Geometry of Surfaces
Springer Science & Business Media, 3 févr. 1995 - 236 pages
Geometry used to be the basis of a mathematical education; today it is not even a standard undergraduate topic. Much as I deplore this situation, I welcome the opportunity to make a fresh start. Classical geometry is no longer an adequate basis for mathematics or physics-both of which are becoming increasingly geometric-and geometry can no longer be divorced from algebra, topology, and analysis. Students need a geometry of greater scope, and the fact that there is no room for geometry in the curriculum un til the third or fourth year at least allows us to assume some mathematical background. What geometry should be taught? I believe that the geometry of surfaces of constant curvature is an ideal choice, for the following reasons: 1. It is basically simple and traditional. We are not forgetting euclidean geometry but extending it enough to be interesting and useful. The extensions offer the simplest possible introduction to fundamentals of modem geometry: curvature, group actions, and covering spaces. 2. The prerequisites are modest and standard. A little linear algebra (mostly 2 x 2 matrices), calculus as far as hyperbolic functions, ba sic group theory (subgroups and cosets), and basic topology (open, closed, and compact sets).
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Algebras angle sum axis boundary called circle at infinity closed geodesic compact surface complex functions complex numbers cone points construction Corollary corresponding coset curvature curve defined desingularization disc distance function elliptic plane equation equidistant euclidean geometry euclidean plane euclidean surface Exercises F-orbit finite fixed point free follows free group fundamental polygon fundamental region geometric surface glide reflection group F H2-isometry H2-length H2-line hence homeomorphic homotopy class hyperbolic geometry hyperbolic plane hyperbolic surface identified integer intersection inversion isometries of R2 Killing-Hopf theorem Klein bottle lemma length lift limit rotation line segment local isometry modular group neighborhood nontrivial orbifold orbit map orientable surface orientation-preserving orientation-reversing isometry orthogonal pairs permutation Proof pseudosphere punctured sphere quotient rotations of S2 Section sequence sheets Show shown in Figure sides Sn/F spherical stereographic image subgroup tessellation theory tile topological torus translation triangle twisted cylinder unit circle vertex vertices x-axis y-axis
Page 203 - Mat. pura appl., ser. 1, 7, 185-204. In his Opere Matematiche 1: 262-280. Beltrami, E. (1868a). Saggio di interpretazione della geometria non-euclidea. Giorn. Mat., 6, 284-312. In his Opere Matematiche 1 : 262-280, English translation in Stillwell (1996). Beltrami, E. ( 1 868b). Teoria fondamentale degli spazii di curvatura costante. Ann. Mat. pura appl., ser. 2, 2, 232-255. In his Opere Matematiche 1: 406-429, English translation in Stillwell (1996). Bernoulli, D. (1743). Letter to Euler, 4 September...