Differential Geometry: Curves - Surfaces - Manifolds
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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 helpalong 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. Booknews Designed for a one-semester course in classical differential geometry followed by a one-semester course on Riemannian geometry, this text introduces the geometry of curves, surfaces, and general manifolds. Special topics include Frenet frames, ruled surfaces, minimal surfaces, the Gauss-Bonnett theorem, constant curvature, and Einstein manifolds. The text is intended for students who have completed an undergraduate course in calculus and linear algebra. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Preface to the German Edition1Notations and Prerequisites from Analysis12Curves in IR[superscript n]72AFrenet curves in IR[superscript n]72BPlane curves and space curves142CRelations between the curvature and the torsion202DThe Frenet equations and the fundamental theorem of the local theory of curves262ECurves in Minkowski space IR[actual symbol not reproducible]322FThe global theory of curves363The Local Theory of Surfaces533ASurface elements and the first fundamental form543BThe Gauss map and the curvature of surfaces643CSurfaces of revolution and ruled surfaces753DMinimal surfaces943ESurfaces in Minkowski space IR[actual symbol not reproducible]1113FHypersurfaces in IR[superscript n+1]1184The Intrinsic Geometry of Surfaces1274AThe covariant derivative1284BParallel displacement and geodesics1344CThe Gaussian equation and the Theorema Egregium1394DThe fundamental theorem of the local theory of surfaces1464EThe Gaussian curvature in special parameters1514FThe Gauss-Bonnet Theorem1594GSelected topics in the global theory of surfaces1745Riemannian Manifolds1895AThe notion of a manifold1905BThe tangent space1975CRiemannian metrics2045DThe Riemannian connection2106The Curvature Tensor2256ATensors2256BThe sectional curvature2346CThe Ricci tensor and the Einstein tensor2427Spaces of Constant Curvature2537AHyperbolic space2547BGeodesics and Jacobi fields2647CThe space form problem2797DThree-dimensional Euclidean and spherical space forms2848Einstein Spaces2978AThe variation of the Hilbert-Einstein functional3008BThe Einstein field equations3098CHomogeneous Einstein spaces3138DThe decomposition of the curvature tensor3198EThe Weyl tensor3298FDuality for four-manifolds and Petrov types337Bibliography347Index of Notation351Index353
