##
Graduate lecture course, 10 hours, Michaelmas Term 2018, week 1.

##
Professor Joyce

## Introduction to Differential Geometry

## EPSRC CDT in Partial Differential Equations, foundation module.

### Tuesday 9th October 9.0am-11.0am, room C3

### Tuesday 9th October 1.45pm-4.0pm, room C3

### Wednesday 10th October 1.45pm-4.0pm, room C3

### Thursday 11th October 1.45pm-4.00pm, room C3

### Friday 12th October 1.45pm-4.0pm, room C3,

### plus problem class, Thursday 18 October (2nd week) 2.0-3.30pm, room
C1.

### Work for the problem class should
be handed in to Jonathan Whyman by 12.0 on Wednesday 17 October.

###

### Overview

The aim of the course is to familiarize students with the basic
language of differential geometry, and the beginnings of Riemannian
geometry.

### Important note for students:

Nobody seems very happy with this course. On the one hand, people
(particularly lecturers of subsequent courses) often ask me to put more
material in, to provide background for later courses. On the other
hand, students who have not studied manifolds before tend to find the
pace of the course too fast, may not get much out of it, and
ask me to take material out. I can't please everybody. This year I am
expanding from 8 hours to 10, in an effort to go more slowly. For
comparison, the first half of the course (4-5 hours) covers essentially
the same material I will cover in my 16 hour lecture course on C3.3
Differentiable Manifolds this term (to which you are welcome to come).
It is intended as a quick review, not to teach material from scratch.

Anyway: I hope that students who have already done an
undergraduate course on manifolds and differential geometry (and
haven't forgotten it) will cope. For students who haven't studied
manifolds before, my strong
recommendation is that you should do some advanced reading on
manifolds before the course, e.g. from the reading list below (the
Hitchin lecture notes are free to download), and you could look through
the PDF slides below.

### Synopsis (2 lectures = one 2.5 hour slot)

Lecture 1. Definition of
smooth manifolds X
and smooth maps f : X → Y by atlas of
charts. Examples. Vector bundles. The tangent bundle TX
and cotangent bundle T^{*}X.

Lecture 2. Immersions,
embeddings, and submanifolds. Submersions, Transverse fibre products.

Lecture 3. Tensors
and index notation T^{ab...}_{cd...},
behaviour under change of coordinates. Lie brackets of vector fields
and Lie derivatives. Exponentiating vector fields.

Lecture 4. Exterior
forms, de Rham differential d (definition in coordinates, and
coordinate independence). Brief introduction to homology and
cohomology. De Rham cohomology. Examples.

Lecture 5. Orientations on
manifolds.
Interpretation of integration in Differential Geometry, as integration
of a smooth n-form over an
oriented n-dimensional
manifold. Stokes' Theorem. Applications to de Rham cohomology. Examples.

Lecture 6. Connections
on
vector bundles: what they are, and why we need a
connection to differentiate something. Curvature of connections.
Connections on TX and
torsion.

Lecture 7. Riemannian
metrics g_{ij}.
Explanation in terms of lengths of curves.
The Levi-Civita connection ∇ and the Fundamental Theorem of Riemannian
geometry. Riemann curvature, Ricci curvature, and scalar curvature.
Volume forms on (oriented) Riemannian manifolds, and integrating
functions. Lebesgue spaces and Sobolev spaces.

Lecture 8. Examples. Riemannian
2-manifolds and surfaces in R^{3}.
Geodesics.

Lecture 9. Lie groups and Lie
algebras. Examples of Lie groups. Lie algebras of Lie groups.
Fundamental group, simply-connected spaces, universal covers.

Lecture 10. Lie's
theorems relating Lie algebras to connected, simply-connected Lie
groups. The classification of simple Lie algebras over C. Real
forms of Lie algebras. Principal bundles, frame bundles, and G-structures.

### Reading

- John M. Lee, ‘Introduction
to Smooth Manifolds’,
second edition, Graduate Texts in Mathematics 218, Springer, 2013.
- Nigel Hitchin, ‘Differentiable
manifolds’, Oxford lecture notes for course C3.3, 2016, PDF file

- Manfred P. do Carmo, ‘Riemannian
Geometry',
Birkhäuser, 1992.
** **

**Further reading**

** **
- Frank W. Warner, ‘Foundations
of differentiable
manifolds and Lie groups’, Scott, Foresman and Co., 1971.
- Jurgen Jost, ‘Riemannian
Geometry and Geometric
Analysis’, Universitext, Springer, 1995-2011.
- John M. Lee, ‘Riemannian
manifolds’, Graduate
Texts in Mathematics 176, Springer, 1997.
- Antoni A. Kosinski, ‘Differential
Manifolds’,
Academic Press, 1993.
- Roger Carter, Graeme Segal and Ian MacDonald, 'Lectures on Lie Groups and Lie Algebras',
L.M.S. Student Texts 32, Cambridge University Press, 1995.

**PDF files to download:**

**Synopses**

You may wish to print the slides out and bring them to the
lectures. The slides have occasional gaps where I intend to draw a
picture on the board; the gap is for you to draw the picture on the
printout.

**Slides for first session**

**Slides for second session**

**Slides for third session**

**Slides for fourth session**

**Slides for fifth session**

**Problem sheet**