Introductory description

Smooth manifolds are generalisations of the notion of curves and surfaces R^3 and provide a rigorous mathematical concept of space as well as a natural setting for analysis. They form a fundamental part of modern mathematics and are used widely in pure and applied subjects such as differential geometry, general relativity and partial differential equations. Almost all the manifolds discussed in the course will be assumed to have a smooth structure. This allows us to perform the standard operations of differentiation in a very general context. We will begin by discussing manifolds embedded in euclidean space, R^n. We develop some of the basic notions such as smooth maps and tangent spaces in this context. We go on to describe the notion of an abstract manifold, and explain how these concepts generalise. We discuss basic concepts of orientability, differential forms and integration. Time permitting, we will aim to give a proof of the general form of Stokes's Theorem. We may briefly touch on subjects such as riemannian manifolds and Lie groups. These are the subject of dedicated courses in the 4th year.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Manifolds in euclidean space, smooth maps, tangent spaces, immersions and submersions, tangent and normal bundles, orientations, abstract manifolds, vector bundles, partitions of unity, differential forms, integration, Stokes's theorem.

Learning outcomes

By the end of the module, students should be able to:

• Understand one important ideas that underlie the theory of differentiable manifolds and to develop skills in manipulation of differentiable objects.
• Prove that various sets are manifolds by explicit construction of charts.
• Construct manifolds using the implicit function theorem.
• Understand the definition of tangent space of a manifold, and perform various constructions related to vector fields.
• Manipulate differential forms and understand Stokes theorem.

Indicative reading list

Tu, L. W. An Introduction to Manifolds, Springer-Verlag
Lee, J .M. Introduction to Smooth Manifolds, Springer-Verlag
Warner, F. Foundations of differentiable manifolds and Lie groups, Springer-Verlag
Boothby, W. An introduction to differentiable manifolds and Riemannian geometry, Academic Press

Subject specific skills

The course will expose students to a wide range of ideas in
a subject of broad applicability in mathematics and the sciences.
The students will learn to familiarise themselves with abstract
concepts, and to relate them to physical intuition.

Transferable skills

They will learn to follow complex reasoning, to construct logical arguments, to think independently, and to develop research skills.