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Riemannian Geometry - MAT00101M

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  • Department: Mathematics
  • Module co-ordinator: Dr. Ian McIntosh
  • Credit value: 20 credits
  • Credit level: M
  • Academic year of delivery: 2024-25
    • See module specification for other years: 2023-24

Module summary

Riemannian geometry extends the study of differential geometry beyond curves and surfaces to look at geometry in any number of dimensions. The key concepts studied here are the Riemannian metric, which enables arc length to be defined and leads to the study of geodesics, and the Levi-Civita connection, which enables parallel transport and leads to the Riemannian curvature tensor.

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

While this module can be taken from a general background in pure mathematics, a first course in curves and surfaces would be an advantage.

 

(UG students, eg those from Natural Sciences or Physics, who have not taken Differential Geometry should consult the lecturer first if they wish to take this module).

Module will run

Occurrence Teaching period
A Semester 1 2024-25

Module aims

Riemannian geometry extends the study of differential geometry beyond curves and surfaces to look at geometry in any number of dimensions. The key concepts studied here are the Riemannian metric, which enables arc length to be defined and leads to the study of geodesics, and the Levi-Civita connection, which enables parallel transport and leads to the Riemannian curvature tensor.

Module learning outcomes

By the end of the module students will be able to demonstrate understanding of, and be able to calculate with:

  1. The concept of a manifold, including the intrinsic idea of tangent vector fields and differential 1-forms, and how these provide a framework for differential calculus.

  2. The notion of a Riemannian metric, and how it generalises the first fundamental form of surfaces in Euclidean space.

  3. Connections (or covariant derivatives) and parallel transport.

  4. Riemannian curvature and related measures of curvature.

  5. Geodesics and their relationship to length minimising curves.

Module content

The basic differential geometry of manifolds; connections, Riemannian metric and the Fundamental Theorem; Riemannian curvature and parallel transport; geodesics and the Hopf-Rinow theorem.

Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Riemannian Geometry		 3 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Riemannian Geometry		 3 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

1. W M Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry

2 I Chavel, Riemannian geometry - a modern introduction, CUP 1995


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.