Geometries and Transformations

A readable exposition of how Euclidean and other geometries can be distinguished using linear algebra and transformation groups.

Norman W. Johnson (Author)

9781107103405, Cambridge University Press

Hardback, published 7 June 2018

452 pages
24.1 x 16.2 x 2.7 cm, 0.81 kg

'This is a book written with a passion for geometry, for complete lists, for consistent notation, for telling the history of a concept, and a passion to give an insight into a situation before going into the details.' Erich W. Ellers, zbMATH

Euclidean and other geometries are distinguished by the transformations that preserve their essential properties. Using linear algebra and transformation groups, this book provides a readable exposition of how these classical geometries are both differentiated and connected. Following Cayley and Klein, the book builds on projective and inversive geometry to construct 'linear' and 'circular' geometries, including classical real metric spaces like Euclidean, hyperbolic, elliptic, and spherical, as well as their unitary counterparts. The first part of the book deals with the foundations and general properties of the various kinds of geometries. The latter part studies discrete-geometric structures and their symmetries in various spaces. Written for graduate students, the book includes numerous exercises and covers both classical results and new research in the field. An understanding of analytic geometry, linear algebra, and elementary group theory is assumed.

Introduction
1. Homogenous spaces
2. Linear geometries
3. Circular geometries
4. Real collineation groups
5. Equiareal collineations
6. Real isometry groups
7. Complex spaces
8. Complex collineation groups
9. Circularities and concatenations
10. Unitary isometry groups
11. Finite symmetry groups
12. Euclidean symmetry groups
13. Hyperbolic coxeter groups
14. Modular transformations
15. Quaternionic modular groups.

Subject Areas: Algebraic topology [PBPD], Geometry [PBM], Algebra [PBF]