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.
Study Plan
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 2025/2026The 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.
1° Year
| Modules | Credits | TAF | SSD |
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| Modules | Credits | TAF | SSD |
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| Modules | Credits | TAF | SSD |
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1 module between the following1 module between the following3 modules among the followingLegend | 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.
Differential geometry (2019/2020)
Teaching code
4S003196
Teacher
Coordinator
Credits
6
Language
English
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
| 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).
