## 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.

### MSC:

 53C65 Integral geometry 53C22 Geodesics in global differential geometry
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.