Module description
Syllabus
Definition and examples of topological spaces and manifolds; functions between manifolds; the tangent space; the tangent bundle; vector fields; Lie derivatives; tensor fields; differential forms; exterior calculus; integration on manifolds; affine connections; torsion; curvature; covariant derivatives; parallel transport; manifolds with metrics; the Levi-Civita connection. If time permits, additional topics such as de Rham cohomology will be discussed.
Prerequisites
Knowledge of multivariate calculus, basic linear algebra and topology is sufficient. You are advised to have taken 5CCM211A and 5CCM226A Metric Spaces and Topology.
Assessment details
2 hour written examination.
Semester 1 only students will be set an alternative assessment in lieu of in-person exams in January.
Full year students will complete the standard assessment.
Educational aims & objectives
The module aims to provide an introduction to differential geometry both for students whose interests are in pure mathematics and for those who are studying theoretical physics and other areas of applied mathematics. The basic objects of study are manifolds, which allow one to translate familiar ideas from vector calculus to curved space. Applications to topology and theoretical physics will be discussed as time allows.
Teaching pattern
Two hours of lectures and one hour of tutorial per week throughout the term
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists