B3.2 Geometry of Surfaces (2023-24)
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 - to appear) 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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This chapter includes background material from Prelims Geometry, Part A Complex Analysis, Topology, Projective Geometry and Multidimensional Analysis and Geometry.
Only elements of each of the above courses are helpful. Sheet 0 will give you a sense of what topics are relevant. Solutions are available to that sheet.
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This is an introductory lecture giving a sense of the course as a whole. In particular it focuses on the different structures - topological, differential, orientable, metric, complex - that a surface can be endowed with and discusses the role of transition maps in ensuring the consistency of such structures. ERRORS NOTED IN LECTURE NOW CORRECTED
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The focus of this chapter is the classification of compact, connected surfaces up to homeomorphism: each homeomorphism class can be uniquely determined by its orientability and its Euler characteristic.
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This chapter discusses the local metric surfaces and also introduces the notion of an abstract geometric surface or Riemannian 2-manifold. The chapter concludes with a significant theorem showing that the curvature of a surface is an intrinsic quantity, dependent only on the surface's metric structure and no dependent on how the surface has been situated in Euclidean space.
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Geodesics on surfaces generalize the notion of lines in the plane. They are (locally) the curves of least distance.
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Here we meet some global theory of surfaces. The topology of a surface puts constraints on what properties functions, curvature and vector fields can, or must, have on a surface. One startling result is that the total curvature - an ostensibly geometric quantity -is determined by the surface's topology.
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We begin with a discussion of the hyperbolic plane, famously discovered as an example of a non-Euclidean geometry. Hyperbolic surfaces - geometric surfaces with constant Gaussian curvature -1 - can be formed by appropriately quotienting the plane by groups of isometries. It turns out the theory of hyperbolic surfaces is much richer than that of elliptic or Euclidean surfaces.
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In this, the final chapter, we study Riemann surfaces, surfaces with complex structure, or equally 1-dimensional complex manifolds. Their nature is quite different to that of smooth real surfaces. To begin Riemann surfaces are orientable, as biholomorphic maps are orientation-preserving. But biholomorphic maps are more 'rigid' than diffeomorphisms and this means that uncountably many different complex structures exist on the torus.
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