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Module MA3429: Differential Geometry I
 Credit weighting (ECTS)

5 credits
 Semester/term taught

Michaelmas term 201415
 Contact Hours

11 weeks, 3 lectures including tutorials per week

 Lecturer Prof Sergy Frolov http://www.maths.tcd.ie/~frolovs/DifGeom/DGI.html

 Learning Outcomes
 On successful completion of this module, students will be able to:
 Obtain a coordinateinduced 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 multilinearity or by showing that the components transform according to the tensor transformation law;
 Construct manifestly chartfree 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 LeviCivita connection and the metric curvature tensor and compute the components of each of these tensors given a particular lineelement;
 Define tensor densities, construct chartinvariant volume and surface elements for curved Lorentzian manifolds and hence construct welldefined 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
 None
 Assessment Detail
 This module will be examined in a 2hour examination in Trinity term.