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Riemannian Geometry

Unit code: MATH31082
Credit Rating: 10
Unit level: Level 3
Teaching period(s): Semester 2
Offered by School of Mathematics
Available as a free choice unit?: N



Additional Requirements

MATH31082 pre-requisites


The course unit unit aims to introduce the basic ideas of Riemannian geometry.


Basis ideas of Riemannian geometry such as Riemannian metric, covariant differentiation, geodesics and curvature belong to the core of mathematical knowledge and are widely used in applications that range from general relativity in physics to mechanics and engineering. Besides that, this subject is one of the most beautiful in mathematics, containing such gems as Gauss's Theorema Egregium and the Gauss-Bonnet Theorem providing a link with the topology of surfaces.The course introduces these ideas, building on the course unit MATH20222 Introduction to Geometry.


Learning outcomes

On completion of this unit successful students will be able to:     

  •     deal with various examples of Riemannian metrics;
  •     work practically with connection and curvature;
  •     appreciate the relation between geodesics and variational principle;
  •     apply the ideas of Riemannian geometry to other areas.


Future topics requiring this course unit

Riemannian geometry is used in almost all areas of mathematics and its applications, including physics and engineering.


Assessment methods

  • Other - 20%
  • Written exam - 80%

Assessment Further Information

  • Coursework (take-home) 20%
  • End of Semester exam, 2 hours duration (3 hours for MATH41082), 80%.




The notion of Riemannian metric in a domain of Rn.

Angle and length of tangent vetors. Arc length of a curve.

Examples of metric: sphere and other quadrics in R3; Lobachevsky plane.

Idea of Killing vectors.

Volume element corresponding to Riemannian metric. 




Definition of a covariant derivative.  Expression in local coordinates.

Christoffel symbols.

Examples: covariant differentiation in Rn in curvilinear coordinates.

Covariant differentiation on surfaces in Rn .

Relations between covariant differentiation and Riemannian metric. Levi-Civita connection.




Idea of parallel transport.  Infinitesimal parallel transport.

Equation of parallel transport.  Geodesics.

Geodesics and Riemannian metric.

Examples of geodesics.




Induced Riemannian metric (First quadratic form).

Gauss-Weingarten derivation formulae.  

Second quadratic form and shape (Weingarten) operator.

Geometric meaning of Gaussian curvature and Theorema Egregium.




Infinitesimal parallel transport over a closed contour.

Definition of curvature tensor.

Gaussian curvature of surfaces and scalar curvature.

Application in Gravity theory. Einstein - Hilbert equations. 

 Idea of Gauss-Bonnet theorem




Triangulation of surfaces and Euler characteristic. Examples.

Gauss-Bonnet Theorem.

Two-dimensional gravity.

Recommended reading

No particular textbook is followed. Students are advised to keep their own lecture notes. There are many good sources available treating various aspects of differentiable manifolds on various levels and from different viewpoints. Below is a list of texts that may be useful. More can be found by searching library shelves.


R. Abraham, J. E. Marsden, T. Ratiu. Manifolds, tensor analysis, and applications. Springer-Verlag, 1996. ISBN 0387967907.

B.A. Dubrovin, A.T. Fomenko, S.P. Novikov. Modern geometry, methods and applications. Part I: The Geometry of Surfaces, Transformation Groups, and Fields, Vol. 93, 1992,

Barret O Neill. Elementary Differential geometry, Revised Second Edition, Academic Press (Elsevier), 2006, ISBN-10: 0120887355.

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework also provides an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

  • Lectures - 22 hours
  • Tutorials - 11 hours
  • Independent study hours - 67 hours

Teaching staff

Hovhannes Khudaverdyan - Unit coordinator

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