One service mathematics has rendered the 'Et moi, ... , si j'avait su comment en revenir, je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Matht"natics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics seNe as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Inhalt
I. Basic Concepts of Differential Geometry.- II. Tensors. Riemannian Geometry.- III. Basic Elements of Topology.- Appendices.- Appendix 1 The Simplest Groups of Transformations of Euclidean and Non-Euclidean Spaces.- Appendix 2 Some Elements of Modern Concepts of the Geometry of the Real World.- Appendix 3 Crystallographic Croups.- Appendix 4 Homology Groups and Methods of their Calculation.- Appendix 5 The Theory of Geodesics, Second Variation and Variational Calculus.- Appendix 6 Basic Geometric Properties of the Lobachevskian Plane.- Appendix 7 Selected Exerices on the Material of the Course.- Additional Material.- References.