OVERVIEW

Lectures are held in Italian or English, at the students' choice. The course is addressed to students in mathematics. but it can also be also attended by students in physics.

AIMS AND CONTENT

LEARNING OUTCOMES

The objective of the course is to provide students with the fundamental notions and methods of modern differential geometry.

AIMS AND LEARNING OUTCOMES

The fundamental notions discussed in the course are those of smooth structure, tangent and cotangent bundle, vector field and tensor field, algebra of differential forms, and de Rham cohomology. Connections between the topics covered and mathematical analysis, topology, algebra, mathematical physics, algebraic topology and algebraic geometry are highlighted.

TEACHING METHODS

The course follows a traditional approach.

SYLLABUS/CONTENT

1. Smooth atlases; smooth structures; topological issues

2. Smooth manifolds

3. The inverse mapping theorem

4. Partitions of unity

5. Quotient manifolds

6. Sheaf of smooth functions

7. Cotangent space and tangent space; differential of a smooth map

8. Tangent bundle and vector fields

9. Flow of a vector field; Lie derivative

10. Multilinear algebra (tensor product of R-Modules; exterior algebra)

11. Tensor fields

12. Differential forms and Cartan differential

13. de Rham cohomology

14. Orientability; integration of differential forms and Stokes' theorem.

RECOMMENDED READING/BIBLIOGRAPHY

Detailed notes will be made available to students on the Aulaweb site.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/corsi/11907/studenti-orario

Class schedule

DIFFERENTIAL GEOMETRY

EXAMS

EXAM DESCRIPTION

Oral exam (2 questions).

ASSESSMENT METHODS

The oral exam consists of 2 question (to be answered at the blackboard); each question is evaluated from 0 up to 15/30 (30-point grading scale; highest result: 30 e lode; pass result 18)

FURTHER INFORMATION

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)

• Additional time (+50% disability/invalidity)

• Additional time during oral exams to organise the answer

• Calculator (programmable and graphing calculators are not allowed)

• Conceptual Maps

• Tables and/or Forms

• Take the exam in written form

• Take the exam in oral form

• Tutor reader (for written tests only)

• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it