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This course is part of the MAGIC core.


This course is designed for PhD students in pure or in applied mathematics. It should also be of interest to those in mathematical physics. In what follows, and in the course itself, I will write `smooth manifolds' since `smooth' is a shorter word (and because there has never been agreement as to whether the other word should be `differentiable' or `differential').

Smooth manifolds underlie a great deal of modern mathematics: differential geometry (of course), global work in differential equations, the theory of Lie groups, geometric mechanics and much else, as well as large areas of mathematical physics.

The main part of this course will cover the basic theory of smooth manifolds and smooth maps, vector fields and differential forms, the tangent and cotangent bundles and the general notion of vector bundle. These are irreducible requirements for work with smooth manifolds.

After that the course will cover one or both of (i) connections in vector bundles, and (ii) Poisson manifolds and their symplectic leaves.

The connection theory of vector bundles is part of differential geometry and is a good way to get a feel for curvature and for its relationship with tensor structures.

Poisson geometry is a relatively recent field. It provides an easy route into symplectic manifolds, and involves multivector fields and the Schouten (or Gerstenhaber) bracket, tools which are of wide use in many parts of mathematics and physics today.

The course will include some detailed proofs, but the main focus will be on giving a feel for the various topics and methods; I hope that at the end of the course you will be able to make use of the literature to learn more of what is particularly important for you in your own work.

The course is presented using a beamer file with support from the visualizer. There are also background notes with detailed references.

The course will be broadly similar to that in spring 2017 but there will be more use of coordinate formulations.

If you are enrolled in the course, or considering enrolling, please feel free to email me any questions or comments about the course.


Spring 2019 (Monday, January 21 to Friday, March 29)


  • Live lecture hours: 20
  • Recorded lecture hours: 0
  • Total advised study hours: 80


  • Mon 10:05 - 10:55
  • Wed 10:05 - 10:55


Calculus of several variables (integration, implicit function theorem). Linear algebra (axioms of a vector space, linear operators, bases).

The differential geometry of curves and surfaces in 3-dimensional Euclidean space is neither a prerequisite nor part of the syllabus, but if you know something about this you will see how it fits in as a special case.


Outline syllabus (details will be added gradually):
Basics of smooth manifolds and smooth maps
Basics of vector bundles
Tangent vectors, vector fields, and the tangent bundle
Differential forms and the cotangent bundle
Integration of differential forms
Connections in vector bundles
Poisson manifolds and their symplectic leaves


Multidimensional Real Analysis I: DifferentiationDuistermaat and Kolk
Multidimensional Real Analysis II: IntegrationDuistermaat and Kolk
Foundations of Mechanics: A Mathematical Exposition of Classical Mechanics ...Abraham and Marsden
A comprehensive introduction to differential geometry, Volume ISpivak
Differentiable Manifolds: A First CourseConlon
Foundations of differentiable manifolds and Lie groupsWarner
Connections, Curvature and Cohomology: Volume IGreub, Halperin and Vanstone
Connections, Curvature, and Cohomology: Volume IIGreub, Halperin and Vanstone
Introduction to Smooth ManifoldsLee
Treatise on analysis, Volume IIIDieudonné
Differential Geometry of Curves and Surfacesdo Carmo
Poisson StructuresLaurent-Gengoux, Pichereau and Vanhaecke
Lectures on the Geometry of Poisson ManifoldsVaisman
Introduction to Symplectic TopologyMcDuff and Salamon
Applicable Differential GeometryCrampin and Pirani
Differential Manifolds and Theoretical PhysicsW D Curtis and F R Miller
Geometrical methods of mathematical physicsSchutz


Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)


No assessment information is available yet.

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