The study of the curvature and topology of Riemannian manifolds is mainstream in differential geometry. Many of the important contributions in this topic go back to the pioneering works by Cohn-Vossen in 1935–6, [19] and [20]. In fact the study of total curvature on complete noncompact Riemannian manifolds made by him contains many fruitful ideas. Many hints in his thoughts lead us to the study of the curvature and topology of Riemannian manifolds.

The well-known Gauss–Bonnet theorem states that the total curvature of a compact Riemannian 2-manifold is a topological invariant. Cohn-Vossen first proved that the total curvature of a finitely connected complete noncompact Riemannian 2-manifold M is bounded above by 2πχ(M), where χ(M) is the Euler characteristic of M. Among many beautiful consequences of this result, he proved the splitting theorem for complete open Riemannian 2-manifolds of nonnegative Gaussian curvature admitting a straight line. The structure theorem for such 2-manifolds was also established by him. He investigated the global behavior of complete geodesics on these 2-manifolds and this gave rise to the study of poles. The Bonnesen-type isoperimetric problem for complete open surfaces admitting a total curvature was first investigated by Fiala [26] for the analytic case and then by Hartman [34] for the C2 case. Here the Cohn-Vossen theorem plays an essential role. The total curvature of infinitely connected complete open surfaces was discussed by Huber from the point of view of complex analysis.