Riemannian Geometry - MAT00052M

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  • Department: Mathematics
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2022-23

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

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.

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Special assessment rules

None

Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 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).