- Department: Mathematics
- Module co-ordinator: Dr. Ian McIntosh
- Credit value: 10 credits
- Credit level: M
- Academic year of delivery: 2022-23
- See module specification for other years: 2021-22
Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
- None
Pre-requisite knowledge for MSc students: standard calculus, first courses on linear algebra, vector calculus, and differential geometry.
Occurrence | Teaching period |
---|---|
A | Spring Term 2022-23 |
To introduce the concept of a manifold as a space with a locally Euclidean smooth structure.
To examine Riemann's notion of an intrinsic method for measuring distance on a manifold.
To study the curvature and geodesics of Riemannian manifolds and obtain some geometric consequences.
The concept of a manifold, the intrinsic idea of tangent vector fields and differential 1-forms, and how these provide a framework for differential calculus with many applications (for example in General Relativity).
The notion of a Riemannian metric, and how it generalises the first fundamental form of surfaces in Euclidean space.
Connections (or covariant derivatives), parallel transport and curvature, and how to apply them in Riemannian geometry.
The local theory of geodesics, and their interaction with the global structure of the manifold.
Simple aspects of the notion of Riemannian curvature.
Syllabus
Manifolds and smooth functions, with examples.
Tangent vector fields and the tangent bundle.
The differential, of maps and 1-forms.
Connections and covariant derivatives of vector fields and other tensors.
Riemannian metrics and the Levi-Civita connection: the fundamental theorem of Riemannian geometry.
Geodesics and parallel transport; geodesic completeness.
Riemannian and sectional curvature.
Task | Length | % of module mark |
---|---|---|
Closed/in-person Exam (Centrally scheduled) Riemannian Geometry |
2 hours | 100 |
None
Task | Length | % of module mark |
---|---|---|
Closed/in-person Exam (Centrally scheduled) Riemannian Geometry |
2 hours | 100 |
Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
I Chavel, Riemannian geometry - a modern introduction, CUP 1995 (S3.85 CHA).