ID:
502207
Duration (hours):
84
CFU:
9
SSD:
GEOMETRIA
Located in:
PAVIA
Year:
2025
Overview
Date/time interval
Secondo Semestre (26/02/2026 - 12/06/2026)
Syllabus
Course Objectives
Being aware of the content and meaning of the basic theoretical results of homotopy theory, fundamental group, and the differential geometry of curves and embedded surfaces.
Being able to reproduce with awareness the various steps of the proof of the main theoretical results.
Being capable of utilizing the theoretical results for computing the fundamental group of topological spaces by adapting tools and techniques seen in class to the context.
Being able to tackle the study of differentiable curves and surfaces (calculating fundamental forms, Gauss curvature, areas, etc.).
Being able to reproduce with awareness the various steps of the proof of the main theoretical results.
Being capable of utilizing the theoretical results for computing the fundamental group of topological spaces by adapting tools and techniques seen in class to the context.
Being able to tackle the study of differentiable curves and surfaces (calculating fundamental forms, Gauss curvature, areas, etc.).
Course Prerequisites
Contents of the courses: Linear algebra, Geometry 1, Algebra 1, Calculus 1 and 2.
Teaching Methods
Lectures and exercise classes.
Assessment Methods
The exam consists of a written and an oral part, to be completed in the same examination session (typically the following week).
The written part consists of two or three exercises, with at least one exercise for each of the two topics covered in the course. The evaluation focuses on the student's skills in applying the main theoretical concepts presented in class, in computing the fundamental group and the homotopy equivalence class of topological spaces, and in studying differentiable curves and surfaces. The exercises will consist of questions of varying difficulty aimed at assessing the depth of acquisition of these skills.
During the written part, notes or books are not allowed. Copies of recommended books will be available for consultation. Additionally, students will be provided with reference sheets for use during the written examination.
To proceed to the oral part, students must achieve a minimum score of 15 out of 30 in the written test.
In the oral part, the focus will be mainly on assessing the student's knowledge of the concepts presented during the course, the clarity of their presentation, and their ability to apply them.
The final grade will be determined by considering the overall breadth and depth of learning, as well as the clarity of presentation and problem-solving skills demonstrated. The grade will be obtained through a comparison, not necessarily reduced to an arithmetic average, of the evaluation of the written part and the oral part.
The written part consists of two or three exercises, with at least one exercise for each of the two topics covered in the course. The evaluation focuses on the student's skills in applying the main theoretical concepts presented in class, in computing the fundamental group and the homotopy equivalence class of topological spaces, and in studying differentiable curves and surfaces. The exercises will consist of questions of varying difficulty aimed at assessing the depth of acquisition of these skills.
During the written part, notes or books are not allowed. Copies of recommended books will be available for consultation. Additionally, students will be provided with reference sheets for use during the written examination.
To proceed to the oral part, students must achieve a minimum score of 15 out of 30 in the written test.
In the oral part, the focus will be mainly on assessing the student's knowledge of the concepts presented during the course, the clarity of their presentation, and their ability to apply them.
The final grade will be determined by considering the overall breadth and depth of learning, as well as the clarity of presentation and problem-solving skills demonstrated. The grade will be obtained through a comparison, not necessarily reduced to an arithmetic average, of the evaluation of the written part and the oral part.
Texts
M. Abate, F. Tovena. "Curve e Superfici". Springer
M.P. Do Carmo: "Differential Geometry of curves and surfaces", Prentice-Hall
D. L. Johnson: "Presentation of Groups", Cambridge University press
R. Lyndon, P. Schupp: "Combinatorial Group Theory", Springer
E. Sernesi: "Geometria 2", Bollati Boringhieri.
M. Manetti: "Topologia", Springer.
C. Kosniowski: "Introduzione alla topologia algebrica", Zanichelli.
J. Munkres: "Topology", Pearson.
M.P. Do Carmo: "Differential Geometry of curves and surfaces", Prentice-Hall
D. L. Johnson: "Presentation of Groups", Cambridge University press
R. Lyndon, P. Schupp: "Combinatorial Group Theory", Springer
E. Sernesi: "Geometria 2", Bollati Boringhieri.
M. Manetti: "Topologia", Springer.
C. Kosniowski: "Introduzione alla topologia algebrica", Zanichelli.
J. Munkres: "Topology", Pearson.
Contents
The course is divided into two parts: the first one is dedicated to the study of the differential geometry of curves and embedded surfaces, while the second part is an introduction to the theory of homotopy and the fundamental group of a topological space,
Extended summary
Curves
Regular curves. Arc length parameter. Frenet formulae. Curvature and torsion.
Surfaces
Regular surfaces. Diffeomorphisms of surfaces. Tangent plane. First fundamental form. The Gauss map of an orientable surfaces. Second fundamental form. Normal curvature. Gaussian and mean curvature. Isometries and the Theorema Egregium. Geodesics. The Gauss-Bonnet theorem.
Fundamental group
Homotopy of paths. Concatenation product and fundamental group. Invariance of the fundamental group via homeomorphisms and via homotopic equivalence. Functorial properties. Deformation retracts. Contractible spaces. Van Kampen Theorem. Examples and computations of the fundamental group.
Extended summary
Curves
Regular curves. Arc length parameter. Frenet formulae. Curvature and torsion.
Surfaces
Regular surfaces. Diffeomorphisms of surfaces. Tangent plane. First fundamental form. The Gauss map of an orientable surfaces. Second fundamental form. Normal curvature. Gaussian and mean curvature. Isometries and the Theorema Egregium. Geodesics. The Gauss-Bonnet theorem.
Fundamental group
Homotopy of paths. Concatenation product and fundamental group. Invariance of the fundamental group via homeomorphisms and via homotopic equivalence. Functorial properties. Deformation retracts. Contractible spaces. Van Kampen Theorem. Examples and computations of the fundamental group.
Course Language
Italian
More information
Office hours by appointment (send an email)
The students belonging to the categories of the project on innovative teaching will have the possibility to attend office hours also in the late afternoon and to see the notes of the lectures.
The students belonging to the categories of the project on innovative teaching will have the possibility to attend office hours also in the late afternoon and to see the notes of the lectures.
Degrees
Degrees
MATHEMATICS
Bachelor’s Degree
3 years
No Results Found
People
People (2)
No Results Found