This book evolved from courses about elementary differential geometry which I have taught in Freiburg, Hamburg and Potsdam. The word “elementary” should not be understood as “particularly easy”, but indicates that the development of formalism, which would be necessary for a deeper study of differential geometry, is avoided as much as possible. We will instead approach geometrically interesting problems using tools from the standard fundamental courses in analysis and linear algebra. It is possible to raise interesting questions even about objects as “simple” as plane curves. The proof of the four-vertex theorem, for example, is anything but trivial.

The book is suitable for students from the second year of study onwards and can be used in lectures, seminars, or for private study.

The first chapter is interesting mostly for historical reasons. The reader can here find out how geometric results have been obtained from axioms for thousands of years, since Euclid. In particular, the controversy about the parallel axiom will be explained. In this chapter we will mostly follow Hilbert's presentation of plane geometry, since it is rather close to Euclid's formulation of the axioms and yet meets today's requirements for mathematical rigour. In the mean time the axiomatic system has been simplified significantly [2]. A presentation with only seven axioms can be found in [30].