Differential Geometry: Curves - Surfaces - Manifolds

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American Mathematical Soc., 2006 - Mathematics - 380 pages
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Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in I\!\!R^3 that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas.

With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces.

The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces.

The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces.

The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to help along the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added.

 

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Contents

III
2
V
8
VII
15
VIII
21
IX
28
X
34
XI
38
XII
50
XXXV
210
XXXVI
218
XXXVII
224
XXXVIII
234
XXXIX
240
XLII
249
XLIII
257
XLIV
267

XIII
56
XV
57
XVI
67
XVII
79
XVIII
98
XIX
115
XX
125
XXI
128
XXII
134
XXIV
135
XXV
141
XXVI
147
XXVII
153
XXVIII
159
XXIX
167
XXX
182
XXXI
195
XXXII
202
XXXIV
203
XLV
270
XLVII
271
XLVIII
281
XLIX
297
L
303
LI
312
LII
316
LIII
319
LIV
327
LV
332
LVI
337
LVII
347
LVIII
356
LIX
365
LX
368
LXI
372
LXII
376
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Page 5 - Let U be an open set in JRn and let f: U — > JRn be a continuosuly differentiable mapping with the property that the Jacobian at a fixed point UQ is invertible.

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