School of Mathematics
MA3429 - Differential Geometry 2011-12 (SS Theoretical Physics
JS & SS Mathematics )
Lecturer: Prof. P. Taylor
Requirements/prerequisites:
Duration: Michaelmas Term, 11 weeks
Number of lectures per week: 3 including tutorials
Assessment:
ECTS credits: 5
End-of-year Examination: 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. However there will be separate results for MA3429 and MA4448.

Description:

Textbooks:

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 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, 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.
• Re-derive the geodesic equation from an action principle and compute null, timelike, or spacelike geodesics on a particular space-time.
• Derive the Einstein equations or equations for similar metric theories from an action principle.

Nov 10, 2011

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On 10 Nov 2011, 10:40.