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

### 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 : XY 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 Tab...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 gij. 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 R3. 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.

1. John M. Lee, ‘Introduction to Smooth Manifolds’, second edition, Graduate Texts in Mathematics 218, Springer, 2013.
2. Nigel Hitchin, ‘Differentiable manifolds’, Oxford lecture notes for course C3.3, 2016, PDF file
3. Manfred P. do Carmo, ‘Riemannian Geometry’, Birkhäuser, 1992.

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

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