# DIFFERENTIAL GEOMETRY

## Learning outcomes of the course unit

The goal of the course is to give to the students, the basic tools of Differetial Geometry with special emphasis on Riemannian geometry mainly relationships that exist between local theory and global theory.

Students must demostrate knowledge and understanding in a field of study characterized by the use of advanced textbooks and scientific articles and to possess the ability to find and use data to formulate answer to well-defined problems. In addition, they must demonstrate that they have developed those skills allowing them to study by themselves without a guide.

## Prerequisites

theory curves and surface

## Course contents summary

The first part of the lectures is an introduction about general facts of differential and Riemannian geometry.

The second part of the course is devoted to the local theory of a Riemannian manifold mainly the existence of a geodesic between two points and the existence of closed geodesics.

The third part of the course give us a relation between local theory and the global theory, when you think to a Riemannian manifold at whole.

## Course contents

Riemannian metric, Riemannian distance, a group of isometries, properly discontinuous actions, Riemannian submersions, integral and volume form

Affine connection and Levi-Civita connection, parallel transport, geodesics, the first variation formula, Gauss's lemma, the existence of a convex neighborhoods.

Curvature, sectional curvature, scalar curvature, Ricci curvature, Riemannian Laplacian, Killing fields, harmonic forms, Hodge theorem, techniques of Bochner

Jacobi fields, conjugate points, focal points.

Theorem of Hopf-Rinof, Hadamard theorem.

Manifolds with constant sectional curvature, A Theorem of Cartan, classification of space form.

Homogeneous Riemannian manifold, O'Neil's formula, symmetric spaces

Second variation formula, Theorem of Bonnet-Meyer and theorem Weinstein-Synge.

Index (Focal) Lemma index, Rauch comparison theorem, Comparison Theorem of Berger-Rauch and corollaries.

Morse index theorem, cut points.

Existence of closed geodesics, Theorem of Preissmann.

## Recommended readings

Manfredo do carmo, Riemannian Geometry, Birkauser

Cheeger-Ebin ''Comparison theorems in Riemannian geometry, North-Holland

Chavel, Riemannian Geometry: A modern introduction, Cambridge Univ. Press, Cambridge 1984.

Sakai, Riemannian Geometry, Translations of Mathematical Monographs vol. 149.

## Teaching methods

During the lessons we develop all the topics of the program. We also give some class of exercises to provide students the solutions to homeworks that have been left during the class.

## Assessment methods and criteria

The exam consist of an oral exam at the end of the course. The aim is to evalue the level of knowledge and understanding acquired by students during the class. Moreover, the students should also be able to explain correctly the arguments exposed in the class to another person.