# Introduction to differential geometry

## Summary

This course will serve as a first introduction to the geometry of Riemannian manifolds, which form an indispensible tool in the modern fields of differential geometry, analysis and theoretical physics.

## Content

Differentiable manifolds appearing in fields ranging from PDEs to theoretical physics usually come equipped with a Riemannian metric; in the language of the course "Introduction to differentiable manifolds", this is simply a symmetric tensor defining an inner product on the space of tangent vectors over each point of the manifold. In contrast to a simple differentiable manifold which only carries topological information, a Riemannian manifold also contains a geometric structure: A Riemannian metric allows us to define notions such as the length of a curve or the distance between two points in the manifold.

In this course, we will introduce the basic concepts associated to Riemannian geometry, such as the Riemannian metric, the curvature tensor and the notion of a geodesic curve. We will then proceed to explore the geometric properties of these objects, in many cases extending ideas and results from Euclidean geometry to this more general setting. We will also discuss the consequences of certain geometric assumptions on the topology of the Riemannian manifolds.

The course will cover the following topics:

- Riemannian metrics
- Riemannian connections and geodesics
- Curvature
- The metric structure of a Riemannian manifold and the Hopf-Rinow theorem
- The geometry of hypersurfaces
- Spaces of constant curvature
- Comparison theorems and topological applications

## Keywords

Differential geometry; Riemannian metric; Curvature tensor; geodesics

## Learning Prerequisites

## Required courses

Introduction to differentiable manifolds (MATH-322), Analysis I-IV.

## Important concepts to start the course

Differentiable manifold, tangent space, vector fields, tensors.

## Learning Outcomes

By the end of the course, the student must be able to:

- Define the central objects of Riemannian geometry (Riemannian metric, geodesics etc.)
- Use these objects together with the fundamental identities satisfied by them to solve problems
- Prove the main theorems appearing in the course

## Transversal skills

- Assess progress against the plan, and adapt the plan as appropriate.
- Demonstrate a capacity for creativity.
- Demonstrate the capacity for critical thinking
- Access and evaluate appropriate sources of information.

## Teaching methods

2h lectures + 2h exercises

## Expected student activities

Attending lectures and solve problems sheets; interacting in class; revise course content.

## Assessment methods

Final exam.

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

## Supervision

Office hours | No |

Assistants | Yes |

Forum | Yes |

## Resources

## Virtual desktop infrastructure (VDI)

No

## Bibliography

There are many introductory books on Riemannian geometry, unfortunately most of them intended for an audience of graduate students. We will follow closely the exposition of the following two books (which are also available at the EPFL library):

do Carmo, Manfredo; Riemannian geometry. Birkhäuser Boston, Inc., Boston, MA, 1992

Petersen, Peter; Riemannian geometry. Springer-Verlag New York 2006

## Ressources en bibliothèque

## Notes/Handbook

Written notes will be provided

## In the programs

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Introduction to differential geometry**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks