Differential Geometry: Curves - Surfaces - Manifolds
by Wolfgang Kuhnel

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Book Reviews of Differential Geometry: Curves - Surfaces - Manifolds

Book Review: A beautiful geometry
Summary: 5 Stars

Summary of Differential Geometry: Curves - Surfaces - Manifolds

With just the basic tools from multi-variable 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.

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