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 involving 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;

Main Textbook

Modern Geometry, Methods and Applications. Part I and II, BA. Dubrovin, A.T. Fomenko, S.P. Novikov.

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

Module Prerequisite

MA2322 Calculus on Manifolds

Assessment Detail

This module will be examined in a 2-hour examination in Michaelmas term.