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MAU34301 Differential geometry

Module Code MAU34301
Module Title Differential geometry
Semester taught Semester 1
ECTS Credits 5
Module Lecturer Prof. Sergey Frolov
Module Prerequisites MAU23206 Calculus on manifolds

Assessment Details

  • This module is examined in a 2-hour examination at the end of Semester 1.
  • Students are assessed based on the exam alone.
  • Any failed components are reassessed, if necessary, by an exam in the reassessment session.

Contact Hours

11 weeks of teaching with 3 lectures and 1 tutorial per week.

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 the 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, and 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 for a given line element.
  • Define tensor densities, construct chart-invariant volume and surface elements for curved Lorentzian manifolds, and thus construct well-defined covariant volume and surface integrals for such manifolds.

Recommended Reading

  • Modern geometry, methods and applications, parts I and II, by Dubrovin, Fomenko and Novikov.
  • Geometrical methods of mathematical physics by B. Schutz.
  • Differential geometry of manifolds by S. Lovett.
  • Lecture notes on general relativity by S.M. Carroll (available here).
  • Advanced general relativity by S. Winitzki (available here).