Bounded geodesics in manifolds of negative curvature
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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 \(\geq 3\).
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