Lecturer: Damiano Testa

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 hours

Assessment: 100% 3 hour examination

Formal registration prerequisites: None

Assumed knowledge:

MA259 Multivariable Calculus:

• Basic theory of differentiation, including statements (though not proofs) of Inverse and Implicit Function Theorems

MA222 Metric Spaces: 

• Integrations in several variables and familiarity with basic notions of point-set topology and metric spaces
• Topologies
• Continuity
• Compactness
• Connectedness
• Completeness

Useful background: MA254 Theory of ODEs - We will use some results about the existence, uniqueness and dependence on parameters of solutions to ODEs in one or two places of the course. These results will be stated and could be taken on faith, but some prior acquaintance might be useful.

Synergies: The theory of manifolds is fundamental in many areas of modern mathematics. Modules that go well with this Module are (of course some choice should be made depending on whether your tastes are more analytic, geometric or topological):

• MA3D9 Geometry of Curves and Surfaces
• MA3F1 Introduction to Topology
• MA3B8 Complex Analysis
• MA3H6 Algebraic Topology 
• MA3G1 Theory of Partial Differential Equations

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA4C0 Differential Geometry
• MA4E0 Lie Groups
• MA424 Dynamical Systems
• MA4A5 Algebraic Geometry

Content: The course will start by introducing the concept of a manifold (without recourse to an embedding into an ambient space). In the words of Hermann Weyl (Space, Time, Matter, paragraph 11):

“The characteristic of an n-dimensional manifold is that each of the elements composing it (in our examples, single points, conditions of a gas, colours, tones) may be specified by the giving of n quantities, the “co-ordinates,” which are continuous functions within the manifold. This does not mean that the whole manifold with all its elements must be represented in a single and reversible manner by value systems of n co-ordinates (e.g. this is impossible in the case of the sphere, for which n = 2); it signifies only that if P is an arbitrary element of the manifold, then in every case a certain domain surrounding the point P must be representable singly and reversibly by the value system of n co-ordinates.”

Thus the points on the surface of a sphere form a manifold. The possible configurations of a double pendulum (one pendulum hung off the pendulum bob of another) is a manifold that is nothing but the surface of a two-torus: the surface of a donut (a triple pendulum would give a three-torus etc.) The possible positions of a rigid body in three-space form a six-dimensional manifold. Colour qualities form a two-dimensional manifold (cf. Maxwell’s colour triangle).

It becomes clear that manifolds are ubiquitous in mathematics and other sciences: in mechanics they occur as phase-spaces; in relativity as space-time; in economics as indifference surfaces; whenever dynamical processes are studied, they occur as “state-spaces” (in hydrodynamics, population genetics etc.)

Moreover, in the theory of complex functions, the problem of extending one function to its largest domain of definition naturally leads to the idea of a Riemann surface, a special kind of manifold.

Although it seems so natural from a modern vantage point, it took some time and quite a bit of work (by Gauss, Riemann, Poincare, Weyl, Whitney, …) till mathematicians arrived at the concept of a manifold as we use it today. It is indispensable in most areas of geometry and topology as well as neighbouring fields making use of geometric methods (ordinary and partial differential equations, modular and automorphic forms, Arakelov theory, geometric group theory…)

Some buzz words suggesting topics which we plan to cover include:

• The notion of a manifold (in different setups), examples of constructions of manifolds (submanifolds, quotients, surgery)
• The tangent space, vector fields, flows/1-parameter groups of diffeomorphisms
• Tangent bundle and vector bundles
• Tensor and exterior algebras, differential forms
• Integration on manifolds, Stokes’ theorem
• de Rham cohomology, examples of their computation (spheres, tori, real projective spaces...)
• Degree theory, applications: argument principle, linking numbers, indices of singularities of vector fields.

We will also discuss a lot of concrete and interesting examples of manifolds in the lectures and work sheets, such as for example: tori, n-holed tori, spheres, the Moebius strip, the (real and complex) projective plane, higher-dimensional projective spaces, blow-ups, Hopf manifolds…

The nature of the material makes it inevitable that considerable time must be devoted to establishing the foundations of the theory and defining as well as clarifying key concepts and geometric notions. However, to make the content more vivid and interesting, we will also seek to include some attractive and non-obvious theorems, which at the same time are not too hard to prove and natural applications of the techniques introduced, such as, for instance, Ehresmann's theorem on differentiable fibrations, or that a sphere cannot be diffeomorphic to a product of (positive-dimensional) manifolds.

This Module is mathematically closely related to, but formally completely independent of MA3D9 Geometry of Curves and Surfaces.

Books:

Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer (1983) (esp. Chapters 1, 2, 4)

M. Berger, B. Gostiaux, Differential Geometry: Manifolds, Curves, and Surfaces, Springer (1988) (esp. Chapters 2-6)

Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1, Publish or Perish, Inc. (2005) (esp. Chapters 1-8)

John Lee, Introduction to Smooth Manifolds, Springer (2012)

Loring W. Tu, An Introduction to Manifolds, Springer (2011)

Additional Resources