Chinburg, Ted; Reid, Alan W. Closed hyperbolic 3-manifolds whose closed geodesics all are simple. (English) Zbl 0783.53028 J. Differ. Geom. 38, No. 3, 545-558 (1993). Let \(M\) be a complete orientable hyperbolic \(n\)-manifold of finite volume. A closed geodesic in \(M\) is called simple if it has no self- intersections. Otherwise a closed geodesic is called nonsimple. In the case of \(n=2\) any hyperbolic 2-manifold has nonsimple closed geodesics, indeed most closed geodesics in the case \(n = 2\) are nonsimple. The question has been asked in the case of \(n = 3\) as to whether there exist finite volume hyperbolic 3-manifolds all of whose closed geodesics are simple. The main result of this paper is to give a construction of infinitely examples of such manifolds. The examples are all closed. The construction is arithmetic in nature and uses the theory of quaternion algebras to interpret the existence of a nonsimple closed geodesic in terms of the Hilbert symbol of a particular quaternion algebra associated to a finite volume hyperbolic 3-manifold. Reviewer: A.W.Reid (Austin) Cited in 3 ReviewsCited in 8 Documents MSC: 53C22 Geodesics in global differential geometry 53C20 Global Riemannian geometry, including pinching Keywords:finite volume; quaternion algebras; Hilbert symbol PDFBibTeX XMLCite \textit{T. Chinburg} and \textit{A. W. Reid}, J. Differ. Geom. 38, No. 3, 545--558 (1993; Zbl 0783.53028) Full Text: DOI