B3.2 Geometry of Surfaces (2024-25)
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- Lecturer: Profile: Richard Earl
Course information
General Prerequisites:
Part A Topology is recommended. Multidimensional Analysis and Geometry 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. Orientability and the Euler characteristic. Classification theorem for compact surfaces (the proof will not be examined).
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.
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.
The questions cover the various background material that is useful to the course - first year geometry, second year complex analysis, topology, projective geometry and multivariable calculus.
Do not be put off by this list of courses as only elements of each course are relevant. Supporting reading appears in the background material in Chapter 0 of the lecture notes.
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This sheet covers the classification theorem for topological surfaces and also has some early examples of smooth, geometric surfaces. Hand in solutions to the Part B exercises. Solutions are provided for the Part A (routine and optional) and Part C (optional extension) exercises.
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This sheet covers the local geometry of smooth surfaces, the fundamental forms, curvature and the Theorema Egregium. Hand in solutions to the Part B exercises. Solutions are provided for the Part A (optional and mainly calculations) and Part C (optional extension) exercises.
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Exercises on geodesics, the Gauss-Bonnet theorem and applications and the hyperbolic plane. Hand in solutions to Part B. There are also optional more routine questions (Part A) and optional extension questions (Part C ) if helpful.
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This sheets covers the later of the material on hyperbolic surfaces, and the material on Riemann surfaces including the Riemann-Hurwitz formula.
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I will not lecture the synopsis in the order it is published. Instead the Riemann surface material will be moved to the end of the course and, after the classification theorem, I will move directly on to the local and global real differential geometry.
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Registration start: Monday, 7 October 2024, 12:00 PMRegistration end: Friday, 8 November 2024, 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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