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Riemannian Geometry - MAT00052M

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• 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

Related modules

Additional information

Pre-requisite knowledge for MSc students: standard calculus, first courses on linear algebra, vector calculus, and differential geometry.

Module will run

Occurrence	Teaching period
A	Spring Term 2022-23

Module aims

• 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.

Module learning outcomes

• 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.

Module content

 

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.

Assessment

Task	Length	% of module mark
Closed/in-person Exam (Centrally scheduled) Riemannian Geometry	2 hours	100

Special assessment rules

None

Reassessment

Task	Length	% of module mark
Closed/in-person Exam (Centrally scheduled) Riemannian Geometry	2 hours	100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

I Chavel, Riemannian geometry - a modern introduction, CUP 1995 (S3.85 CHA).