Geometry of Surfaces

Couverture
Springer Science & Business Media, 3 févr. 1995 - 236 pages
1 Commentaire
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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Table des matières

II
1
III
2
IV
5
V
9
VI
11
VII
14
VIII
18
IX
21
XXXVIII
108
XXXIX
111
XL
112
XLI
113
XLII
118
XLIII
122
XLIV
126
XLV
129

X
22
XI
25
XII
26
XIII
29
XIV
33
XV
34
XVI
36
XVII
39
XVIII
41
XIX
45
XX
48
XXI
50
XXII
52
XXIII
56
XXIV
60
XXV
63
XXVI
65
XXVII
67
XXVIII
69
XXIX
75
XXX
80
XXXI
85
XXXII
88
XXXIII
92
XXXIV
96
XXXV
99
XXXVI
101
XXXVII
105
XLVI
130
XLVII
132
XLVIII
135
XLIX
138
L
140
LI
143
LII
145
LIII
147
LIV
153
LV
154
LVI
156
LVII
160
LVIII
163
LIX
167
LX
172
LXI
178
LXII
182
LXIII
185
LXIV
189
LXV
190
LXVI
194
LXVII
196
LXVIII
198
LXIX
201
LXX
203
LXXI
207
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Fréquemment cités

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...
Page 204 - Gray [1982] From the history of a simple group. Math. Intelligencer 4, 59-67.

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