Course No.：S070101ZJ003
Course Category：Professional Basic Course
Period/Credits： 40/2
Prerequisites： Advanced Algebra, Linear Algebra, Curve Surface Theory, point Set Topology, Differential Geometry I.
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 is the improvement of Differential Geometry I, mainly introduce the operator theory, geodesic, the preliminary of complex manifold and complex geometry.
Through this course, hoping the student can master the basic concept and basic skills, understand the modern development of differential geometry, as a basis for further study of modern mathematics and specialty research.
Primary Coverage：
Chapter 1, Geometric operators on the Riemann manifold
Hodge star operator; Laplace-Beltrami operator; Hodge theorem and its application
Chapter 2, Geodesic and its application
the second order variational formulation of arc length; Jacobi field; conjugate point; index lemma; Hessian comparison theorem; Laplacian comparison theorem; Volume comparison theorem
Chapter 3, The preliminary of complex manifold and complex geometry
the concept of complex mainfold; almost complex manifold; Hermite and Kaehler metric; Ricci form; holomorphic sectional curvature; structure equation; Chen class; Kaehler submanifold.
Textbook:
Bai Zhengguo, Shen Yibing at el, Preliminary of Riemann geometry (revised edition), Higher Education Press,Beijing,2004.
References:
1. Kosxul J, Zou Yiming, Introduction of symplectic geometry, Science Press, Beijing, 1999.
2. O’Neil B., Semi-Riemannian geometry: with applications to relativity, New York : Academic Press, 1983.
3. S. Kobayashi and K. Nomizu, Foudations of differential geometry, VOL I, II, Interscience publishers, 1969.
4. 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：April 22, 2012
