00474 - Differential Geometry

Learning outcomes

At the end of the course, the student acquires advanced knowledge on smooth manifolds and differential calculus with particular regard to de Rham cohomology and Morse theory. It is able to apply the acquired notions for solving problems and building demonstrations.

Course contents

Differentiable manifolds and smooth functions.
Vector bundles.
Tangent space and tangent bundles.
Submanifolds.
Vector fields and flows.
Riemannian metrics.
Geodesics and exponential map.
Differential forms.
Integration and Stokes theorem.
De Rham cohomology.

Prerequisites

All compulsory courses of the bachelor degree, in particular linear algebra (Geometria 1), analysis in several variables (Analisi 2), topology (Geometria 2). It will be useful, but not necessary, some knowledge of the theory of curves and surfaces immersed in R^3 (first module of Geometria 3).

Readings/Bibliography

Lee, An introduction to smooth manifolds, Springer

Lee, An introduction to Riemannian manifolds, Springer

Teaching methods

Frontal Lessons, weekly exercise sheets.

Assessment methods

The exam related to this component of the integrated course (00474 - DIFFERENTIAL GEOMETRY - 6 credits) consists of a written test followed by a short oral exam. The exam mark attributed to the student is given by the average of the marks awarded for the two modules of the course, with rounding to the upper unit.

Examination methods for this component 

Written and oral exams must be taken in the same round. Access to the oral exam is granted by scoring at least 16/30 in the written exam. The final grade is determined by the oral exam keeping the result of the written exam in consideration.

The written exam lasts 2 hours. No reference material, such as books or notes, or electronic devices are allowed.

Teaching tools

Information about the course, and weekly exercise sheets will be posted on the Virtuale platform.

Office hours

See the website of Maria Beatrice Pozzetti