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MA3H5 content

Content:
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.

Syllabus:
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.

Books:
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