Non-elementary examples of compact Anosov manifolds Anosov manifolds (Riemannian manifolds whose geodesic flow is Anosov) are a natural generalisation of negatively curved compact manifolds. I'm wondering how good the generalisation is. In particular, is there any Anosov manifold that has some positivity in its curvature?
 A: Yes, there are compact surfaces in $R^3$ (hence, whose curvature is positive somewhere) with Anosov geodesic flow:
Donnay, Victor J.; Pugh, Charles C., Anosov geodesic flows for embedded surfaces, de Melo, Welington (ed.) et al., Geometric methods in dynamics (II). Volume in honor of Jacob Palis. In part papers presented at the international conference on dynamical systems held at IMPA, Rio de Janeiro, Brazil, July 2000, to celebrate Jacob Palis’ 60th birthday. Paris: Société Mathématique de France (ISBN 2-85629-139-2/pbk). Astérisque 287, 61-69 (2003). ZBL1054.37009.
A: You can also have a look at the two papers by P. Eberlein: When is a geodesic flow of Anosov type? I, II, J. Differential Geom. 8(3): 437-463 (1973) (end of the first paper), where some examples of nonpositively-curved manifolds with sectional curvature vanishing on an open neighborhood are constructed. Since the Anosov property is open, you can always perturb these examples to allow some (small) positive sectional curvature somewhere.
