Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
Academic year in which the course will be held: 
Course type: 
Compulsory subjects, characteristic of the class
First Semester
Standard lectures hours: 
Detail of lecture’s hours: 
Lesson (80 hours)

It is required that students have already acquired the basic concepts of the theory of abstract differentiable manifolds and that they have a thorough knowledge of the pointset topology and of the fundamental group of a topological space. Finally, it is highly recommended that students have had some experience with the notion of curvature of a smooth surface in the 3-dimensional Euclidean space.

Final Examination: 

Verification of learning will consist of two parts:

1) A traditional oral exam, during which the student will have to show that she/he has acquired the basic notions, the proofs of the main theorems, and the ability to analyze concrete examples.

2) A seminar to be held in front of the class, showing that the student has acquired the ability to understand an advanced and recent topic, and to complete all the details of the arguments.

Voto Finale

The purpose of the teaching is multifold and includes:

1) the acquisition of the notions, theorems and basic techniques from classical Riemannian geometry, starting from the problem of the existence of a Riemannian metric on a generic differentiable manifold up to the Bonnet-Myers and Cartan-Hadamard theorems on the link between curvature bounds and the topology of the underlying manifold.

2) the acquisition of the ability to verify, on concrete examples, the basic geometric properties of a Riemannian manifold. Given the very abstract nature of the topics discussed, students will be guided to building and studying concrete examples both during the lectures and at home through exercises.

3) to acquire the ability to re-elaborate topics and proofs, in order to go ahead independently in the study of more advanced and more recent aspects and themes, being able to complete the details of the various arguments.

At the end of the course, students are expected to acquire:

1) the notions and the basic results of classic Riemannian Geometry.

2) the ability to verify, on concrete examples, the main geometrical properties of Riemannian manifolds.

3) the ability to re-elaborate what is seen during the lectures and to proceed independently to the study of advanced and recent aspects of the theory, being able to complete the details of the various arguments.

Starting from the problem of the existence of a Riemannian metric on a generic differentiable manifold, we will move to the notion of Levi-Civita derivation and corresponding parallel transport, which will allow us to define the concept of a geodesic curve as a smooth curve with zero acceleration. The intrinsic metric structure of a Riemannian manifold, in relation to the existence in the large of geodesics curves and their minimization property, will be analyzed in detail. The study of the Riemann curvature tensor and its traces will precede the culmination of the course, dedicated to the link between the sign of the curvature and the topology of a complete Riemannian manifold. The main topics of the course can be summarized in the following list:

1) Tensor bundles and tensor Fields
2) Definition and existence of Riemannian metrics
3) Levi-Civita connection and parallel transport
4) Geodesics and exponential mapping
5) The metric structure of a Riemannian manifold
6) Global theory of geodesics and completeness
7) The Curvature of a Riemannian manifold
8) First and Secondary Variation of Functional Energy
9) Jacobi fields and conjugate couples
10) Curvature and Topology: the theorems of Bonnet-Myers and Cartan-Hadamard

Basic textbooks:

M. P. do Carmo, Riemannian geometry. Birkhäuser Boston, Inc., Boston, MA, 1992.

J. M. Lee, Riemannian manifolds. An introduction to curvature. Graduate Texts in Mathematics 176. Spinger, New York, 1997.

Textbooks for deepening the preparation:

S. Gallot, D. Hulin, J. Lafontaine, Riemannian geometry. Third edition. Universitext. Springer-Verlag, Berlin, 2004.

P. Petersen, Riemannian Geometry. Graduate Texts in Mathematics, 171. Springer, 2006.

In view of the main objectives of the course, the teaching method will consist in frontal lectures, in homework, and in the individual study of an advanced and recent topic.

Office hours: every Monday from 2 pm to 4 pm or any other day by appointment.