Aspects of Differential Geometry I

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

微分幾何學是一個廣泛的領域。我們選擇專注於某些適合入門的方面，並沒有試圖進行百科全書式的處理。在第一冊中，我們專注於基礎知識。第1章介紹多變量微積分，並討論反函數定理、隱函數定理、黎曼積分理論和變量變換定理。第2章介紹光滑流形、切空間和余切空間，以及斯托克斯定理。第3章是黎曼幾何的入門。介紹了李維-奇維塔連接、測地線、雅可比算子，並證明了高斯-波內定理。這些內容適合本科課程。我們提供了一些與傳統證明不同的證明，並且這些卷中還有一些新的材料。例如，對於帶邊界的拟黎曼流形的切-高斯-波內定理的處理是新的。

目錄：前言/致謝/基本概念/流形/黎曼和拟黎曼幾何/參考文獻/作者簡介/索引