Geometry of Manifolds II

This is an advanced graduate course on the geometry of manifolds. It requires familiarity with rudiments of manifolds and differential geometry. The official prerequisite is Geometry of Manifolds I.

Class meetings: Saturday and Monday 3:30pm-5:30pm in Math 101 (Dr Mehri's hall).

Text: We will use the following texts and possibly more (TBA).

• [Sh] Geometry of Manifolds, lectures by Dr Siavash Shahshahani. (Chapters on connection).
• [P] Poor; Differential Geometric Structures, McGraw-Hill, 1981.
• [M] Milnor; Morse Theory, Princeton University Press. (For Geodesics and the Exponential map.)
• [dC] do Carmo; Riemannian Geometry, Birkhauser. (Chapter on Geodesics and the Exponential map.)
• [GHL] Gallot, Hulin, Lafontaine; Riemannian Geometry. (Parts on Hopf-Rinow theorem.)
• [Ch] Chern; Connections on a Vector Bundle, in: Global Differential Geometry, edited by Chern, MAA Studies in Mathematics, vol. 72? (For the discussion of chern classes.)
• [FM] Gauge theory and the topology of 4-manifolds, edited by R Friedman and J Morgan, IAS/Park City Mathematics Series, vol. 4, AMS & IAS, 1998.

Syllabus: We tentatively plan to cover theory of connections and curvature on the tangent bundle of a manifold, on a vector bundle and on principal bundles. Then we may tentatively continue with gauge theory, but other topics are also under consideration.

Homework: Homework will regularly be assigned in class and posted here . It is important that you do the homework as we proceed, not only because it helps you learn the material better and benefit from class discussions, but also because it carries a lot of weight in your grades! You are allowed (and even encouraged) to do the homework collectively, but please write the solutions down individually!

Exam: The final exam is officially on Tir 11, 1380, in the afternoon. No make-up will be given for the exams unless under very exceptional circumstances.

Grades: The grading system will be based mainly (or maybe even exclusively) on the collected homeworks, plus possibly a take-home final exam with a minimal share of the final grade.

Annoncements: Announcements about the course will be made in class and put on the website: http://sharif.ac.ir/~safari/courses/ManifoldsII/