Course Code: B12006H
Course Name: Differential Geometry
Credits: 4.0
Level: Undergraduate
Pre-requisite: Calculus, Linear Algebra, Complex Variables
Lecture Time: 80 periods
Course Description
This course is a required basic course for the undergraduate students with major in mathematics. Through this course, students are required to learn and master the knowledge of local theory of curves and surfaces in three dimensional Euclidean space, transformation group theory, and the intrinsic geometry of surfaces.
Topics and Schedule
1. Geometry in Regions of a Space
1.1.Coordinate systems
1.2.Euclidean space
1.3.Riemannian and pseudo- Riemannian spaces
1.4. The simplest groups of transformations of Euclidean space
1.5. The Serret-Frenet formulae
1.6. Pseudo-Euclidean spaces
2. The Theory of Surfaces
2.1.Geometry on a surface in space
2.2.The second fundamental form
2.3.The metric on the sphere
2.4.Space-like surfaces in pseudo-Euclidean space
2.5.The language of complex numbers in geometry
2.6. Analytic functions
2.7. The conformal form of the metric on a surface
3. Geometry of the Gauss Map
3.1.Definition of the Gauss map and its fundamental properties
3.2.The Gauss map in local coordinates
3.3.Vector fields
3.4. Minimal surfaces
4. Intrinsic Geometry
4.1.Isometrics; Conformal maps
4.2.Geodesics; Parallel transport
4.3.The exponential map; Geodesic polar coordinates
5. The Elements of the Calculus of Variations
5.1.One-dimensional variational problems
5.2.Conservation laws
5.3.Hamiltonian formalism
5.4.The geometrical theory of phase space
5.5.The second variation for the equation of the geodesics
Textbook
B.A.Dubrovin, A.T.Fomenko, Modern Geometry-Methods and Applications The Geometry of Surfaces, Transformation Groups, and Fields, the first volume, Second Edition, Springer-Verlag, 1999
References
Manfredo P.do Carmo, Differential Geometry of Curves and Surfaces, China Machine Press, 2005 (Chinese Version)
