# Unit information: Geometry of Manifolds in 2021/22

Unit name Geometry of Manifolds MATHM0037 20 M/7 Teaching Block 1 (weeks 1 - 12) Dr. Hassannezhad Not open MATH20006 Metric Spaces and MATH20004 Introduction to Geometry None School of Mathematics Faculty of Science

## Description including Unit Aims

Lecturers: Asma Hassannezhad and Viveka Erlandsson

Unit Aims

The aim of the unit is to study foundations of the geometry of manifolds including the concept of an abstract manifold, Riemannian metric and curvature, and to provide students with a firm grounding in the theory and techniques in this area and to offer students ample opportunity to build on their problem-solving ability.

Unit Description

The study of manifolds is fundamental in many important areas of modern mathematics. Manifolds provide a natural setting for research in various areas including geometry, analysis, partial differential equation and mathematical physics. It generalises the notion of curves and surfaces in R3 as well as many concepts from linear algebra. This unit builds a foundation of abstract differentiable and Riemannian manifolds. A differentiable manifold locally looks like the Euclidean space Rn and we can generalise the notion of the inner product in Rn to an inner product on the tangent space of the manifold. This gives rise to the definition of a Riemannian metric. A differentiable manifold equipped with a Riemannian metric is called a Riemannian manifold. In this unit, we study the various differentiable and geometric structures of a manifold including differential forms, Lie derivative, geodesics and curvatures.

Syllabus

Topics covered will include:

• Differentiable Manifolds: definition of abstract manifolds, partition of unity, tangent spaces, vector fields on manifolds, Lie derivative, Lie groups and Lie algebras
• Riemannian manifolds: Riemannian metric, covariant derivative, exponential map, geodesics, curvatures and spaces of constant curvature

Relation to Other unit

This unit can be considered as a continuation and as an advance version of the second-year unit MATH20004 Introduction to Geometry. It is related to the 3rd-year unit MATH30018 Fields, Forms and Flows. It can be also complementary to the unit MATH30001/MATHM0008 Topics in Modern Geometry.

## Intended Learning Outcomes

By the end of the unit, students should have developed an understanding of basic definitions and results regarding differentiable manifolds, like tangent space, Riemannian metric and different types of curvatures. They should be able to solve routine problems and to apply the techniques of the unit to unseen situation

## Teaching Information

Lectures (3 hrs per week) and recommended problems.

## Assessment Information

The final mark is calculated as follows:

• 80% Exam
• 20% Coursework (2 assignments worth 10% each)

## Resources

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATHM0037).

How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours of study to complete. Your total learning time is made up of contact time, directed learning tasks, independent learning and assessment activity.