Differential Geometry

Author:              Zhiming Ou

                         Mattermatics Learning Center

Publisher:       3265 Public Way

ISBN: 978-1-0677109-4-1

Summary

Differential geometry is a mathematical discipline that uses the techniques of calculus and linear algebra to study the properties of curves, surfaces, and higher-dimensional spaces called differential manifolds. The key concepts involve curvature, torsion, tangents, differential forms.

Differential geometry is essential in physics and various applied sciences, such as general relativity, computer vision, motion planning.

In this book, I summarized the known results, and proposed new problems, and topics for research. It is good for beginners to learn this subject, and for scientists to apply and further study.

 

Content

Chapter 1 Curves

• 1.1 Concepts 3

Examples of 3D Curves: straight line, helix, suspension bridge

Curves as locus of a moving particle           5

Features of curves: curvature, torsion, contact plane, least arclength         6

• 1.2 Arclength Parametrization 9
• 1.3 Frenet-Serret Formulas 10
• 1.4 Non-Unit Speed Curves 14
• 1.5 the Isoperimetric Inequality 16

 

Chapter 2 Surfaces          19

• 2.1 Describing a Surface 19

Examples: revolution, helix, Enneper’s surface; surface with rulings.

Tangent Planes     22

• 2.2 Curvatures of Surfaces 24

 

Chapter 3 Fundamental Theorems on Surfaces         31

• 1 Motion of the Natural Frame on a Surface 31

The Natural Frame          32

Gauss-Codazzi Equations       33

• 2 Fundamental Theorems 35

 

Chapter 4 Curves on a Surface          35

• 1 Geodesic Curvature and Geodesic torsion 35
• 2 Geodesic Lines 39
• 3 Calculus of Variations 39
• 4 Gauss-Bonnet Theorem 42

 

Chapter 5 Differential Manifolds       44

• 1 Differential Forms 45

Exterior Product (wedge product), External Forms     

• 2 Calculus on Differential Manifolds 52

Exterior derivative

Stokes' theorem

 

Unsolved Problems in Differential Geometry     54