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 R^{n}, 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.

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

Lecture 3. Tensors and index notation T

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

Lecture 8. Examples. Riemannian 2-manifolds and surfaces in R

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.

- Nigel Hitchin, ‘Differentiable manifolds’, Oxford lecture notes, 2014, PDF file. Used with permission.
- Nigel Hitchin, ‘Geometry of Surfaces ’, Oxford lecture notes, 2013, PDF file. Used with permission.
- 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.

- 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.) - 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. - 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.

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

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

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

A series of five problem sheets on Differentiable Manifolds:

A sheet of miniprojects on Differentiable Manifolds:

A series of four problem sheets on Lie groups:

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

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.