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Classical Geometries in Modern Contexts - Geometry of Real Inner Product Spaces Third Edition (Hardcover, 3rd ed. 2012)
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Classical Geometries in Modern Contexts - Geometry of Real Inner Product Spaces Third Edition (Hardcover, 3rd ed. 2012)
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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
Moebius 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 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. New to this third edition is a chapter
dealing with a simple and great idea of Leibniz that allows us to
characterize, for these same spaces X, hyperplanes of euclidean,
hyperbolic geometry, or spherical geometry, the geometries of
Lorentz-Minkowski and de Sitter, and this through finite or
infinite dimensions greater than 1. Another new and fundamental
result in this edition concerns the representation of hyperbolic
motions, their form and their transformations. Further we show that
the geometry (P,G) of segments based on X is isomorphic to the
hyperbolic geometry over X. Here P collects all x in X of norm less
than one, G is defined to be the group of bijections of P
transforming segments of P onto segments. The only prerequisites
for reading this book are basic linear algebra and basic 2- and
3-dimensional real geometry. This implies that mathematicians who
have not so far been especially interested in geometry could study
and understand some of the great ideas of classical geometries in
modern and general contexts.
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