Aspects of Differential Geometry I (Synthesis Lectures on Mathematics and Statistics), by Peter Gilkey, JeongHyeong Park, Ramón Vázquez-Lorenzo
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Aspects of Differential Geometry I (Synthesis Lectures on Mathematics and Statistics), by Peter Gilkey, JeongHyeong Park, Ramón Vázquez-Lorenzo
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Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. In Book I, we focus on preliminaries. Chapter 1 provides an introduction to multivariable calculus and treats the Inverse Function Theorem, Implicit Function Theorem, the theory of the Riemann Integral, and the Change of Variable Theorem. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and Stokes' Theorem. Chapter 3 is an introduction to Riemannian geometry. The Levi-Civita connection is presented, geodesics introduced, the Jacobi operator is discussed, and the Gauss-Bonnet Theorem is proved. The material is appropriate for an undergraduate course in the subject. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the Chern-Gauss-Bonnet Theorem for pseudo-Riemannian manifolds with boundary is new.
Table of Contents: Preface / Acknowledgments / Basic Notions and Concepts / Manifolds / Riemannian and Pseudo-Riemannian Geometry / Bibliography / Authors' Biographies / Index
Aspects of Differential Geometry I (Synthesis Lectures on Mathematics and Statistics), by Peter Gilkey, JeongHyeong Park, Ramón Vázquez-Lorenzo- Amazon Sales Rank: #3349107 in Books
- Published on: 2015-03-01
- Original language: English
- Number of items: 1
- Dimensions: 9.25" h x .35" w x 7.50" l, .62 pounds
- Binding: Paperback
- 154 pages
About the Author University of Oregon
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1 of 1 people found the following review helpful. Excellent in all respects By Ekaterina Puffini Wonderful book - great for an advanced undergraduate or for a first year student who wants to the basics of differential geometry. The second volume is more advanced. The first two sections cover a bit more material than is done in Spivak - calculus on manifolds. A nice feature of this book are the pictures of the mathematicians who proved the relevant theorems - makes mathematics have a human face. I understand a 3rd volume is being written that will be more of a research monograph treating advanced topics and recent research developments.
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