Classical Geometries in Modern Contexts: Geometry of Real Inner Product Spaces Third Edition

The focus of this book and its geometric notions is on real vector spaces X that are finite or infinite inner product spaces of arbitrary dimension greater than or equal to 2. It characterizes both euclidean and hyperbolic geometry with respect to natural properties of (general) translations and general distances of X. Also for these spaces X, it studies the sphere geometries of Mobius and Lie as well as geometries where Lorentz transformations play the key role. Proofs of newer theorems characterizing isometries and Lorentz transformations under mild hypotheses are included, such as for insta.

Classical Geometriesin Modern Contexts; Geometry of Real Inner Product Spaces; Contents; Preface; Preface to the Second Edition; Preface to the Third Edition; Chapter 1 Translation Groups; 1.1 Real inner product spaces; 1.2 Examples; 1.3 Isomorphic, non-isomorphic spaces; 1.4 Inequality of Cauchy-Schwarz; 1.5 Orthogonal mappings; 1.6 A characterization of orthogonal mappings; 1.7 Translation groups, axis, kernel; 1.8 Separable translation groups; 1.9 Geometry of a group of permutations; 1.10 Euclidean, hyperbolic geometry; 1.11 A common characterization.

1.12 Other directions, a counterexampleChapter 2 Euclidean and Hyperbolic Geometry; 2.1 Metric spaces; 2.2 The lines of L.M. Blumenthal; 2.3 The lines of Karl Menger; 2.4 Another definition of lines; 2.5 Balls, hyperplanes, subspaces; 2.6 A special quasi-hyperplane; 2.7 Orthogonality, equidistant surfaces; 2.8 A parametric representation; 2.9 Ends, parallelity, measures of angles; 2.10 Angles of parallelism, horocycles; 2.11 Geometrical subspaces; 2.12 The Cayley-Klein model; 2.13 Hyperplanes under translations; 2.14 Lines under translations; 2.15 Hyperbolic coordinates.

2.16 All isometries of (X, eucl), (X, hyp)2.17 Isometries …