Казарян Максим Эдуардович: Differential Geometry/Дифференциальная геометрия (на английском языке)

Учебные материалы к курсам Math In Moscow (MiM).

Classical differential geometry is often considered as the "art of manipulating indices." In these lectures we develop a more geometric approach by explaining the true mathematical meaning of all introduced notions. Where possible, we try to avoid coordinates totally. But the correspondence to the traditional coordinate presentation is also explained. The usage of invariant language not only simplifies many arguments but also reduces the amount of computations in concrete problems. The best example is a simple formula for the Gaussian curvature of a surface based on the concept of connection 1-form.