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AuthorNikulin, Viacheslav V. author
TitleGeometries and Groups [electronic resource] / by Viacheslav V. Nikulin, Igor R. Shafarevich
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1994
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Descript VIII, 254 p. 1 illus. online resource


This book is devoted to the theory of geometries which are locally Euclidean, in the sense that in small regions they are identical to the geometry of the Euclidean plane or Euclidean 3-space. Starting from the simplest examples, we proceed to develop a general theory of such geometries, based on their relation with discrete groups of motions of the Euclidean plane or 3-space; we also consider the relation between discrete groups of motions and crystallography. The description of locally Euclidean geometries of one type shows that these geometries are themselves naturally represented as the points of a new geometry. The systematic study of this new geometry leads us to 2-dimensional Lobachevsky geometry (also called non-Euclidean or hyperbolic geometry) which, following the logic of our study, is constructed starting from the properties of its group of motions. Thus in this book we would like to introduce the reader to a theory of geometries which are different from the usual Euclidean geometry of the plane and 3-space, in terms of examples which are accessible to a concrete and intuitive study. The basic method of study is the use of groups of motions, both discrete groups and the groups of motions of geometries. The book does not presuppose on the part of the reader any preliminary knowledge outside the limits of a school geometry course


I. Forming geometrical intuition; statement of the main problem -- ยง1. Formulating the problem -- ยง2. Spherical geometry -- ยง3. Geometry on a cylinder -- ยง4. A world in which right and left are indistinguishable -- ยง5. A bounded world -- ยง6. What does it mean to specify a geometry? -- II. The theory of 2-dimensional locally Euclidean geometries -- ยง7. Locally Euclidean geometries and uniformly discontinuous groups of motions of the plane -- ยง8. Classification of all uniformly discontinuous groups of motions of the plane -- ยง9. A new geometry -- ยง10. Classification of all 2-dimensional locally Euclidean geometries -- III. Generalisations and applications -- ยง11. 3-dimensional locally Euclidean geometries -- ยง12. Crystallographic groups and discrete groups -- IV. Geometries on the torus, complex numbers and Lobachevsky geometry -- ยง13. Similarity of geometries -- ยง14. Geometries on the torus -- ยง15. The algebra of similarities: complex numbers -- ยง16. Lobachevsky geometry -- ยง17. The Lobachevsky plane, the modular group, the modular figure and geometries on the torus -- Historical remarks -- List of notation -- Additional Literature

Mathematics Group theory Geometry Topology Mathematics Geometry Group Theory and Generalizations Topology


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