微分幾何中的初等論題（簡體書）

In the past decade there has been a significant change in the freshman/sophomore mathematics curriculum as taught at many， if not most， of our colleges. This has been brought about by the introduction of linear algebra into the curriculum at the sophomore level. The advanta8;es of using linear algebra both in the teaching of differential equations and in the teaching of multivariate calculus are by now widely recognized. Several textbooks adopting this point of view are now available and have been widely adopted. Students completing the sophomore year now have a fair preliminary understanding of spaces of many dimensions. 本書是一部講述微分幾何的教程。著重講述了n+1維空間中n維旋轉曲面的幾何。通過光滑函數水平集的曲面的講述，這樣不需要更多的預備章節，直接引入全域的概念。向量微積分成為研究該理論的最基本工具。書中講述了測地線，平行輸運、曲率和凸性這些基礎知識，微分形式的介紹僅是為了在積分中的應用。這本教程需要有線性代數、多變量微積分和微分方程的預備知識，供中高級水平的本科生作為一個學期使用。

Chapter 1
Graphs and Level Sets
Chapter 2
Vector Fields
Chapter 3
The Tangent Space
Chapter 4
Surfaces
Chapter 5
Vector Fields on Surfaces; Orientation
Chapter 6
The Gauss Map
Chapter 7
Geodesics
Chapter 8
Parallel Transport
Chapter 9
The Weingarten Map
Chapter 10
Curvature of Plane Curves
Chapter 11
Arc Length and Line Integrals
Chapter 12
Curvature of Surfaces
Chapter 13
Convex Surfaces
Chapter 14
Parametrized Surfaces
Chapter 15
Local Equivalence of Surfaces and Parametrized Surfaces
Chapter 16
Focal Points
Chapter 17
Surface Area and Volume
Chapter 18
Minimal Surfaces
Chapter 19
The Exponential Map
Chapter 20
Surfaces with Boundary
Chapter 21
The Gauss-Bonnet Theorem
Chapter 22
Rigid Motions and Congruence
Chapter 23
Isometries
Chapter 24
Riemannian Metrics
Bibliography
Notational Index
Subject Index