Normal contact structures on 3-manifolds
A closed 3-manifold \(M\) is called a contact manifold if it carries a global differential 1-form \(\eta\) such that \(\eta \wedge d\eta\) is a volume form. \(M\) admits a global nonvanishing vector field \(\xi\) such that \(\eta(\xi)=1\) and \(d\eta(\xi,X)=0\) for all vector fields \(X\) on \(M\). On a contact distribution \({\mathcal D}=\ker \eta\) one can find an endomorphism \(J\colon{\mathcal D}\to{\mathcal D}\) compatible with \(d\eta\) in the sense that \(d\eta(JX,JY)=d\eta(X,Y)\). \(J\) is defined uniquely up to homotopy. The triple \((J,\xi,\eta)\), in other words, the reduction of the structure group of \(M\) to \(\text{U}(1)\times 1\), is called an almost contact structure on \(M\). If an almost complex structure \(\overline J\) defined on \(M\times \mathbb{R}\) by \(\overline J(X,t)=(JX-t\xi,\eta(X))\) is integrable, then the almost contact structure \((J,\xi,\eta)\) and the contact form \(\eta\) are called normal. In this paper, the author determines manifolds diffeomorphic to closed 3-manifolds that admit normal contact or almost contact forms. In Theorem 1, the author proves that a closed 3-manifold admits a normal contact form if and only if it is diffeomorphic to a manifold of the form \(\Gamma \backslash X\), where \(X\) is either \(S^3\), \(\widetilde{\text{SL}}_2\) (the universal covering of \(\text{SL}_2(\mathbb{R})\)), or \(\text{Nil}^3\) (the Heisenberg group), and \(\Gamma\) is a (torsion free) subgroup of the identity component \(\text{Isom}_0 X\) of \(X\). The manifolds in this theorem are the Seifert fibred 3-manifolds with nonzero Euler number over orientable base orbifolds. \textit{H. Sato} [Tôhoku Math. J., II. Ser. 29, 577-584 (1977; Zbl 0382.53031)] proved that if \(M\) admits a normal almost contact structure, then \(\pi_2(M)=0\) or \(M\) is diffeomorphic to \(S^2 \times S^1\). In Theorem 2, the author completes the investigation begun by Sato and proves that \(M\) admits a normal almost contact structure if and only if \(M\) is diffeomorphic to one of the manifolds listed in Theorem 1 or to either \(\Gamma \backslash (H^2 \times E^1)\) with \(\Gamma\subset \text{Isom}_0 (H^2 \times E^1)\), or a \(T^2\)-bundle over \(S^1\) with periodic monodromy (a Euclidean 3-manifold that admits a Seifert fibration over an orientable 2-orbifold), or \(S^2 \times S^1\). The proofs of both theorems are based on the results by \textit{H.-T. Geiges} and \textit{J. Gonzalo} [Invent. Math. 121, No. 1, 147-209 (1995)], where the compact complex surfaces diffeomorphic to the product \(M\times S^1\) of a 3-manifold \(M\) and \(S^1\) are determined. If \(M\) admits a normal almost contact structure, then the integrable almost complex structure \(\overline J\) on \(M\times \mathbb{R}\) defined as above induces such a complex surface, which is contained in the class of complex surfaces with geometric structures discussed by \textit{C. T. C. Wall} [Topology 25, No. 2, 119-153 (1986; Zbl 0602.57014)]. To prove the theorems the author examines the possibilities of \(M\times S^1\) under the existence of the contact form on \(M\).
scientific article; zbMATH DE number 1140478