Module MA3429

Differential Geometry I


Michaelmas Term



Dr. Sergey Frolov

Office: 3.4 Hamilton Building
Office Hours: e-mail for appointment
Phone:  01 896 4027

Tutors / Graders

Paul Ryan,

                                                                             Main Textbook

B.A. Dubrovin, A.T. Fomenko, S.P. Novikov. Modern geometry, methods and applications. Part I and II

Course Description

Definition of a manifold, mappings of manifolds, tensors on manifolds: Part II, §1.1, 1.2
Algebraic operations on tensors, tensors of type (0,k): Part I, §17.2, 18.1, 18.2
Embeddings and immersions of manifolds, surface in Euclidean space, transformation groups as manifolds, projective spaces, elements of Lie groups, complex manifolds, homogeneous spaces: Part II, §1.3 - 5.3; Part I, §14.3
Vector bundles on a manifold, the concept of a smooth fibre bundle: Part II, §7.1, 7.2, 24.1
The behaviour of tensors under mappings, vector fields, the Lie derivative, Lie algebras and vector fields, left-invariant fields, invariant metrics: Part I, §22.1 - 24.5
Covariant differentiation, parallel transport of vector fields, geodesics, connections compatible with the metric, connections compatible with a complex structure, the curvature tensor: Part I, §28.1 - 30.3 Modern Geometry. Methods and Applications Part 2 The Geometry and Topology of Manifolds (Graduate Texts in Mathematics 93).djvu Modern Geometry. Methods and Applications Part 2 The Geometry and Topology of Manifolds (Graduate Texts in Mathematics 93).djvu

Recommended Reading

Geometrical Methods of Mathematical Physics, B. Schutz, (Cambridge University Press 1980);

Differential Geometry of Manifolds, S. Lovett, (AK Peters, Ltd. 2010)

Applied Differential Geometry, W.L. Burke, (Cambridge University Press 1985)

Lecture Notes on GR, Sean M. Carroll. Available here

Advanced General Relativity,Sergei Winitzki. Available here

Class Meetings

 Tutorials: Wednesday
Sniam Lecture Theatre Sniam




Salmon Theatre, Hamilton Building





LLoyd Building


There will be tutorials every week starting with 19 September 2016.

Learning Outcomes

On successful completion of this module, students will be able to:

Obtain a coordinate-induced basis for the tangent space and cotangent space at points of a differentiable manifold, construct a coordinate induced basis for arbitrary tensors and obtain the components of tensors in this basis;

Determine whether a particular map is a tensor by either checking multi-linearity or by showing that the components transform according to the tensor transformation law;

Construct manifestly chart-free definitions of the Lie derivative of a function and a vector, to compute these derivatives in a particular chart and hence compute the Lie derivative of an arbitrary tensor;

Compute, explicitly, the covariant derivative of an arbitrary tensor;

Define parallel transport, derive the geodesic equation and solve problems invloving parallel transport of tensors;

Obtain an expression for the Riemann curvature tensor in an arbitrary basis for a manifold with vanishing torsion, provide a geometric interpretation of what this tensor measures, derive various symmetries and results involving the curvature tensor;

Define the metric, the Levi-Civita connection and the metric curvature tensor and compute the components of each of these tensors given a particular line-element;

Define tensor densities, construct chart-invariant volume and surface elements for curved Lorentzian manifolds and hence construct well-defined covariant volume and surface integrals for such manifolds;



Two hour End of year Final examination.


  • Homework


  • Final Exam



Slides on Differentiable Manifolds

The file is here

Slides on Covariant Differentiation, Parallel Transport and Curvature Tensor

The file is here

Homework/Tutorial/Examination Questions 

These questions are here and the file should be checked regularly for updates

Solutions to some Homework Questions

The file is here, and it should be checked regularly for updates

Sample Final Exam Questions

The file is here, and it might be updated

Solutions to 2015 Final Exam Questions

The file is here

Solutions to 2016 Final Exam Questions

The file is here

Solutions to 2017 Final Exam Questions

The file is here

Solutions to 2018 Final Exam Questions

The file is here



Your Comments

You are welcome to submit any comments you have about this class here.