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E-raamat: Differential Geometry

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This carefully written book is an introduction to the beautiful ideas and results of differential geometry. The first half covers the geometry of curves and surfaces, which provide much of the motivation and intuition for the general theory. The second part studies the geometry of general manifolds, with particular emphasis on connections and curvature. The text is illustrated with many figures and examples. The prerequisites are undergraduate analysis and linear algebra. This new edition provides many advancements, including more figures and exercises, and--as a new feature--a good number of solutions to selected exercises.

Arvustused

The book provides an excellent introduction to the differential geometry of curves, surfaces and Riemannian manifolds that should be accessible to a variety of readers Frankly, this is a great book even after only two chapters the reader is exposed to a lot of interesting mathematics. If you are looking to teach or learn differential geometry at an introductory level, Differential Geometry CurvesSurfacesManifolds is a great resource to have on hand." - Jason M. Graham, MAA Reviews

Preface to the English Edition ix
Preface to the German Edition xi
Chapter 1 Notations and Prerequisites from Analysis
1(6)
Chapter 2 Curves in IRn
7(48)
2A Frenet curves in IRn
7(7)
2B Plane curves and space curves
14(6)
2C Relations between the curvature and the torsion
20(7)
2D The Frenet equations and the fundamental theorem of the local theory of curves
27(6)
2E Curves in Minkowski space IR31
33(4)
2F The global theory of curves
37(18)
Exercises
50(5)
Chapter 3 The Local Theory of Surfaces
55(78)
3A Surface elements and the first fundamental form
56(10)
3B The Gauss map and the curvature of surfaces
66(11)
3C Surfaces of rotation and ruled surfaces
77(19)
3D Minimal surfaces
96(17)
3E Surfaces in Minkowski space IR31
113(9)
3F Hypersurfaces in IRn+1
122(11)
Exercises
125(8)
Chapter 4 The Intrinsic Geometry of Surfaces
133(64)
4A The covariant derivative
134(6)
4B Parallel displacement and geodesics
140(5)
4C The Gauss equation and the Theorema Egregium
145(7)
4D The fundamental theorem of the local theory of surfaces
152(5)
4E The Gaussian curvature in special parameters
157(8)
4F The Gauss-Bonnet Theorem
165(15)
4G Selected topics in the global theory of surfaces
180(17)
Exercises
192(5)
Chapter 5 Riemannian Manifolds
197(36)
5A The notion of a manifold
198(7)
5B The tangent space
205(7)
5C Riemannian metrics
212(6)
5D The Riemannian connection
218(15)
Chapter 6 The Curvature Tensor
233(32)
6A Tensors
233(9)
6B The sectional curvature
242(8)
6C The Ricci tensor and the Einstein tensor
250(15)
Chapter 7 Spaces of Constant Curvature
265(44)
7A Hyperbolic space
266(10)
7B Geodesics and Jacobi fields
276(15)
7C The space form problem
291(5)
7D Three-dimensional Euclidean and spherical space forms
296(13)
Exercises
306(3)
Chapter 8 Einstein Spaces
309(52)
8A The variation of the Hilbert-Einstein functional
312(9)
8B The Einstein field equations
321(4)
8C Homogenous Einstein spaces
325(6)
8D The decomposition of the curvature tensor
331(10)
8E The Weyl tensor
341(9)
8F Duality for four-manifolds and Petrov types
350(11)
Exercises
358(3)
Solutions to selected exercises 361(30)
Bibliography 391(4)
List of notation 395(2)
Index 397
Wolfgang Kuhnel, University of Stuttgart, Germany.
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