Torsion and conformally Anosov flows in contact Riemannian geometry (Q811829)

Let \(M\) be a \(3\)-dimensional compact manifold equipped with a non-Sasakian contact metric structure \((\phi,\xi,\eta,g)\). Assume that \(g\) is critical for the Chern-Hamilton functional \(E(g) = \frac{1}{2}\int_M \| \tau\| ^2\,dv\), where \(\tau = L_\xi g\) is the torsion. The author proves that \(\xi\) is conformally Anosov and there exists a smooth curve in the contact distribution of conformally Anosov flows. It was shown by \textit{D. E. Blair} and the author in [Balkan J. Geom. Appl. 3, No. 2, 33--46 (1998; Zbl 0955.53044)] that if a compact contact metric \(3\)-manifold has all the \(\xi\)-sectional curvatures negative, then \(\xi\) is conformally Anosov. A consequence of the above result is that the converse does not hold. In the final part of the paper the author studies contact metric \(3\)-manifolds with constant \(\xi\)-sectional curvature.