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In 2021, the topic of this course is DIFFERENTIAL GEOMETRY

Introduction

Differential geometry is a classical subject. It is the mathematical study of geometry using calculus and differential equations. The motivation for the subject comes from navigation and map making. In 1569, the cartographer Mercator experimentally discovered his celebrated map projection in which courses of constant bearing are represented by straight lines. This was a hundred years before Newton and Leibniz introduced the differential calculus needed to understand Mercator's projection fully. Nowadays, there are many map projections with various subtle and useful properties but none is totally accurate. The fundamental reason for this is Gauss' Theorema Egregium proved in 1821, which says that a quantity now called the Gaussian curvature is intrinsically defined by measurements of distance alone. As the Gaussian curvature of the unit sphere is everywhere 1 and that of the plane is everywhere 0, any map of the earth necessarily distorts distance. In 1854, Riemann extended Gaussian curvature to higher dimensions. The resulting Riemann curvature tensor was developed and extended over the next 50 years and the differential geometry of smooth metrics was born. This theory is central in both mathematics and physics; for example, Einstein's equations of general relativity are just restrictions on the curvature of space-time. The terminology of modern differential geometry reflects its historical roots: one speaks of a manifold defined by an atlas of smooth charts et cetera and even the word geometry comes from Greek, roughly meaning measurement of earth.

Topics 

* Motivation (and for smooth manifolds#)
~ Navigation and cartography
~ Celestial mechanics
~ Mathematical physics

* Surfaces in three-space
~ Inverse and Implicit Function Theorems in three dimensions
~ Euclidean normalisation: Gaussian and mean curvature
~ Affine normalisation: the Pick invariant
~ Statement of Gauss' Theorema Egregium

* Smooth manifolds#
~ Flora and fauna: vector fields, tangent and co-tangent bundles
~ Calculus on manifolds: exterior derivative, de Rham complex
~ Vector bundles: connections and curvature

* Riemannian geometry
~ Torsion and curvature
~ Proof of Theorema Egregium
~ The Levi-Civita connection and Riemannian curvature
~ Examples
- Spaces of constant curvature: hyperbolic space
- Kaehler geometry: complex projective space

* Non-Riemannian geometry
~ Lorentzian geometry: general relativity, Einstein's equations
~ Projective differential geometry: invariants and curvature
~ Conformal differential geometry: invariants and curvature
~ Other differential geometries

# There will be some overlap with Dr David Baraglia's Topics B course on Lie Algebras.

Learning Outcomes

On successful completion of this course, students will be able to
1. Define Riemannian manifolds, and understand curvature both conceptually and computationally;
2. Define other differential geometric structures and understand key examples;
3. State and prove Gauss' Theorema Egregium;
4. Understand the construction of the Levi-Civita connection and its role in Riemannian differential geometry;
5. Construct the basic invariants of projective and conformal differential geometry.

Prerequisites

The course requires an adequate knowledge of linear algebra and multivariable calculus. It is recommended, but not required, that students in this course also take PURE MTH 4012 - Pure Mathematics Topic B - Honours on Lie Algebras given by Dr David Baraglia, in which many further examples of smooth manifolds will occur in the context and mathematical study of symmetry.

There are no required resoures for this course but lecture notes will be provided.

There are many excellent books on differential geometry in the Barr Smith Library. Browse the shelves, especially around 514.764, or consult from the following short selection.

* T. Aubin, A Course in Differential Geometry, American Mathematical Society 2001.
* S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, Springer 1987, 1990, 2004.
* S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-time, Cambridge University Press 1973.
* S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society 2001.
* J.M. Lee, Manifolds and Differential Geometry, American Mathematical Society 2009.
* J.M. Lee, Introduction to Smooth Manifolds, Springer 2003, 2013.
* P.W. Michor, Topics in Differential Geometry, American Mathematical Society 2008.

This course will have an active MyUni website.

The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

The following table is a guide to the workload for each component of the course. 

Activity	Quantity	Workload hours
Lecture	30	90
Assignments	6	66
Total		156

1) Introduction to smooth manifolds and associated gadgets (10 Lectures)
2) Riemannian geometry (10 lectures)
3) Projective, conformal, and other differential geometries (10 lectures)

None.

Assessment task	Task type	Due	Weighting	Learning outcomes
Examination	Summative	Examination period	70%	all
Homework assignment	Formative and summative	One week after assigned	30%	all

An aggregate score of 50% is required to pass the course.

There will be a total of 6 homework assignments, given out at intervals of about two weeks.

Homework assignments must be given to the lecturer in person or emailed as a pdf. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.