Programme

We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim of this course is to prove Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using geometric properties of geodesics. These are the curves which, at least locally, minimize the distance on a Riemannian manifold. Time permitting, we will give an introduction to abstract Riemannian geometry in arbitrary dimension.

Core Documentation

Differential Geometry of Curves & Surfaces, by Manfredo Do Carmo. Second edition.
Curves and Surfaces, by Marco Abate and Francesca Tovena.

Type of delivery of the course

In-class lectures.

Attendance

Students are advised to attend classes regularly and to keep up to date with class content and exams.

Type of evaluation

There will be two written homework assignments. Additionally, there will be an oral exam, where the students will give a presentation on a topic decided in advance with the instructor and I will ask for the proof of a result proven in class. The theorem to prove will be chosen from a list built during the course.

Programme

We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim of this course is to prove Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using geometric properties of geodesics. These are the curves which, at least locally, minimize the distance on a Riemannian manifold. Time permitting, we will give an introduction to abstract Riemannian geometry in arbitrary dimension.

Core Documentation

Differential Geometry of Curves & Surfaces, by Manfredo Do Carmo. Second edition.
Curves and Surfaces, by Marco Abate and Francesca Tovena.

Type of delivery of the course

In-class lectures.

Attendance

Students are advised to attend classes regularly and to keep up to date with class content and exams.

Type of evaluation

There will be two written homework assignments. Additionally, there will be an oral exam, where the students will give a presentation on a topic decided in advance with the instructor and I will ask for the proof of a result proven in class. The theorem to prove will be chosen from a list built during the course.

Programme

We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim of this course is to prove Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using geometric properties of geodesics. These are the curves which, at least locally, minimize the distance on a Riemannian manifold. Time permitting, we will give an introduction to abstract Riemannian geometry in arbitrary dimension.

Core Documentation

Differential Geometry of Curves & Surfaces, by Manfredo Do Carmo. Second edition.
Curves and Surfaces, by Marco Abate and Francesca Tovena.

Type of delivery of the course

In-class lectures.

Attendance

Students are advised to attend classes regularly and to keep up to date with class content and exams.

Type of evaluation

There will be two written homework assignments. Additionally, there will be an oral exam, where the students will give a presentation on a topic decided in advance with the instructor and I will ask for the proof of a result proven in class. The theorem to prove will be chosen from a list built during the course.