On geodesics which intersect a convex domain of nonpositive curvature. (Sur les géodésiques qui coupent un convexe en courbure négative ou nulle.) (French) Zbl 0828.53063

The author shows that for a simply connected complete Riemannian manifold \(X\) of nonpositive curvature the space of oriented geodesics is an exact symplectic variety and the measure of geodesics of \(X\) intersecting a convex region \(K\) is given by the surface area of \(\partial K\) times the volume of the \((n -1)\)-dimensional unit ball.


53C65 Integral geometry
53C22 Geodesics in global differential geometry
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[1] Besse, A.L.) .- Manifolds all of whose Geodesics are Closed, Springer-Verlag, 1978. · Zbl 0387.53010
[2] Santalo, L.A.) .- Integral geometry and geometric probability. In Gian-Carlo Rota, editor, Encyclopedia of Mathematics and its Applications, volume I. Addison-Wesley, 1976. · Zbl 0342.53049
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