### Teachings

Select Academic Year:     2016/2017 2017/2018 2018/2019 2019/2020 2020/2021 2021/2022
Professor
ANDREA LOI (Tit.)
Period
First Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/65]  MATHEMATICS [65/00 - Ord. 2020]  PERCORSO COMUNE 9 72

### Objectives

The aim of this course is to provide students with the basic elements of differential geometry such as differentiable manifolds, vecotr fields and differentiable forms, fundamental in a master course in Mathematics.
The theoretical structure of the teaching consists in the development of the themes of the program, through the introduction of fundamental concepts and the development of a series of theorems with related demonstrations, supported by significant examples, exercises and applications.
In particular, the teaching aims to acquire a good knowledge of the theory differentiable manifolds, smooth maps between differentiable manifolds (immersions, embedding, submersions), Lie groups and Lie algebras.

- APPLIED KNOWLEDGE AND UNDERSTANDING
At the end of the course the student must: be able to apply the concepts and techniques learned both to standard exercises and to the resolution of new problems, which require the independent elaboration of a strategy, or of small rigorous demonstrations, not identical to those already known but inspired by them.

- JUDGMENT AUTONOMY
Knowing how to recognize when a logical procedure is correct. Learn the standard demonstration techniques of differentiable manifolds.

- COMMUNICATION SKILLS
The student will be able to explain and solve problem solving; it will also be able to discuss and demonstrate correctly the most relevant results concerning differential geometry.

- ABILITY TO LEARN
Ability to learn how to solve complex exercises and problems independently. Ability to be able to read and understand an advanced text of mathematics.

### Prerequisites

The contents of first degree in Mathematics.

### Contents

SEE loi.unica.it for the detailed programme.
EUCLIDEAN SPACES
Smooth maps; Tangent vectors in R^n as derivates;
MANIFOLDS
Topological manifolds and smooth manifolds; smooth maps on manifolds; the inverse function theorem; quotient spaces, the real projective space and the grassmannian.
THE TANGENT SPACE
The tangent space at a point; the differential of a map; the chain rule; Immersions and Submersions; Submanifolds; the regular level set theorem; examples of regular submanifolds; the rank of a smooth map; the Tangent Bundle; vector fields; integral curves.
LIE GROUPS AND LIE ALGEBRAS
Examples of Lie groups; Lie subgroups; Lia Algebras; the Lie algebra of a Lie group, the exponential map.

### Teaching Methods

The exams will be held in person if the conditions of the Covid-19 health emergency allow it. Otherwise, the exams will be delivered on the Teams platform or on an alternative online platform previously agreed between the teacher and the student.

### Verification of learning

Compatibly with the modality of exams foreseen in the Manifesto of Studies for the A.Y. 2020-21 following the COVID-19 emergency, the exams will be held either in the presence or on the Teams platform or on an alternative telematic platform previously agreed between the teacher and the student.

The oral test is about 45 minutes long, with questions about the main parts of the program. The final vote, expressed in thirty-eight, takes account of the student's preparation on each of the topics dealt with.

### Texts

W. Boothby,
An introduction to differentiable manifolds and Riemannian Geometry, Academic Press.

L. Conlon, Differentiable Manifolds, Nodern Birkhauser Classics.

Lorin, W. Tu, An Introduction to Manifolds, Springer Verlag.

I. Madsen, J. Tronehave, From calculus to cohomology,

M. Spivak, Calculus on Manifolds, Addison-Wesley Publishing Company.