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00474 - Differential Geometry

Academic Year 2025/2026

                
                        Docente:
                        Maria Beatrice Pozzetti
                    
                        Credits:
                        6
                    
                        SSD:
                        MAT/03
                    
                        Language:
                        Italian
                    
                        Teaching Mode:
                        In-person learning (entirely or partially)
                        
                            Campus:
                            Bologna
                        
                            Corso:
                            Second cycle degree programme (LM) in
                            Mathematics (cod. 6730)

                            Teaching resources on Virtuale
                        
                                    Course Timetable
                                
from Sep 17, 2025 to Dec 19, 2025

Learning outcomes

The course will be in Italian, in case of discrepancies the italian version is the correct one.

At the end of the course, the student acquires advanced knowledge on smooth manifolds and differential calculus with particular regard to de Rham cohomology. They are able to solve problems and construct original proofs with the acquired notions.

Course contents

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.

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

Abate, Tovena, Geometria differenziale, Springer Unitext

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