Overview
- Dimension-free presentation
- Inclusion of proofs of newer theorems characterizing isometries and Lorentz transformations under mild hypotheses
- Common presentation for finite and infinite dimensional real inner product spaces X on an elementary basis, i.e., avoiding transfinite methods
- Highlights like the projective approach to dimension-free hyperbolic geometry or the priniple of duality are developed
- Includes supplementary material: sn.pub/extras
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Table of contents (5 chapters)
Keywords
About this book
This book is based on real inner product spaces X of arbitrary (finite or infinite) dimension greater than or equal to 2. With natural properties of (general) translations and general distances of X, euclidean and hyperbolic geometries are characterized. For these spaces X also the sphere geometries of Möbius and Lie are studied (besides euclidean and hyperbolic geometry), as well as geometries where Lorentz transformations play the key role. The geometrical notions of this book are based on general spaces X as described. This implies that also mathematicians who have not so far been especially interested in geometry may study and understand great ideas of classical geometries in modern and general contexts.
Proofs of newer theorems, characterizing isometries and Lorentz transformations under mild hypotheses are included, like for instance infinite dimensional versions of famous theorems of A.D. Alexandrov on Lorentz transformations. A real benefit is the dimension-free approach to important geometrical theories. Only prerequisites are basic linear algebra and basic 2- and 3-dimensional real geometry.
Reviews
This book is based on real inner product spaces X of arbitrary (finite or infinite) dimension greater than or equal to 2. With natural properties of (general) translations and general distances of X Euclidean, hyperbolic translations and distances, respectively, are characterized. For these spaces X also the sphere geometries of Möbius and Lie are studied besides Euclidean and hyperbolic geometry), as well as geometries where Lorentz transformations play the key role. The geometrical notions of this book are based on general spaces X as described. This implies that also mathematicians who have not so far been especially interested in geometry may study and understand great ideas of classical geometries in modern and general contexts. Proofs of newer theorems, characterizng isometries and Lorentz transformations under mild hypotheses are included, like for instance infinite dimensional versions of famous theorems of A.D. Alexandrov on Lorentz transformations. A real benefit is the dimension-free approach to important geometrical theories. Only prerequisites are basic linear algebra and basic 2- and 3-dimensional real geometry.
-- L'Enseignement Mathématique
Authors and Affiliations
Bibliographic Information
Book Title: Classical Geometries in Modern Contexts
Book Subtitle: Geometry of Real Inner Product Spaces
Authors: Walter Benz
DOI: https://doi.org/10.1007/978-3-7643-8541-5
Publisher: Birkhäuser Basel
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Birkhäuser Basel 2007
eBook ISBN: 978-3-7643-8541-5Published: 15 December 2007
Edition Number: 2
Number of Pages: XII, 277
Topics: Geometry, Mathematical Methods in Physics