Bounded geodesics in manifolds of negative curvature. (English) Zbl 0990.53038

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 Burns and Pollicott (unpublished) gave a positive answer if \(M\) has constant negative curvature.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 \(M\).


53C22 Geodesics in global differential geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53D25 Geodesic flows in symplectic geometry and contact geometry
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