Differential Geometry (FIRST PART)
A.Y. 2024/2025
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.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. Differentiable manifolds (notation and basic definitions).
2. Riemannian metrics.
3. Linear connections on manifolds.
4. The Levi-Civita connection and the curvature tensor.
5. Introduction to the moving frame formalism.
6. Geodesics and exponential map.
7. Geodesics and distance; complete Riemannian manifolds.
8. Jacobi Fields - second variation of the length functional.
9. Curvature and topology.
10. Riemannian submanifolds.
11. Introduction to Global Analysis.
2. Riemannian metrics.
3. Linear connections on manifolds.
4. The Levi-Civita connection and the curvature tensor.
5. Introduction to the moving frame formalism.
6. Geodesics and exponential map.
7. Geodesics and distance; complete Riemannian manifolds.
8. Jacobi Fields - second variation of the length functional.
9. Curvature and topology.
10. Riemannian submanifolds.
11. Introduction to Global Analysis.
Prerequisites for admission
Geometria 1, 2, 3 e 4; Analisi 1 e 2.
Teaching methods
42 hours of classroom lessons, 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"; J. M. Lee, "Introduction to Riemannian 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.
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
Shifts:
Turno
Professor:
Mastrolia PaoloProfessor(s)