Bounded geodesics in manifolds of negative curvature
This paper is stimulated by the following question: does the unit tangent bundle to a compact manifold \(M\) of negative sectional curvature contain a proper closed subset which projects onto \(M\) under the bundle projection and is invariant under the geodesic flow? Burns and Pollicott (unpublished) gave a positive answer if \(M\) has constant negative curvature. The present paper gives a positive answer if \(\dim(M)\geq 3\). The main theorem says that if \(M\) is a complete manifold of sectional curvature \(\leq -1\) and \(\dim(M)\geq 3\), then for any unit tangent vector \(v\) to \(M\) and any point \(x\) of \(M\), there is a complete geodesic through \(x\) whose tangent vector never comes close to \(v\) (with respect to the standard Sasaki metric). The proof also can be used to show that, if \(M\) is a finite volume complete manifold of pinched negative curvature and \(\dim(M)\geq 3\), then there is a complete geodesic through each point of \(M\) which is a bounded subset of \(M\).
scientific article; zbMATH DE number 1559194