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# Module MA3429: Differential Geometry I

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2014-15
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 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;

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 2-hour examination in Trinity term.