Maximal ergodic theorems and applications to Riemannian geometry

We prove new ergodic theorems in the context, of infinite ergodic theory, and give some applications to Riemannian and Kahler manifolds without conjugate points. One of the consequences of these ideas is that a complete manifold without conjugate points has nonpositive integral of the infimum of Ricci curvatures, whenever this integral makes sense. We also show that, a complete Kahler manifold with nonnegative holomorphic curvature is flat if it has no conjugate points.