# DIFFERENTIAL GEOMETRY

## Learning outcomes of the course unit

The goal of the course is to give to the students, by means of frontal class, an introduction to Riemannian Geometry.

The goal of the course is to give to the students, by means of frontal class, an introduction to Riemannian Geometry.

## Prerequisites

DIFFERENTIAL GEOMETRY

DIFFERENTIAL GEOMETRY

## Course contents summary

RIEMANNIAN GEOMETRY

RIEMANNIAN GEOMETRY

## Course contents

Riemannian metrics, Affine connections, Riemannian connections, geodesics, minimizing properties of geodesics, convex neighborhoods, curvatura, sectional curvature, Ricci curvature, Jacobi equation, conjugate points, Hopf-Rinow's Theorem, Theorem of Hadamard, Bochner techniques, Theorem of Bonnet-Meyer, Theorem of Synge, The Morse index Theorem

Riemannian metrics, Affine connections, Riemannian connections, geodesics, minimizing properties of geodesics, convex neighborhoods, curvatura, sectional curvature, Ricci curvature, Jacobi equation, conjugate points, Hopf-Rinow's Theorem, Theorem of Hadamard, Bochner techniques, Theorem of Bonnet-Meyer, Theorem of Synge, The Morse index Theorem

## Recommended readings

ALEXANDRINO, BETTIOL ''LIE GROUPS AND GEOMETRICAL ASPECTS OF ISOMETRIC ACTIONS, MANFREDO DO CARMO ''RIEMANNIAN GEOMETRY''

ALEXANDRINO, BETTIOL ''LIE GROUPS AND GEOMETRICAL ASPECTS OF ISOMETRIC ACTIONS, MANFREDO DO CARMO ''RIEMANNIAN GEOMETRY''

## Teaching methods

The course counts 9CFUs which corresponds to 48 hours of lectures. The didactic activities is given by frontal class.

The course counts 9CFUs which corresponds to 48 hours of lectures. The didactic activities is given by frontal class.

## Assessment methods and criteria

Verification of the knowledges is achieved by an oral exam.

Verification of the knowledges is achieved by an oral exam.