MAU34301 Differential geometry

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.