Translated from, the First Russian Edition by Leo F. BoronThe lectures of the author on differential geometry in the Physics-Mathematics Department of the Kharkov State University form the basic material of the present book. The author’s aim is to present a rigorous discussion of the fundamentals of differential geometry and of the methods of investigation which are character­ istic of this branch of mathematics, without disturbing well- established tradition in the process. A large amount of factual material concerning differential geometry has been relegated to exercises and problems, the solution of which ought to be considered obligatory for serious students of geometry.