, \eta(X) = g(\xi, X), d\eta(X, Y) = g(X, \phi Y)$ We refer to ($\eta$, g) or ($\phi. \xi, \eta$, g) as a $contact metric structure$. When $\phi$ is Killing, the contact metric structure is said to be $K-contact$. A contact metric structure on $M$ naturally gives rise to an almost complex structure on the product M x R and if this aimost complex strueture is integrable, the given contact metric structure is $Sasakian$. A Sasakian manifold is always K-contact and in dimension 3 a K-contact manifold is Sasakian. For a generai reference to these ideas, see e.g. (2). In (6) Okumura showed that a conformally flat Sasakian manifold of dimension $\ge$ 5 is of constant curvature +1 and in (11, 12) Tanno extended this result to the K-contact case and for dimensions $\ge$ 3. Now in dimension $\ge$ 5 a contact metric structure of constant curvature must be of constant curvature +1 and the structure Sasakian [Olszak (7)]. In dimension 3 a contact metric structure of constant curvature must be of constant curvature 0 or 1; in fact these cases are the only 3-dimensional locally symmetric contact metric manifolds (4). Turning to the question of conformally flat contact metric manifolds, K. Bang (1) showed that in dimension $\ge$ 5 there are no conformally flat contact metric structures with $R_x\xi \xi$= 0$, even though this is a large class of contact metric manifolds. In the case of the standard contact metric structure on the tangent sphere bundle, the metric is conformally flat if and only if the base manifold is a surface of constant Gaussian curvature O or 1 (3) and in which case the tangent sphere bundle has constant curvature O or 1, respectively. In view of these strong curvature results we may ask if there are any conformally flat contact metric structures which are not of constant curvature. We show here that in fact they do exist. This study takes a direct approach to the question in dimension 3. The analysis involved is interesting in its own right and gives another solution to the standard «force-free» model equations of solar physics. The author expresses his appreciation to Professors Zhengfang Zhou, Marcel Goossens and Themis Koufogiorgos for helpful conversations dunng this work