Classical Geometries in Modern Contexts - Geometry of Real Inner Product Spaces
The main hero of this book is a real vector space X (possibly infinite dimensional) with a chosen (positive definite) scalar product. The approach of the author follows the spirit of the Klein Erlangen program, where geometry of a space is encoded in the appropriate transformation group G. In the cases studied in the book, the group G is a semidirect product of an orthogonal group O (defined by means of a given scalar product) and a translation group T specific for a given geometry. The whole approach is coordinate free and covers uniformly finite as well as infinite dimensional cases. The first chapter is devoted to a description of the translation group T. The main result here shows that a chosen definition of the translation group implies that the resulting geometry is either Euclidean or hyperbolic. The next chapter is devoted to a study of invariants (under the group G) for these two geometries. The third chapter treats the Lie sphere geometries. The last chapter contains a discussion of the geometry of Minkowski space (including the proof of the Alexandrov theorem, which characterizes the conformal Lorentz transformations and a description of the Einstein and de Sitter space-times). Almost no prerequisites are needed to read the book.