Chapter 1 Curve
1.1 Overview
1.2 Parameterized curves
1.3 Regular curves and arc lengths
1.4 Vector product in R^3
Local theory of curves parameterized by the length of 1.5 arcs
1.6 Local Standard*
1.7 Global properties of plane curves*

Chapter 2 Regular Surfaces
2.1 Overview
2.2 Regular Surface: Inverse of Regular Value
2.3 Transformation of parameters: differentiable functions on surfaces
2.4 Tangent plane: Differentiation of a map
2.5 First Basic Form: Area
2.6 Curved Scent*
2.7 Characteristics of compact surfaces that can determine the direction*
2.8 Geometric definition of area*
Appendix: Overview of Continuity and Differentiability

Chapter 3: Geometry of Gaussian Maps
3.1 Overview
3.2 Definition and basic properties of Gaussian maps
3.3 Gaussian mapping in local coordinates
3.4 Vector field*
3.5 Straight and minimally curved surfaces*
Appendix: Self-contained linear mappings and quadratic forms

Chapter 4: Intrinsic Geometry of Surfaces
4.1 Overview
4.2 Isodistance and Conformal Mapping
4.3 Gauss's theorem and compatible equations
4.4 Parallel translations and geodesics
4.5 Gauss-Bonnet theorem and its applications
4.6 Exponential mapping and geodetic polar coordinates
4.7 Other properties of geodesics: convex neighborhoods*
Appendix: Proof of fundamental theorems on local theory of curves and surfaces

Chapter 5: Global Differential Geometry
5.1 Overview
5.2 Rigidity of the sphere
5.3 Complete Surfaces and the Hope-Linow Theorem
5.4 First and second sides of the length of a circle: Bonnet's theorem
5.5 Jacobian Field and Conjugate Points
5.6 Cover Space: Hadamard's Theorem
5.7 Theorem of the Curve: Fairley-Milner Theorem
5.8 Surfaces with zero Gaussian curvature
5.9 Jacobi theorem
5.10 Abstract Surfaces: Further Generalizations
5.11 Hilbert's theorem
Appendix: General Topology for Euclidean Spaces

References and related notes

An introductory book that clearly explains the theory of differential geometry using a geometric approach.
This book is written by Mannfredo P., known as an authority in the field of differential geometry.
This is the first Korean translation of do Carmo's famous work, 『Differential Geometry of Curves and Surfaces』.
This is a bible book that covers all the essential theories of differential geometry, introducing the differential geometry of curves and surfaces from both local and global perspectives.
It allows for a clearer and more intuitive understanding of concepts because it emphasizes fundamental geometric properties rather than mechanically covering details.
Additionally, when you first study this book, we've marked out topics that you can skip, so you'll be helpfully informed of what content you need to study.
The first section of each chapter explains what you will learn in that chapter and how it connects to other chapters.
Complex and abstract concepts are presented more clearly and easily using various pictorial materials.
And we have included various types of problems to help you develop problem-solving and application skills.
By following the rigorous and clear explanations in this book, you can master differential geometry.