Differential geometry (first part)

A.Y. 2020/2021
Overall hours
Learning objectives
To provide the students with an introduction to the modern theory of differentiable and Riemannian manifolds.
Expected learning outcomes
We provide a theoretical/technical background aimed at the understanding and resolutionf of geometrical problems, with the aid of analytical tecniques.
Course syllabus and organization

Single session

Lesson period
First semester
Synchronous/asynchronous lectures via Zoom app + recorded lessons and lecture notes available on Ariel and on the personal webpage.
Also the exams will be conducted using Zoom (unless otherwise communicated).
Course syllabus
1. Topological preliminaries.
2. Differentiable manifolds.
3. The tangent space.
4. Multilinear algebra.
5. The rank theorem.
6. The Frobenius theorem. (*)
7. Riemannian metrics.
8. Connections on manifolds.
9. The Levi-Civita connection and the curvature tensor.
10. The exponential map.
11. Isometric immersions.
12. Local minimization of geodesics.
13. Complete Riemannian manifolds.
14. Introduction to the moving frame formalism. (*)

(*): depending on available time and students request.
Prerequisites for admission
Geometria 1, 2, 3 e 4; Analisi 1 e 2.
Teaching methods
42 hours of classroom lessons (normally using the blackboard, right now using a virtual blackboard via Zoom), in 2-hours blocks, focused on the theory but also accompanied by examples and exercises.
Teaching Resources
Lecture notes; textbooks suggested in class (in particular: J. M. Lee, "Introduction to differentiable manifolds"; W. M. Boothby, "An introduction to differentiable manifolds and Riemannian geometry").
Assessment methods and Criteria
The final examination consists only of an oral exam,where the student will be required to illustrate (and demonstrate) results and computations presented during the course.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
Professor: Mastrolia Paolo