## Differential Geometry

גיאומטריה דיפרנציאלית

### Fall semester 2018/19

: Bo'az Klartag**Lecturer**

: Monday, 15:15 - 18:00, Wolfson auditorium.**Classes**The Wolfson auditorium is not in the usual Ziskind building, but in the "Wolfson building for biological research". From the main entrance to Ziskind building, you need to cross the lawn with the fountain, enter the Wolfson building, and it is the first room to your right.

The semester begins on November 4, 2018 and ends on February 8, 2019.

There will be no class on December 10.

: Please fill this form in order to join the course mailing list. Students that plan to submit their homework must join the mailing list.**Mailing list**__Syllabus__This course consists of two parts.

The first part is "Analysis on Manifolds", starting from the definition a differentiable manifold, vectors fields, differential forms, integration, Stokes theorem.

The second part is "Curvature" and it focuses on Riemannian manifolds and submanifolds of Euclidean space: Riemannian metric, parallel transport, connections, geodesics, curvature, isoperimetric inequalities.

**Prerequisites**Familiarity with multivariate calculus (say, the divergence threorem) and point-set topology (say, a Hausdorff topological space).

What is the relation to other courses taught at Weizmann and at other Israeli universities?**A frequently asked question:**This class is a longer version (3 hours per week in place of 2) of the "Analysis on Manifolds" class that was taught in the last two years by Sergei Yakovenko. The extra one third will be devoted to Riemannian geometry ("curvature").

In some Israeli universities most of this material is the topic of a graduate course (e.g., the Hebrew University), and in others it is an elective undergraduate course (e.g., Tel Aviv University). If you have already taken "Analysis on Manifolds" as an undergrad at Tel Aviv University, then many parts of this class should be familiar to you, though we will try to teach at a slightly higher level and maybe using different examples.

**Final grade**It will be based on the solution of the homework exercises.

**Related literature**

- Lee, J. M, Introduction to Smooth Manifolds, 2003.
- Flanders, Differential forms with applications to the physical sciences, 1963.
- DoCarmo, Riemannian Geometry, 1992.
- Chavel, Riemannian geometry - a modern introduction, 1994.
- A short appendix by Gromov, "Isoperimetric Inequalities in Riemannian Manifolds" in the book by Milman and Schechtman, "Asymptotic Theory of Finite Dimensional Normed Spaces", 2001.

**Comments**- The proof of Jörgens theorem that I showed in class appears on page 7 of Robert Bryant's notes:

Nine lectures on exterior differential systems. - The definition we used of the Gauss curvature in two dimensions using moving frames, is treated also in Singer and Thorpe, Lecture notes on elementary topology and Geometry, 1967.
- Another good reference text for this class is Hitchin's notes on Differentiable Manifolds.

- The proof of Jörgens theorem that I showed in class appears on page 7 of Robert Bryant's notes:
**Homework assignments**