Geometry of SurfacesSpringer Science & Business Media, 1995 M02 3 - 236 páginas 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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II | 1 |
III | 2 |
IV | 5 |
V | 9 |
VI | 11 |
VII | 14 |
VIII | 18 |
IX | 21 |
XXXIX | 108 |
XL | 111 |
XLI | 112 |
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XLIII | 118 |
XLIV | 122 |
XLV | 126 |
XLVI | 129 |
X | 22 |
XI | 25 |
XII | 26 |
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XV | 34 |
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XXIX | 69 |
XXX | 75 |
XXXI | 80 |
XXXII | 85 |
XXXIII | 88 |
XXXIV | 92 |
XXXV | 96 |
XXXVI | 99 |
XXXVII | 101 |
XXXVIII | 105 |
XLVII | 130 |
XLVIII | 132 |
XLIX | 135 |
L | 138 |
LI | 140 |
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LIV | 147 |
LV | 153 |
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LVII | 156 |
LVIII | 160 |
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LXI | 172 |
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LXIV | 185 |
LXV | 189 |
LXVI | 190 |
LXVII | 194 |
LXVIII | 196 |
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Términos y frases comunes
a₁ Algebras angle sum boundary circle at infinity closed geodesic compact surface complex functions cone points construction Corollary coset curvature curve defined desingularization disc distance function edges equation equidistant euclidean geometry euclidean plane euclidean surface Exercises finite fixed point free follows free group fundamental polygon fundamental region genus geometric surface glide reflection H2-isometry H²-length H2-line hence homeomorphic homotopy class hyperbolic geometry hyperbolic plane hyperbolic surface identified intersection inversion isometries of S2 Killing-Hopf theorem Klein bottle lemma lift line segment local isometry modular group neighborhood orbifold orbit map orientation-preserving orientation-reversing isometries p₁ pairs path permutation polygon Proof pseudosphere punctured sphere quotient Section sequence sheets Show shown in Figure side-pairing sides space spherical subgroup symmetry T-orbit tessellation TMTL topological torus translation triangle unit circle vertex vertices x-axis y-axis
Pasajes populares
Página 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...
Página 204 - Gray [1982] From the history of a simple group. Math. Intelligencer 4, 59-67.