Dominic Joyce, Introduction to Differential Geometry, Graduate Summer School, Nairobi 2019

Overview

Differential Geometry is the study of (smooth) manifolds. Manifolds are multi-dimensional spaces that locally (on a small scale) look like Euclidean n-dimensional space Rn, but globally (on a large scale) may have an interesting shape (topology). For example, the surface of a football (sphere) and the surface of a donut (torus) are 2-dimensional manifolds. Often one studies manifolds with a geometric structure, such a Riemannian metric, which tells you the lengths of curves on a manifold.

Manifolds are the language in which much of theoretical physics and physical applied mathematics is written. For example, Einstein's General Relativity models the universe as a 4-dimensional manifold U with a Lorentzian metric g, which encodes distance in space and duration in time. Things like electromagnetic fields, pizza and people are given by additional geometric structures on U.

I hope to explain topics including manifolds and submanifolds, smooth maps, submanifolds, vector bundles, tensors, de Rham cohomology, connections on vector bundles and curvature, Lie groups.

I will be aiming the course at mathematics MSc and PhD students, so people who don't have a good background in geometry and topology may find the course goes a bit fast for them. If you think this may apply to you, I encourage you to do some reading in advance of the summer school, for example from the reading list below, and to look at the slides of the lectures below.

What follows is material for a 10 hour graduate lecture course. In the summer school I will probably do rather less than this, depending on what people want; for example, I could just cover the first 5 lectures, and then move on to complex manifolds.

Synopsis

Lecture 1. Definition of smooth manifolds X and smooth maps f : X → Y by atlases 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.

Reading

Freely available online:

  1. Nigel Hitchin, ‘Differentiable manifolds’, Oxford lecture notes, 2014, PDF file. Used with permission.
  2. Nigel Hitchin, ‘Geometry of Surfaces ’, Oxford lecture notes, 2013, PDF file. Used with permission.
  3. Rob van der Vorst, 'Introduction to differentiable manifolds', PDF file, free online lecture notes available at https://www.few.vu.nl/~vdvorst/notes-2012.pdf.

Freely available, good background though less directly relevant:

  1. Renzo Cavalieri, 'Introduction to Topology', PDF file, available free at the author's webpage at https://www.math.colostate.edu/~renzo/teaching/Topology10/Notes.pdf. (Essential background if you haven't studied topology before.)
  2. Alexander Kirillov, 'Introduction to Lie groups and Lie Algebras', PDF file, available free at the author's webpage at https://www.math.stonybrook.edu/~kirillov/mat552/liegroups.pdf.
  3. Allen Hatcher, 'Algebraic Topology', PDF file. A classic, available free at the author's webpage at https://pi.math.cornell.edu/~hatcher/AT/AT.pdf.

Books:

  1. John M. Lee, ‘Introduction to Smooth Manifolds’, second edition, Graduate Texts in Mathematics 218, Springer, 2013. (My favourite book on manifolds.)
  2. Manfred P. do Carmo, ‘Riemannian Geometry', Birkhäuser, 1992.
  3. Loring W. Tu, 'An Introduction to Manifolds', Springer, 2011.

Further reading:

  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. (My favourite book for learning about Lie groups.)

Some nearly infallible books by me, fairly advanced, but which will make you into a better person if you read them:

  1. D. Joyce, 'Compact manifolds with special holonomy', Oxford Mathematical Monographs series, OUP, 2000.
  2. D. Joyce, 'Riemannian holonomy groups and calibrated geometry', Oxford Graduate Texts in Mathematics 12, OUP, 2007.

PDF files of the lecture course to download:

Slides for first session.

Slides for second session.

Slides for third session.

Slides for fourth session.

Slides for fifth session.

Question sheets:

One short problem sheet on the whole course: Problem sheet -1.

A series of five problem sheets on Differentiable Manifolds:

Problem sheet 0.

Problem sheet 1.

Problem sheet 2.

Problem sheet 3.

Problem sheet 4.

A sheet of miniprojects on Differentiable Manifolds:

Miniprojects.

A series of four problem sheets on Lie groups:

Problem sheet 1.

Problem sheet 2.

Problem sheet 3.

Problem sheet 4.

Other possibly useful stuff:

A handout on fundamental groups of topological spaces and simple-connectedness, needed to state theorems about Lie groups PDF file

A handout on Lie groups PDF file

LaTeX source files of the lecture course:

If some of you would like to go back to your own universities and give a lecture course on differential geometry using my course materials, which you may edit and adapt as you wish, you are welcome to do so. They are written in LaTeX using the "beamer" documentclass. You can download the LaTeX source files, as a directory compressed as a ZIP file, here: LaTeX source files.