Geometries and transformations

Description

"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"--  Read more... Abstract: This readable exposition uses linear algebra and transformation groups to differentiate and connect both Euclidean and other geometries. Written for graduate students, the book includes numerous exercises and covers both classical results and new research in the field.  Read more... Content: Homogenous spaces -- Linear geometries -- Circular geometries -- Real collineation groups -- Equiareal collineations -- Real isometry groups -- Complex spaces -- Complex collineation groups -- Circularities and concatenations -- Unitary isometry groups -- Finite symmetry groups -- Euclidean symmetry groups -- Hyperbolic coxeter groups -- Modular transformations -- Quaternionic modular groups.