Introduction to Classical Geometries

This book develops the geometric intuition of the reader by examining the symmetries of the space in question. This approach introduces in turn all the classical geometries: Euclidean, affine, elliptic, projective and hyperbolic. Pre-requisites are calculus, linear algebra and basic analytic geometry.

This book follows Klein's proposal for studying geometry by looking at the symmetries (or rigid motions) of the space in question. This approach introduces in turn all the classical geometries: Euclidean, affine, elliptic, projective and hyperbolic. Numerous illustrations are included to reinforce understanding. Once plane geometry is well understood, it is much easier to go into higher dimensions. Therefore, the main focus of this text is on the mathematically rich two-dimensional case, although some aspects of 3- or $n$-dimensional geometries are included. The book appeals to, and helps to further develop the geometric intuition of the reader. Some basic notions of algebra and analysis are also used to convey more complete understanding of various concepts and results. Concepts of geometry are presented in a very simple way, so that they become easily accessible: the only pre-requisites are calculus, linear algebra and basic analytic geometry.

Presents all the classical geometries in a unified way, with many illustrations, thus making their understanding easier Though the proofs and arguments are in general absolutely rigorous, emphasis is put on the concepts and the ideas, not on the technical difficulties Presentation of several advanced concepts of geometry in a very simple way, so that they become easily accessible: the only pre-requisites are calculus, linear algebra and basic analytic geometry Many interesting problems given as exercises Wide bibliography for further reading Using elementary mathematics, more advanced concepts such as manifolds and Lie groups, group actions, curvature and foliations are introduced Includes supplementary material: sn.pub/extras

Klappentext

This book follows Felix Klein's proposal of studying geometry by looking at the symmetries (or rigid motions) of the space in question. In this way the classical geometries are studied: Euclidean, affine, elliptic, projective and hyperbolic. For simplicity the focus is on the two-dimensional case, which is already rich enough, though some aspects of the 3- or n-dimensional geometries are included. Once plane geometry is well understood, it is much easier to go into higher dimensions.

The fundamental ideas of the classical geometries are presented in a clear and elementary way, making them accessible to a wide audience, and relating them to more advanced topics in modern geometry, such as manifolds, Lie groups, the Gaussian curvature, group actions, and foliations.

The book appeals to, and develops, the geometric intuition of the reader. The only prerequisites are calculus, linear algebra and basic analytic geometry. After studying the material, the reader will have a good understanding of basic geometry as well as a clear picture of the relations of this beautiful subject to other branches of mathematics. This is supported by more than 100 carefully chosen illustrations and a large number of exercises. While mainly addressed to students at advanced undergraduate level, the text can be of interest to anyone wanting to learn classical geometry.

Inhalt
Euclidean geometry.- Affine geometry.- Projective geometry.- Hyperbolic geometry.- Appendices.