C3.3 Differentiable Manifolds (2026-27)
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- Lecturer: Profile: Jason Lotay
Course information
General prerequisites:
A5: Topology and ASO: Multidimensional Analysis and Geometry are strongly recommended. (Notions of Hausdorff, open covers, smooth functions on Rⁿ will be used without further explanation.) Useful but not essential: B3.2 Geometry of Surfaces.
Course term: Michaelmas
Course lecture information: 16 lectures.
Course weight: 1
Course level: M
Assessment type: Written Examination
Course overview:
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.
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.
Learning outcomes:
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.
Course synopsis:
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.
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.
Section outline
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These are my Differentiable Manifolds lecture notes, which is what the lecture course will be based on.
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These are Nigel Hitchin's 2014 Differentiable Manifolds lecture notes. You may find them helpful as an additional learning resource.
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This is an introductory sheet which will not be marked. It covers properties of smooth maps between Rᵐ and Rⁿ, and examples of manifolds defined by atlases.
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Opened: Wednesday, 14 October 2026, 11:10 AMThis sheet covers the topics: definition of manifolds, smooth maps, regular value theorem, tangent spaces and quotients.
Section A contains more elementary questions. Section B contains material to test understanding of the course. Section C contains further questions which are optional for the class, but useful for further learning. No solutions should be submitted for marking.
Solutions to the sheet will be provided before the class. -
These sheet concerns various types of smooth maps, vector fields and differential forms.
Section A contains more elementary questions. Section B contains material to test understanding of the course. Section C contains further questions which are optional for the class, but useful for further learning.
Only Section B should be submitted for marking.
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Registration start: Monday, 5 October 2026, 12:00 PMRegistration end: Friday, 6 November 2026, 12:00 PM
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Class Tutor's Comments Assignment
Class tutors will use this activity to provide overall feedback to students at the end of the course.
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