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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

Requisites

Prerequisite

Additional Requirements

MATH31082 pre-requisites

Aims

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

Overview

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:     

  • state the definition of a Riemannian manifold M and calculate the length of a curve, and area of a domain in M,
  • calculate the Riemannian metric on surfaces embedded in E3,
  • define a connection on a manifold, state the Levi-Civita theorem, and calculate the connection for surfaces of cylinder, sphere and cone in E3,  and for Lobachevsky (hyperbolic)  plane,
  • state the properties of geodesics on a Riemannian manifold, and calculate the parallel transport of vectors along a geodesics for the sphere and cylindre in E2 and for Lobachevsky plane,
  • State the definition of the Riemann curvature tensor. Calculate the Riemann curvature tensor for some 2-dimensional Riemannian manifolds.

 

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%.

Syllabus

RIEMANNIAN METRIC ON A DOMAIN OF Rn.

  • 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. 

 COVARIANT DIFFERENTIATION.

  • 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.

GEODISICS AND PARALLEL TRANSPORT.

  • Idea of parallel transport.  Infinitesimal parallel transport.
  • Equation of parallel transport.  Geodesics.
  • Geodesics and Riemannian metric.
  • Examples of geodesics.

THEORY OF SURFACES.

  • 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.

 CURVATURE TENSOR.

  • 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

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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