Preprint Series # 835
Sabau, Sorin V and Shimada, Hideo Riemann-Finsler surfaces. (2007);
This paper study the Gauss-Bonnet theorem for Finsler surfaces
with smooth boundary. This is a natural generalization of the
Gauss-Bonnet theorem for Riemannian surfaces with smooth boundary
as well as an extension of the Gauss-Bonnet theorem for boundaryless
Finsler surfaces. The paper starts with an introduction in the Finsler
geometry of surfaces with emphasis on the Berwald and Landsberg surfaces.