Bounded geodesics in manifolds of negative curvature

Abstract.

Let M be a complete Riemannian manifold with sectional curvature $\leq -1$ and dimension $\geq 3$. Given a unit vector $v\in T^1M$ and a point $x\in M$ we prove the existence of a complete geodesic through x whose tangent vector never comes close to v. As a consequence we show the existence of a bounded geodesic through every point in a complete negatively pinched manifold with finite volume and dimension $\geq 3$.