000			03545nam a22004455i 4500
001			978-3-0348-0420-2
003			DE-He213
005			20140220083254.0
007			cr nn 008mamaa
008			120813s2012 sz | s |||| 0|eng d
020			_a9783034804202 _9978-3-0348-0420-2
024	7		_a10.1007/978-3-0348-0420-2 _2doi
050		4	_aQA440-699
072		7	_aPBM _2bicssc
072		7	_aMAT012000 _2bisacsh
082	0	4	_a516 _223
100	1		_aBenz, Walter. _eauthor.
245	1	0	_aClassical Geometries in Modern Contexts _h[electronic resource] : _bGeometry of Real Inner Product Spaces Third Edition / _cby Walter Benz.
250			_a3rd ed. 2012.
264		1	_aBasel : _bSpringer Basel : _bImprint: Birkhäuser, _c2012.
300			_aXVII, 309 p. _bonline resource.
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
347			_atext file _bPDF _2rda
505	0		_aPreface -- 1 Translation Groups -- 2 Euclidean and Hyperbolic Geometry -- 3 Sphere Geometries of Möbius and Lie -- 4 Lorentz Transformations -- 5 δ–Projective Mappings, Isomorphism Theorems -- 6 Planes of Leibniz, Lines of Weierstrass, Varia -- A Notation and symbols -- B Bibliography -- Index.
520			_aThe 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 Möbius 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.
650		0	_aMathematics.
650		0	_aGeometry.
650		0	_aMathematical physics.
650	1	4	_aMathematics.
650	2	4	_aGeometry.
650	2	4	_aMathematical Methods in Physics.
710	2		_aSpringerLink (Online service)
773	0		_tSpringer eBooks
776	0	8	_iPrinted edition: _z9783034804196
856	4	0	_uhttp://dx.doi.org/10.1007/978-3-0348-0420-2
912			_aZDB-2-SMA
999			_c101746 _d101746