00474 - Differential Geometry

Learning outcomes

At the end of the course, the student acquires advanced knowledge on differential calculus on smooth manifolds and knowledge on the main problems and methods that arise from the study of Riemannian structures, as well as differential forms and de Rham cohomology. The student can apply the acquired notions to solve problems and construct original proofs.

Course contents

The course will be in Italian, unless all the students agree otherwise, in case of discrepancies the italian version is the correct one.

Topics

1. Manifolds and submanifolds: Differentiable manifolds and smooth functions, Tangent space, Submanifolds.

2. Vector bundles. Vector bundles, tangent and cotangent bundle, sections, vector fields and flows, Lie braket and Lie derivative.

3. Differential forms: orientation, integration, Stokes theorem,
De Rham cohomology. (Prof. Migliorini)

4. Introduction to Riemannian geometry: Riemannian metrics,
Geodesics and exponential map, completeness.

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, and the basics of Riemann surfaces (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, in case of less than three students registered for an exam, the exam will be only oral.

The exam mark of the integrated course is 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 and AI 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

See the website of Luca Migliorini