On Curves Counting In Negative curvature

Abstract

Weconsder a complete rimannian manifold M with negative curvature ,and two properly immersed convex sudsets and of M ,then we discuss the asymptotic behavior of the number of common perpendiculars of length at most S from to , giving error terms and counting with weights .also we describe the relationship with counting problem in circle packings. Also we illustrate several airthemetic applications on the asymptotics of the number of representations of integer by binary quadratic or Hamiltonian forms.