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, 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.
Table of contents:
Preface.- Translation Groups.- Euclidean and Hyperbolic Geometry.- Sphere Geometries of Möbius and Lie.- Lorentz Transformations.- Bibliography.- Notation and Symbols.- Index.
Table of contents:
Preface.- Translation Groups.- Euclidean and Hyperbolic Geometry.- Sphere Geometries of Möbius and Lie.- Lorentz Transformations.- Bibliography.- Notation and Symbols.- Index.
The precise and clear style of the presented matter enables the reader (with some basic knowledge) to obtain some profound insight into this field of geometry. [...] The book can be recommended to students and teachers who want to get into the state of the art of this theory. Review of the 2nd edition by O. Röschel (Graz) Internationale Mathematische Nachrichten Nr. 212, Dez. 2009 [...] The book on the one hand brings together many elder results scattered through the literature and on the other hand leads to the frontier of research. Thus it is highly welcomed and can be recommended warmly to anyone interested in this topic. Review of the first edition by G. Kowol, Vienna Monatshefte für Mathematik Vol. 150, No. 3/2007 "... The mathematical prerequisites are minimal - the rudiments of linear algebra suffice - and all theorems are proved in detail. Following the proofs does not involve more than following the lines of a computation, and the author makes every effort to avoid referring to a synthetic geometric understanding, given that he aims at attracting readers with a distaste for synthetic geometry, which, given the academic curricula of the past decades, represent the overwhelming majority of potential readers of any mathematical monograph. One of the lessons of this monograph is that there is a coordinate-free analytic geometry, which significantly simplifies computations and frees the mind from redundant assumptions. the author makes every effort to avoid referring to a synthetic geometric understanding, given that he aims at attracting readers with a distaste for synthetic geometry, which, given the academic curricula of the past decades, represent the overwhelming majority of potential readers of any mathematical monograph. One of the lessons of this monograph is that there is a coordinate-free analytic geometry, which significantly simplifies computations and frees the mind from redundant assumptions. ..." Review of the first edition by Victor V. Pambuccian, Mathematical Reviews 2006