MTH5111 - Differential geometry - 2019

Manifolds are topological spaces that are locally homeomorphic to Euclidean space. A differentiable structure on a manifold makes it possible to generalize many concepts from calculus in Euclidean spaces to manifolds. This is an course on differentiable manifolds and related basic concepts, which are the common ground for differential geometry, differential topology, global analysis, i.e. calculus on manifolds including geometric theory of integration, and modern mathematical physics.

Foundational topics covered in the unit include: Smooth manifolds and coordinate systems, tangent and cotangent bundles, tensor bundles, tensor fields and differential forms, Lie derivatives, exterior differentiation, connections, covariant derivatives, curvature, and Stokes's Theorem.

This unit will also cover advanced topics and applications such as: Degree Theory, de Rham cohomology, symplectic geometry, classical mechanics, the Hopf-Rinow theorem, Lie Groups and homogeneous spaces.

On completion of this unit students will be able to:

1. Apply expert differential geometric techniques to solve problems that arise in pure and applied mathematics.
2. Construct coherent and precise logical arguments.
3. Develop and extend current techniques in differential geometry so that they can be applied to new situations in novel ways.
4. Communicate complex ideas effectively.
5. Independently learn and assimilate new mathematical ideas and techniques.

NOTE: From 1 July 2019, the duration of all exams is changing to combine reading and writing time. The new exam duration for this unit is 3 hours and 10 minutes.

Examination (3 hours): 60% (Hurdle)

Continuous assessment: 40%

Hurdle requirement: To pass this unit a student must achieve at least 50% overall and at least 40% for the end-of-semester exam.

This unit is offered at both Level 4 and Level 5, differentiated by the level of the assessment. Students enrolled in MTH5111 will be expected to demonstrate a higher level of learning in this subject than those enrolled in MTH4111. The assignments and exam in this unit will use some common items from the MTH4111 assessment tasks, in combination with several higher level questions and tasks.

• 3 hours of lectures and 1 hour tutorial per week
• 10 hours of independent study per week

See also Unit timetable information

Master of Mathematics