Special directions on contact metric three-manifolds (Q1841871)

\textit{D. E. Blair} [Ann. Fac. Sci. Toulouse (6) Math. 7, 365-378 (1998; Zbl 0918.53012)] has introduced special directions on a contact metric manifold with negative \(\xi\)-sectional curvature as the eigenvectors of the covariant derivative of the characteristic vector field \(\xi .\) In this paper the author introduces the notion of Anosov-like special directions as those special directions for which the plane section containing the characteristic vector field \(\xi\) and a special direction is integrable. If both special directions on a \(3\)-dimensional contact metric manifold are Anosov-like, then \(\xi\) is called Anosov-like. The author proves the following results: (1) In a \(3\)-dimensional contact metric manifold with negative \(\xi \)-Ricci curvature, \(\xi \) is Anosov-like if and only if the torsion \(\tau=L_{\xi} g\) (where \(L_{\xi} g\) is the Lie derivative of the contact metric \( g\)) is \(\xi\)-parallel, i.e. \(\nabla_{\xi }\tau =0\). (2) In a \(3\)-dimensional contact metric manifold, if \(\xi \) is Anosov-like and \(\delta\tau \) is \(\xi \)-parallel, where \(\delta \) is the Berger-Ebin operator, then \(\xi \) is Anosov and the universal covering of the manifold is the Lie group \({\widetilde {SL}}(2,\mathbb{R})\). (3) The standard contact metric structure of the tangent sphere bundle of a surface admits special directions (which are never Anosov-like) if and only if the Gaussian curvature of the base surface is either negative or greater than \(2\).