GEOMETRIA DIFFERENZIALE

The aim of the course is to present the elements of the theory of differentiable manifolds, Lie groups and Lie algebras. The course also aims to train the strudent's abstraction ability and his problem solving skills.

More specifically, these are the course objectives:

Knowledge and understanding: Knowledge of the main results and techniques of the theory of differentiable manifolds. Being able to read and understand topics in the theory of differentiable manifolds and to present them in a clear fashion. Being able to understand the statement of a problem.

Applying knowledge and understanding: Being able to solve exercises and problems in differential geometry, recognizing the most suitable techniques. Being able to show that every hypothesis in a theorem is essential, by means of constructing examples and counterexamples.

Making judgements: Being able to discuss whether a mathematical argument is corrent and complete. Being able to propose a plan of attack for exercises and problems in the theory of differentiable manifolds.

Communication skills: Being able to present in a clear fashion the solution of a mathematical problem, motivating the techniques employed.

Learning skills: Reading and undestanding topics on the theory of differentiable manifolds. Being able to move on to more advanced material in differential geometry at the end of the course.

The course is made up of lectures and recitation sessions. Exercises and problems related to the material covered will regularly be published on the course webpage. The students will be allowed to discuss the problems with the lecturer and the other students, but everyone will have to write up each solution on their own. Students will then will be invited to present their solutions at the blackboard. In case the course will be held online, we reserve the right to make some small modifications to the course structure.

1. Differentiable manifolds.
2. The tangent space and the differential.
3. Regular and immersed submanifolds.
4. Elements on Lie groups.
5. Sard's Theorem.
6. Whitney's Theorem.
7. Vector Fields.
8. Elements on Lie Algebras.

1. Loring W. Tu, "An Introduction to Manifolds", Universitext, Springer.
2. John M. Lee, "An Introduction to Smooth Manifolds", Graduate Texts in Mathematics, Springer.