# Differential Geometry: Curves - Surfaces - Manifolds

American Mathematical Soc., 2006 - Mathematics - 380 pages

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 Copyright

### Popular passages

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.