Differential forms and applications

by Manfredo Perdigão do Carmo

The book treats differential forms and uses them to study some local and global aspects of the differential geometry of surfaces. Differential forms are introduced in a simple way that will make them attractive to "users" of mathematics. A brief and elementary introduction to differentiable manifolds is given so that the main theorem of differential forms, namely Stokes' theorem, can be presented in its natural setting. The applications consist in developing the method of moving frames of E. Cartan to study the local differential geometry of immersed surfaces in R[superscript 3] as well as the intrinsic geometry of surfaces. Everything is then put together in the last chapter to present Chern's proof of the Gauss-Bonnet theorem for compact surfaces.