Course No.： S070100XJ004
Course Category：Basic Course of Subject
Period/Credits : 40/2
Prerequisites：Multivariate calculus; Point Set Topology
Aims & Requirements：
This is the basic course for the master and doctor in correlation specialty of mathematics. It can also be the elective course for the master in correlation specialty such as Physics and Mechanics. Differentiable manifold has become the basic object of the modern mathematical research. This course teaches the basic knowledge of differentiable manifolds and Lie groups. Through this course, hoping students to initially master the basic concepts, methods and techniques of differentiable manifolds. Laying a solid foundation for further study of differential geometry, differential topology, geometric analysis and other related courses.
Primary Coverage：
Chapter 1, Differentiable manifolds
topological manifolds; differentiable manifolds; tangent space; tangent map; submanifold; vector fields; integrability theorem; partition of unity; embedded compact manifold.
Chapter 2, Tensor algebra
vector space and dual space; tensor algebra; symmetric and skew tensor; exterior algebra.
Chapter 3, Exterior differentiation form
cotangent space and linear differential equation; tensor field; Riemannian metric; exterior differential forms.
Chapter 4, Integration and Stokes' theorem on manifolds
the orientation of manifold; the integral of the exterior differential form; manifolds with boundary and induced direction; Stokes' theorem and its application.
References:
1. Warner, F.W., Foundations of Differentiable Manifolds and Lie Groups, GTM Vol.94，Springer-Verlag and China Academic
Publishers, Beijing,1983.
2. S.S.Chen, W.H.Chen，Lectures on Differential Geometry(Second edition)，Peking University Press, Bei Jing，2001.
3. Bai Zhengguo, Shen Yibing at el, Preliminary of Riemann geometry(revised edition),Higher Education Press,Beijing,2004.
Author：Jiagui Peng (School of Mathematical Sciences, GUCAS)
Date：April 22, 2012
