Course No.: S070100XJ005
Course Category: basic course of subject
Period/Credits : 40/2
Prerequisites：Advanced Algebra\Linear Algebra\Curve Surface Theory\Point Set Topology
Aims & Requirements：
This is the specialized 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 theoretical physics. The range or modern differential geometry is wide, this course introduces the basic theory of differentiable manifolds, Riemannian manifold and the geometry of submanifolds. Through this course, hoping students to master the basic concepts and techniques of modern differential geometry, to understand the development of modern differential geometry and to lay the foundation for further study of modern mathematics and professional studies.
Primary Coverage：
Chapter 1, Connection and curvature
affine connection; connection on Rn and submanifolds; affine connection on the manifolds; torsion and curvature; covariant differentiation; Riemann curvature; constant curvature space; the fundamental theorem of Riemann geometry.
Chapter 2, Geodesic and its application
geodesic; exponential map; normal coordinates; the first order variational formulation of arc length; completeness.
Chapter 3, Riemann submanifold
the basic equation of submanifold; moving framing; submanifolds of constant curvature; hypersurface; hypersurface in Euclidean space.
Textbook: Bai Zhengguo, Shen Yibing at el, Preliminary of Riemann geometry(revised edition),Higher Education Press,Beijing,2004.
References:
1. Bai Zhengguo, Shen Yibing at el, Preliminary of Riemann geometry(revised edition),Higher Education Press,Beijing,2004.
2. S. Kobayashi and K. Nomizu, Foudations of differential geometry, VOL I, II, Interscience publishers, 1969.
3. S.S.Chen, W.H.Chen，Lectures on Differential Geometry(Second edition)，Peking University Press, Bei Jing，2001.
Author：Xiaoxiang Jiao (School of Mathematical Sciences, GUCAS)
Date：June, 2009
