# New Scientific Textbook Presents Non-Euclidean Geometries

The regular geometry we learn at school is called Euclidean geometry. However, in mathematical analysis, it is often more natural to use the so-called non-Euclidean geometries. Researchers at the University of Turku, Finland, have published a scientific textbook on non-Euclidean geometries.

The geometry we learn in school, the so-called Euclidean geometry, is more than 2,000 years old. About two centuries ago, the Russian mathematician Lobachewsky established a new kind of geometry for the points within the unit circle. It is called non-Euclidean geometry as it is fundamentally different from the Euclidean geometry.

However, many familiar facts from the Euclidean geometry, such as the Pythagorean theorem, have their counterparts in this geometry. In the non-Euclidean geometry, the points of the unit circle have the same role as the horizon in the Euclidean geometry.

Now, researchers **Parisa Hariri**,** Riku Klén** and** Matti Vuorinen **from the University of Turku have published a scientific textbook on non-Euclidean geometries titled Conformally Invariant Metrics and Quasiconformal Mappings. The monograph deals with the so-called geometric function theory. It studies functions defined on the subsets of the n-dimensional space using methods of mathematical analysis.

While the Euclidean geometry is often the simplest to understand, it turned out that geometries of "non-Euclidean type" would be the most natural to use.

– For example, non-Euclidean geometry is natural in modelling airflow across an airplane wing or in the theory of relativity, explains one of the authors, Assistant Professor Riku Klén.

In a general subset G of the n-dimensional space, one cannot build a geometry as good as the non-Euclidean geometry is on the plane, but one can look for geometries which share some of the main properties. In this study of the "intrinsic geometry of the subset G", one needs to find a notion of distance which takes into account not only the position of the two points of the set with respect to each other, but also their position with respect to the boundary of G. This boundary has the role of the horizon, as it is "infinitely far away". The research of intrinsic geometry is very widely spread and it is studied intensively in many countries.

The textbook is a synthesis of the authors’ and their co-authors’ research during the past three decades. The results previously scattered in the literature are now compiled into one volume and presented in a systematic way.

The theory is developed "ab ovo” from the basic elements and it is intended for a large readership from graduate students to professional mathematicians.

– On one hand, this book is a valuable tool for researchers because it lists many open problems and includes an extensive list of the latest research in the topic. On the other, it contains a large list of exercises for students with full solutions, says Riku Klén.

**More information:**

Assistant Professor Riku Klén, riku.klen@utu.fi