The role of total curvature on complete noncompact Riemannian 2- manifolds
Let M be a connected, complete, noncompact, finitely connected and oriented Riemannian 2-manifold whose total curvature c(M) exists in \({\mathbb{R}}\). By a famous theorem of Cohn-Vossen, the defect \(\delta(M):= \chi(M)-(2\pi)^{-1}c(M)\), \(\chi\) denoting the Euler characteristic, is nonnegative. If M has only one end, the author's main result states that, if \(\delta (M)>\frac{1}{2}\) then every Busemann function on M is non- exhausting, and if \(\delta (M)<\frac{1}{2}\) then every Busemann function on M is exhausting. Several examples, including a recent one of N. H. Kuiper, show that the exhaustion behaviour of Busemann functions is not predictable if \(\delta (M)=\frac{1}{2}\).
scientific article; zbMATH DE number 3877910