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B5531 - VARIETA' DIFFERENZIABILI

Academic Year 2026/2027

                
                        Docente:
                        Luca Migliorini
                    
                        Credits:
                        9
                    
                        SSD:
                        MATH-02/B
                    
                        Language:
                        Italian
                    
                        Moduli:
                        
                            Luca Migliorini
                            (Modulo 1)
                        
                            Luca Battistella
                            (Modulo 2)
                        
                        Teaching Mode:
                        
                                    In-person learning (entirely or partially) (Modulo 1); 
                                
                                    In-person learning (entirely or partially) (Modulo 2)
                                
                            Campus:
                            Bologna
                        
                            Corso:
                            First cycle degree programme (L) in
                            Mathematics (cod. 6061)

                                        Course Timetable
                                    
from Sep 22, 2026 to Dec 17, 2026

                                        Course Timetable
                                    
from Sep 21, 2026 to Dec 14, 2026

Learning outcomes

Upon completion of the course, students will acquire the mathematical interpretation of curves and surfaces and a basic understanding of differentiable varieties of arbitrary dimension. They will be able to use this knowledge to analyze fundamental classical concepts and examples. They will be able to apply this knowledge to other mathematical disciplines and to solve simple problems posed by applied sciences.

Course contents

Part 1: Differential Geometry of Curves and Surfaces:

Curves: Tangent Line Arc-length Parametrization. Curvature and Torsion of a Curve in Space. Frenet Serret Formulas

Surfaces: Tangent Plane. First Fundamental Form.

Curvature of Curves on a Surface. Normal and Geodesic Curvature.

Second Fundamental Form: Principal Curvatures. Mean Curvature and Gaussian Curvature.

Gaussian Map. Geodesics. Parallel Transport. Theorema Egregium

Notions of Spherical and Hyperbolic Geometry. Areas of Geodesic Triangles.

Part 2: Differentiable Manifolds

Notion of Differentiable Manifold. Smooth Functions and Applications between Differentiable Manifolds. Partitions of Unity. Tangent Space. Derivative of a Differentiable Application. Vector Fields. Lie Brackets. Flow of a Vector Field. Complete Vector Fields.
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Readings/Bibliography

First part:

A. Pressley Elementary Differential Geometry

T. Shiffrin : Differential Geometry: A First Course in

Curves and Surfaces

https://math.franklin.uga.edu/sites/default/files/inline-files/ShifrinDiffGeo.pdf

second part :

J.M. Lee Introduction to smooth manifolds

Teaching methods

Lectures at the blackboard.

Assessment methods

Discussion of exercises assigned in class and oral exam.

Students with learning disabilities (LD) or temporary or permanent disabilities:

Please contact the relevant University office promptly
(https://site.unibo.it/studenti-con-disabilita-e-dsa/it); they will be responsible for suggesting any accommodations to the students concerned. These accommodations must be submitted to the teacher for approval 15 days in advance, who will evaluate their suitability, including in relation to the course's learning objectives.

Regarding learning assessments, the use of AI is
prohibited. Any use constitutes a violation of academic integrity.

Teaching tools

Tutoring

Office hours

See the website of Luca Migliorini

See the website of Luca Battistella