Special directions on contact metric manifolds of negative \(\xi\)-sectional curvature
The author introduces an interesting notion of special directions belonging to the contact subbundle of a contact metric manifold \((M,\omega,g)\) with negative sectional curvature for plane sections containing the characteristic vector field \(\xi\) of the contact structure and in the case of 3-dimensional manifolds with \(\xi\) an Anosov vector field, and he compares these with the stable and unstable directions. He shows that if on a 3-dimensional contact metric manifold with negative \(\xi\)-sectional curvature, \(\xi\) is Anosov and the special directions agree with the Anosov directions, then the contact metric structure satisfies the condition \(\nabla_\xi\tau= 0\), where \(\tau\) is the Lie derivation \(L_\xi g\). This condition is quite natural arising in a number of situations; for example, \textit{D. Perrone} [Kōdai Math. J. 13, 88-100 (1990; Zbl 0709.53034)] showed that for a compact contact 3-manifold the condition \(\nabla_\xi\tau= 0\) is the critical point condition for the integral of the scalar curvature considered as a functional on the set of all metrics associated to the contact form. It is a classical example that the characteristic vector field of the contact structure on the tangent sphere bundle of a negatively curved manifold is (twice) the geodesic flow and therefore generates an Anosov flow. In the case of the tangent sphere bundle of a surface, the geodesic flow is closely related to another classical example, namely the standard Anosov flow on the Lie group \(\text{SL}(2,\mathbb{R})\). These examples are closely related from both the topological and Anosov points of view; however, from the Riemannian point of view they are quite different. In fact, in the case of the tangent sphere bundle of a negatively curved surface, the author proves that the special directions never agree with the Anosov directions. Finally, for the group \(\text{SL}(2,\mathbb{R})\), the author exhibits the special directions which do agree with the Anosov directions.
scientific article; zbMATH DE number 1279195