An integral formula for the measure of rays on complete open surfaces (Q1080199)

Let M be a complete Riemannian surface homeomorphic with the plane. For \(p\in M\) let \(\mu\) \(\circ A(p)\) be the natural measure (in the unit tangent circle at p) of the set of all initial vectors of rays emanating from p. Let \((K_ j)\) be a compact exhaustion of M, and assume that M admits positive total curvature. Then \[ \lim_{j}\int_{K_ j}\mu \circ A dM\quad /\int_{K_ j}dM=2\pi -total\quad curvature. \] There are also related results for compact surfaces with finitely many points removed.