B3.2 Geometry of Surfaces (2025-26)
Main content blocks
- Lecturer: Profile: Hulya Arguz
These geometric structures interact in a fundamental way with the topology of the surfaces. A striking example of this is given by the Euler number, which is a manifestly topological quantity, but can be related to the total curvature, which at first glance depends on the geometry of the surface.
The course ends with an introduction to hyperbolic surfaces modelled on the hyperbolic plane, which gives us an example of a non-Euclidean geometry (that is, a geometry which meets all of Euclid's axioms except the axiom of parallels).
Smooth surfaces in Euclidean three-space. Tangent space. Abstract topological and smooth surfaces. The fundamental forms. The concept of a Riemannian 2-manifold; isometries; Gaussian curvature and the Theorema Egregium.
Geodesics. The Gauss-Bonnet Theorem (statement of local version and deduction of global version). Critical points of real-valued functions on compact surfaces. The Poincaré-Hopf Theorem. Morse functions and the gradient field.
The hyperbolic plane, its isometries and geodesics. Compact hyperbolic surfaces.
Riemann surfaces; examples, including the Riemann sphere, the quotient of the complex numbers by a lattice, and double coverings of the Riemann sphere. Holomorphic maps of Riemann surfaces and the Riemann-Hurwitz formula. Elliptic functions.
Section outline
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This is an initial optional problem sheet to get you started before the classes. Do not hand in solutions. It is based on the first three lectures, sections 1-2.5 in the lecture notes, on topological surfaces, Hitchin notes chapter 2. Sample solutions are on the course web page.
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These notes were prepared by TeX-typing Dominic Joyce’s handwritten lecture slides from his recorded lectures in the 2020 COVID year. They closely follow his original material. A secondary reference throughout has been Nigel Hitchin’s lecture notes from 2013.
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Opened: Tuesday, 7 October 2025, 3:00 PMDue: Friday, 24 October 2025, 3:00 PM
This is the problem sheet for the first class in weeks 2 or 3. It is based on the first five lectures on topological surfaces, sections 1-2 in the lecture notes, Hitchin notes chapter 2. Please hand in solutions.
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Opens: Saturday, 25 October 2025, 3:00 PMDue: Friday, 7 November 2025, 3:00 PM
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Opens: Saturday, 8 November 2025, 3:00 PMDue: Friday, 21 November 2025, 3:00 PM
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Opens: Saturday, 22 November 2025, 3:00 PMDue: Sunday, 11 January 2026, 3:00 PM
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Registration start: Monday, 6 October 2025, 12:00 PMRegistration end: Friday, 7 November 2025, 12:00 PM
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Class Tutor's Comments Assignment
Class tutors will use this activity to provide overall feedback to students at the end of the course.
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