# MAGIC063: Differentiable Manifolds

## Course details

A core MAGIC course

### Semester

Autumn 2021
Monday, October 4th to Friday, December 10th

### Hours

Live lecture hours
20
Recorded lecture hours
0
80

### Timetable

Tuesdays
11:05 - 11:55 (UK)
Thursdays
11:05 - 11:55 (UK)

## Description

This course is designed for PhD students in pure or in applied mathematics and should also be of interest to those in mathematical physics.

Smooth manifolds underlie a great deal of modern mathematics: differential geometry, global analysis, the theory of Lie groups, geometric mechanics and much else, as well as large areas of mathematical physics.

The main part of this course will cover the basic theory of smooth manifolds and smooth maps, vector fields and differential forms, and  vector bundles. These are irreducible requirements for work with smooth manifolds.

If time permits the course will also cover basic geometry of vector bundles, and introduction to Riemannian geometry.

The course will concentrate on how to work with smooth manifolds. Most proofs will be only sketched, but references for complete arguments will be provided. I hope that at the end of the course you will be able to make use of the literature to learn more of what is particularly important for you in your own work.

There will be twenty live lecture sessions. The pdf slides will be also available in due course.

If you are enrolled in the course, or considering enrolling, please feel free to email me any questions or comments about the course.

### Prerequisites

Calculus of several variables.

Linear algebra (axioms of a vector space, linear operators in finite dimensions, bases, inner product spaces, dual spaces).

Basic topology of Euclidean spaces (open and closed sets, compactness, open covers).

The differential geometry of curves and surfaces in three dimensional Euclidean space will be used for illustration.

A knowledge of point-set topology is desirable.

### Syllabus

• Overview of the point-set topology. Topological manifolds. Examples.
• Smooth manifolds, smooth maps, tangent vectors.
• Regular Value Theorem, immersions and embeddings.
• Notion of vector bundle. Basic operations and constructions.
• Vector fields and their flows, Lie derivative.
• Tensor fields and differential forms.
• Connections on vector bundles.
• Topics on Riemannian Geometry (if time permits).

## Lecturer

• GK

### Dr Gerasim Kokarev

University
University of Leeds

## Bibliography

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

## Assessment

The assessment for this course will be released on Monday 10th January 2022 at 00:00 and is due in before Sunday 23rd January 2022 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.