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Module MA3429: Differential geometry I

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2012-13
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof. Peter Taylor
 
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;
 
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
None
Assessment Detail
This module will be examined jointly with MA4448 in a 3-hour examination in Trinity term, except that those taking just one of the two modules will have a 2 hour examination. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session.