Select Academic Year:     2016/2017 2017/2018 2018/2019 2019/2020 2020/2021 2021/2022
Professor
BENIAMINO CAPPELLETTI MONTANO (Tit.)
Period
Second Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/65]  MATHEMATICS [65/40 - Ord. 2020]  MATEMATICA PURA 9 72

### Objectives

Riemannian geometry is a branch of differential geometry dealing with a mathematical object - known as "Riemannian manifold" - which models the idea of "curved space" of arbitrary dimension. Many of the usual geometric notions, such as angles, distances, volumes, straight lines (called geodetics) can be defined on it. Moreover, one defines some characteristic concepts, such as curved spaces, curvature and metric tensor.

The main goals of the course of "Riemannian Geometry" are described as follows.

KNOWLEDGE AND UNDERSTANDING ABILITIES
The purpose of the course is to enable students to understand the language, techniques and contents of Riemannian Geometry and related topics.
The preferred teaching tool for achieving these goals are the frontal lessons, where the various topics of the course will be developed, by introducing fundamental concepts and developing a series of theorems with the respective proofs, alongside significant examples, exercises and applications.

APPLIED KNOWLEDGE AND UNDERSTANDING ABILITIES
The student must be able to produce rigorous proofs of results related to the topics treated during the course (also different from the theorems demonstrated during the course). The student should also be able to apply the basic notions on Riemannian metrics, geodetics, curvature etc. also in different contexts, such as physics, probability and statistics, and mathematical analysis.

MAKING JUDGEMENTS
Among the aims of the course one is to make the student as autonomous as possible, to help him to develop his critical sense and ability to identify by himself which ideas are important and which topics would deserve to be studied in subsequent studies.

COMMUNICATION SKILLS
The student will be able to expose and argue the solution of exercises; he will also be able to discuss and prove correctly the most relevant results related to Riemannian geometry through a rigorous logical-mathematical language.

TO LEARN SKILLS
To refine the student's analytical abilities, his mental flexibility in dealing with new and challenging concepts, linking them with those of analysis, classical differential geometry and topology that he already knows from the three-year degree courses. Ability to read by hisself an advanced text of differential geometry.

### Prerequisites

To be familiar with the concepts of Analysis, Geometry and Algebra already treated during the Mathematics First Level Degree. To be familiar with the concept of smooth manifolds, tangent fields, differential forms, etc. treated in a basic course of Differential Geometry, of which this course is the natural continuation.

### Contents

The detailed version of the program is available at the professor's web-page

PROGRAM
Riemannian metrics. Isometries. Examples. Connections. Covariant derivatives. Parallel transport. Levi-Civita connection. Divergence, gradient, Hessian and Laplacian. Geodesics. Exponential map. Normal neighborhoods. Lenght of a curve. Riemannian distance. Theorem of Hopf-Rinow. Riemannian curvature. Sectional curvature. Ricci curvature. Scalar curvature. Einstein manifolds.

### Teaching Methods

The course includes 72 hours of lessons, held at the blackboard. The lesson schedule is indicated on the CdS website in Mathematics.

Some exercises will also be proposed to be done at home, or possibly with the teacher's guidance during office hours. Corrections will be returned to students.

The teacher updates a website dedicated to students where students can find any notes / exsercises of the teacher.

### Verification of learning

An oral exam is to be held on the blackboard, during which the student must demonstrate that he has understood and assimilated the topics of the course and is able to explain the notions and proofs learned during the lessons. To do this, the student, during the oral exam, must write at the blackboard all the steps he is following to arrive at the conclusion of a reasoning. Exposure skills and the ability to relate concepts will also be evaluated.

The exam generally takes about 45 minutes.

The oral exam is passed if the student answers correctly at least two questions on different topics in the program. In any case, an overly inadequate response may compromise the entire oral test.

### Texts

J. Lee - Riemannian manifolds, Springer

Other useful books
D. Perrone – Introduzione alla geometria riemanniana, Aracne
M. Abate, F. Tovena – Geometria differenziale, Springer