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MAU34301 Differential geometry

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.
The module is passed if the overall mark for the module is 40% or more. If the overall mark for the module is less than 40% and there is no possibility of compensation, the module will be reassessed as follows:
1) A failed exam in combination with passed continuous assessment will be reassessed by an exam in the supplemental session;
2) The combination of a failed exam and failed continuous assessment is reassessed by the supplemental exam;
3) A failed continuous assessment in combination with a passed exam will be reassessed by one or more summer assignments in advance of the supplemental session.
Capping of reassessments applies to Theoretical Physics (TR035) students enrolled in this module. See full text at https://www.tcd.ie/teaching-learning/academic-affairs/ug-prog-award-regs/derogations/by-school.php Select the year and scroll to the School of Physics.

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

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.