# C3.3 Differentiable Manifolds - Material for the year 2020-2021

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*A5: Topology* and *ASO: Introduction to Manifolds* are strongly recommended. (Notions of Hausdorff, open covers, smooth functions on **R^n** will be used without further explanation.) Useful but not essential: *B3.2 Geometry of Surfaces*.

16 lectures.

### Assessment type:

- Written Examination

A manifold is a space such that small pieces of it look like small pieces of Euclidean space. Thus a smooth surface, the topic of the Geometry of Surfaces course, is an example of a (2-dimensional) manifold.

Manifolds are the natural setting for parts of classical applied mathematics such as mechanics, as well as general relativity. They are also central to areas of pure mathematics such as topology and certain aspects of analysis.

In this course we introduce the tools needed to do analysis on manifolds. We prove a very general form of Stokes' Theorem which includes as special cases the classical theorems of Gauss, Green and Stokes. We also introduce the theory of de Rham cohomology, which is central to many arguments in topology.

The candidate will be able to manipulate with ease the basic operations on tangent vectors, differential forms and tensors both in a local coordinate description and a global coordinate-free one; have a knowledge of the basic theorems of de Rham cohomology and some simple examples of their use; know what a Riemannian manifold is and what geodesics are.

Smooth manifolds and smooth maps. Tangent vectors, the tangent bundle, induced maps. Vector fields and flows, the Lie bracket and Lie derivative.

Exterior algebra, differential forms, exterior derivative, Cartan formula in terms of Lie derivative. Orientability. Partitions of unity, integration on oriented manifolds.

Stokes' theorem. De Rham cohomology. Applications of de Rham theory including degree.

Riemannian metrics. Isometries. Geodesics.

- M. Spivak,
*Calculus on Manifolds*, (W. A. Benjamin, 1965). - M. Spivak,
*A Comprehensive Introduction to Differential Geometry*, Vol. 1, (1970). - W. Boothby,
*An Introduction to Differentiable Manifolds and Riemannian Geometry*, 2nd edition, (Academic Press, 1986). - M. Berger and B. Gostiaux,
*Differential Geometry: Manifolds, Curves and Surfaces*. Translated from the French by S. Levy, (Springer Graduate Texts in Mathematics, 115, Springer--Verlag (1988)) Chapters 0-3, 5-7. - F. Warner,
*Foundations of Differentiable Manifolds and Lie Groups*, (Springer Graduate Texts in Mathematics, 1994). - D. Barden and C. Thomas,
*An Introduction to Differential Manifolds.*(Imperial College Press, London, 2003.)