Riemannian Geometry - MAT00052M

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  • Department: Mathematics
  • Module co-ordinator: Dr. Chris Wood
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2019-20

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Module will run

Occurrence Teaching cycle
A Spring Term 2019-20

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

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

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
University - closed examination
Riemannian Geometry
2 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
University - closed examination
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).



The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.