Studying at the University of Verona

Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.

This information is intended exclusively for students already enrolled in this course.
If you are a new student interested in enrolling, you can find information about the course of study on the course page:

Laurea magistrale in Mathematics - Enrollment from 2025/2026

The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University.
Please select your Study Plan based on your enrollment year.

CURRICULUM TIPO:

1° Year 

ModulesCreditsTAFSSD

2° Year   activated in the A.Y. 2020/2021

ModulesCreditsTAFSSD
6
B
MAT/05
Final exam
32
E
-
activated in the A.Y. 2020/2021
ModulesCreditsTAFSSD
6
B
MAT/05
Final exam
32
E
-
Modules Credits TAF SSD
Between the years: 1°- 2°
1 module between the following
Between the years: 1°- 2°
1 module between the following
Between the years: 1°- 2°
Between the years: 1°- 2°
Other activities
4
F
-

Legend | Type of training activity (TTA)

TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.




S Placements in companies, public or private institutions and professional associations

Teaching code

4S003196

Teacher

Coordinator

Credits

6

Language

English en

Scientific Disciplinary Sector (SSD)

MAT/03 - GEOMETRY

Period

I semestre dal Oct 1, 2019 al Jan 31, 2020.

Learning outcomes

The course aims to provide students with the basic concepts on Differential Geometry of manifolds. At the end of the course the student will know the main terminology and definitions about manifolds and Riemannian manifolds, and some of the main results. He/she will be able to produce rigorous arguments and proofs on these topics and he/she will be able to read articles and texts of Differential Geometry.

Program

The course consists of lectures. Notes for each lecture will be provided.

-REVIEW GENERAL TOPOLOGY
-SURFACES EMBEDDED IN THE EUCLIDEAN 3-SPACE:
• Differentiable Atlas
• Orientable Atlas
• Tangent plane
• Normal versor
• First Fundamental Form: lengths and area
• Geodesic curvature and normal curvature
• Normal sections and Meusnier Theorem
• Principal Curvatures, Gaussian curvature, Mean curvature: minimal surfaces
• Theorema Egregium
• Geodetics
- TENSOR CALCULUS
• Free vector space
• Tensor product of two vector spaces
• Tensor product of n vector spaces
• Tensor Algebra
• Transformation of the componenents of a tensoriale
• Mixed tensors
• Symmetric tensors
• Antysimmetric (alternating) tensors
• Exterior Algebra
• Determinant
• Area and Volume
-DIFFERENTIAL MANIFOLDS
• Definition and examples
• Classification of 1-manifolds
• Classification of simply-connected 2-manifolds
• Product and quotient spaces
• Differentiable maps
• Tangent space and tangent bundle
• Vector field on a manifold
• Tensor field
• Exterior Algebra on manifolds
• Riemannian Manifolds
• Metric Tensor
• Orientations
• Volume
• Exterior derivative
• De Rham Cohomology
• Homotopy
-AFFINE CONNECTION AND CURVATURE TENSOR
• Affine connection
• Parallel transport
• Levi-Civita connection
• Geodetics
• Riemann curvature tensor
• Bianchi identities

Reference texts
Author Title Publishing house Year ISBN Notes
Do Carmo Differential Geometry of Curves and Surfaces (Edizione 2) 2016
Do Carmo Riemannian Geometry 1992
Jürgen Jost Riemannian Geometry and Geometric Analysis (Edizione 5) Springer 2008

Examination Methods

During the exam, students must show that:
- they know and understand the fundamental concepts of differential geometry
- they have analytical and abstraction abilities
- they support their argumentation with mathematical rigor.

The exam consists of a written test in which the student will have to choose one of two essays in which they provide a broad discussion of one of the topics presented during the lectures (answer approximately 2/3 pages ) and two of three short questions (answer approximately 10 rows).

Students with disabilities or specific learning disorders (SLD), who intend to request the adaptation of the exam, must follow the instructions given HERE