By Walter Benz

The concentration of this ebook and its geometric notions is on genuine vector areas X which are finite or endless internal product areas of arbitrary measurement more than or equivalent to two. It characterizes either euclidean and hyperbolic geometry with recognize to average homes of (general) translations and normal distances of X. additionally for those areas X, it stories the sector geometries of Möbius and Lie in addition to geometries the place Lorentz adjustments play the main role.

Proofs of more recent theorems characterizing isometries and Lorentz ameliorations lower than light hypotheses are integrated, resembling for example countless dimensional models of well-known theorems of A.D. Alexandrov on Lorentz alterations. a true gain is the dimension-free method of vital geometrical theories.

New to this 3rd variation is a bankruptcy facing an easy and nice suggestion of Leibniz that enables us to signify, for those related areas X, hyperplanes of euclidean, hyperbolic geometry, or round geometry, the geometries of Lorentz-Minkowski and de Sitter, and this via finite or endless dimensions more than 1.

Another new and primary bring about this version issues the illustration of hyperbolic motions, their shape and their differences. additional we exhibit that the geometry (P,G) of segments in response to X is isomorphic to the hyperbolic geometry over X. the following P collects all x in X of norm lower than one, G is outlined to be the gang of bijections of P reworking segments of P onto segments.

The simply necessities for examining this ebook are easy linear algebra and uncomplicated 2- and three-d genuine geometry. this suggests that mathematicians who've now not to date been particularly attracted to geometry might examine and comprehend a few of the nice rules of classical geometries in smooth and common contexts.

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