Role of total curvature on rays of non-compact Riemannian 2-manifold

Abstract

It is interesting to study the geometry of total curvature on complete open surfaces. Cohn-Vossen’s inequality states that in every connected noncompact finitely connected complete Riemannian 2-manifold 𝑀 with finite total curvature 𝑐(𝑀) and finite Euler characteristic 𝜒(𝑀), we have 𝑐(𝑀)≤2𝜋𝜒(𝑀). Huber extended this result, if a connected, infinitely connected complete Riemannian 2-manifold 𝑀 without boundary admits a total curvature 𝑐(𝑀), then 𝑐(𝑀)= −∞. The value 2𝜋𝜒(𝑀)−𝑐(𝑀) plays an important role in the study of rays on complete, noncompact Riemannian 2-manifolds. A ray 𝛾:[0,∞]⟶𝑀, on a complete, non-compact Riemannian manifold 𝑀 is by definition a unit speed geodesic every subarc of which is minimizing. Due to the completeness and non-compactness of the Riemannian 2-manifold 𝑀, there exists at least one ray emanating from every point of a manifold. If 𝐴(𝑝) is the collection of all rays emanating from 𝑝∈𝑀 and 𝜇 is the natural measure induced by the Riemannian metric then lim𝑛→∞𝜇𝜊𝐴(𝑝𝑛)⊂𝐴(𝑝) , where {𝑝𝑛} is a sequence of points of 𝑀 converging to 𝑝. Also we have the function 𝜇𝜊𝐴∶𝑀⟶[0,2𝜋] is upper semi-continous and hence Lebesgue integrable. If 𝑀 is connected, finitely connected, complete and non-compact Riemannian 2-manifold, we then investigated the relationship between 𝑐(𝑀) and the function 𝜇𝜊𝐴, proving that if 𝑀 is homeomorphic to 𝑅2 and if Gaussian curvature 𝐺≥0, then 𝜇𝜊𝐴 ≥2𝜋−𝑐(𝑀), and in particular 𝑖𝑛𝑓𝑀𝜇𝜊𝐴=2𝜋−𝑐(𝑀).