Differential Geometry of Curves and Surfaces

In Hilary Term 2026, I offered a directed reading course on the differential geometry of curves and surfaces. This was not a formal lecture course; instead, weekly reading assignments were provided, followed by a one-hour class for discussion of the material.

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