Differential Geometry of Curves and Surfaces

In Hilary Term 2026, I will be offering a directed reading course on the differential geometry of curves and surfaces. The course aims to provide an overview of the field, with the goal of enabling students to appreciate the Gauss-Bonnet Theorem and its applications by the end of the term. This is not a formal lecture course; instead, weekly reading assignments will be provided, followed by a one-hour class for discussion and exploration of interesting problems.

In Trinity Term 2026, students will be encouraged to participate in a presentation session, where they can create a short presentation on their favourite topics from the course.

For further departmental information, please visit: https://www.maths.ox.ac.uk/node/79639.

Please note, this is a provisional syllabus!

WEEK ONE: Curves

• Parameterised curves

• Reparameterisation and arc length

• Curvature

WEEK TWO: Surfaces

• Regular surfaces

• Tangent planes

• Surface area, isometries, and the first fundamental form

WEEK THREE: Curvature I

• The Gauss map

• The Weingarten map

• Gaussian curvature

• Mean curvature

WEEK FOUR: Curvature II

• Normal curvature

• Geometric characterisations of Gaussian curvature

• Some examples!

WEEK FIVE: Geodesics I

• Covariant derivatives

• Basics on geodesics

• Examples!

WEEK SIX: Geodesics II

• Complete surfaces

• Parallel transport

• Geodesics in local coordinates

• Christoffel symbols

WEEK SEVEN: Gauss-Bonnet Theorem

• Local Gauss-Bonnet Theorem

• Global Gauss-Bonnet Theorem

• Proof of the Theorem

WEEK EIGHT: Applications of the Gauss-Bonnet Theorem

This week will be reserved to see some interesting applications of the Gauss-Bonnet Theorem. E.G. Chern-Gauss-Bonnet Theorem, and example uses of the Theorem in proofs