B3.2 Geometry of Surfaces (2021-22)
Main content blocks
- Lecturer: Profile: Dominic Joyce
Course information
General Prerequisites:
Part A Topology is recommended. Introduction to Manifolds would be useful but not essential. Also, B3.2 is helpful, but not essential, for B3.3 (Algebraic Curves).
Course Term: Michaelmas
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: H
Assessment Type: Written Examination
Course Overview:
Different ways of thinking about surfaces (also called two-dimensional manifolds) are introduced in this course: first topological surfaces and then surfaces with extra structures which allow us to make sense of differentiable functions (`smooth surfaces'), holomorphic functions (`Riemann surfaces') and the measurement of lengths and areas ('Riemannian 2-manifolds').
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).
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).
Learning Outcomes:
Students will be able to implement the classification of surfaces for simple constructions of topological surfaces such as planar models and connected sums; be able to relate the Euler characteristic to branching data for simple maps of Riemann surfaces; be able to describe the definition and use of Gaussian curvature; know the geodesics and isometries of the hyperbolic plane and their use in geometrical constructions.
Course Synopsis:
The concept of a topological surface (or 2-manifold); examples, including polygons with pairs of sides identified. Orientation and the Euler characteristic. Classification theorem for compact surfaces (the proof will not be examined).
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.
Smooth surfaces in Euclidean three-space and their first fundamental forms. The concept of a Riemannian 2-manifold; isometries; Gaussian curvature.
Geodesics. The Gauss-Bonnet Theorem (statement of local version and deduction of global version). Critical points of real-valued functions on compact surfaces.
The hyperbolic plane, its isometries and geodesics. Compact hyperbolic surfaces as Riemann surfaces and as surfaces of constant negative curvature.
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.
Smooth surfaces in Euclidean three-space and their first fundamental forms. The concept of a Riemannian 2-manifold; isometries; Gaussian curvature.
Geodesics. The Gauss-Bonnet Theorem (statement of local version and deduction of global version). Critical points of real-valued functions on compact surfaces.
The hyperbolic plane, its isometries and geodesics. Compact hyperbolic surfaces as Riemann surfaces and as surfaces of constant negative curvature.
Section outline
-
-
These are Nigel Hitchin's 2013 Geometry of Surfaces lecture notes. I based my lectures on them, and they are the primary reference for the course.
-
These are the handwritten slides for the pre-recorded lectures, in one file. Divided into Lectures 1-18 and sections 1-5.
-
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 slides, on topological surfaces, Hitchin notes chapter 2. Sample solutions are on the course web page.
-
This is the problem sheet for the first class in weeks 2 or 3. It is based on the first four lectures on topological surfaces, sections 1-2 in the slides, Hitchin notes chapter 2. Please hand in solutions.
-
This is the problem sheet for the second class in weeks 4 or 5. It is based on lectures 5-8 on Riemann surfaces, section 3 in the slides, Hitchin notes chapter 3. Please hand in solutions.
-
This is the problem sheet for the third class in weeks 6 or 7. It is based on lectures 9 - 15 on smooth surfaces and Riemannian metrics as far as the statement of the Gauss-Bonnet Theorem, sections 4.1-4.10 in the slides, Hitchin notes sections 4.1-4.6. Please hand in solutions.
-
This is the problem sheet for the fourth class in week 8 or week 1 of HT22. It is based on lectures 14 - 18 on geodesics, critical points, and the hyperbolic plane, sections 4.9, 4.11 and 5 in the slides, Hitchin notes sections 4.6-4.7 and 5. Please hand in solutions.
-