Torsion and deformation of contact metric structures on 3-manifolds
Let M be a contact manifold with contact form \(\omega\) and characteristic vector field \(X_ 0\). A Riemannian metric g is called an associated metric if there exists a linear transformation field \(\phi\) such that \(\phi^ 2=-1+\omega \otimes X_ 0\), \(\omega (X)=g(X,X_ 0)\) and \(d\omega (X,Y)=g(X,\phi Y)\). Let \(\tau =L_{X_ o}g\) where L denotes Lie differentiation; if \(\tau =0\), the contact metric structure is said to be K-contact. The present paper stems from the work of \textit{S. S. Chern} and \textit{R. S. Hamilton} [Lect. Notes Math. 1111, 279-308 (1985; Zbl 0561.53039)] where \(\int_{M}| \tau |^ 2 vol (M,G)\) as a functional on the set of ``CR-structures'' or equivalently on the set of associated metrics is studied for 3-dimensional compact contact manifolds. They give the critical point condition as \(\nabla_{X_ 0}\tau =0\); unfortunately this is not correct as pointed out by \textit{S. Tanno} [Variational problems on contact Riemannian manifolds (to appear)]. The correct critical point condition is \(\nabla_{X_ 0}\tau =2\tau \circ \phi\) as established in the reviewer's paper [J. Aust. Math. Soc., Ser. A 37, 82- 88 (1984; Zbl 0552.53014)] as well as in Tanno's paper. The authors study a compact 3-dimensional contact metric manifold M for which \(\nabla_{X_ 0}\tau =0\) and give a condition under which M admits an associated metric with positive Ricci curvature. It follows from a well known result of \textit{R. S. Hamilton} [J. Differ. Geom. 17, 255-306 (1982; Zbl 0504.53034)] that the new metric can be deformed to a metric of positive constant curvature; thus if M is simple connected, it is homeomorphic to a 3-sphere. The authors then give a nice corollary: Let M be a compact 3-dimensional contact manifold M with a K-contact structure, if the scalar curvature is \(> -2,\) then M admits an associated metric of positive Ricci curvature.
scientific article; zbMATH DE number 4026275
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