By Peter Gilkey, Jeonghyeong Park, Ramón VázquezLorenzo
, 154 pages
© 2015
ISBN 1627056629
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Price: US $40.00
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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 LeviCivita connection is presented, geodesics introduced, the Jacobi operator is discussed, and the GaussBonnet 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 ChernGaussBonnet Theorem for pseudoRiemannian manifolds with boundary is new.
