Aspects of Differential Geometry I (Synthesis Lectures on Mathematics and Statistics) - Softcover

Gilkey, Peter; Park, JeongHyeong; V�zquez-Lorenzo, Ram�n

9781627056625: Aspects of Differential Geometry I (Synthesis Lectures on Mathematics and Statistics)

Softcover

ISBN 10: 1627056629 ISBN 13: 9781627056625

Publisher: Morgan & Claypool Publishers, 2015

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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

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Review

“The ordering in both books is very good and both books are well written. That is why it is a good idea to use them as textbooks for some courses. While the first book can be thought of as an undergraduate textbook, the second one can be used in the first years of graduate studies and may also be used for some advanced courses with some small adjustments. The notations used in both books are standard. Since the written form is pleasant, both books are quite attractive.” – Bedia Akyar for Mathematical Reviews, October 2016