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Durham University

Department of Mathematical Sciences

MATH4171 Riemannian Geometry IV

In the first term we introduce Riemann's concept of a manifold as a space with locally Euclidean coordinates and an intrinsic method of measuring distances and angles and discuss it in several examples. We also introduce geodesics via an variational approach and discuss parallel transport of vector fields along curves.

The second term is concerned with the notion of curvature and its influence on the geometry of the underlying space. We study spaces of constant curvature, present a global curvature comparison theorem (Bonnet-Myers) and discuss applications.

The aim of this course is to develop acquaintance with more general geometric spaces than just the Euclidean space and to the geometric meaning of curvature.

Outline of Course

Aim: The aim of this course is to develop acquaintance with more general geometric spaces than just the Euclidean space and to the geometric meaning of curvature.

Term 1

  • From submanifolds to abstract manifolds via examples: surfaces of revolution, projective space, Grassman manifolds, hyperbolic space, matrix groups.
  • Tangent vectors and tangent space, computations in examples (e.g. in matrix groups), vectorfields and computations of Lie brackets.
  • Examples of Riemmanian manifolds.
  • Length of curves, Riemannian manifolds as metric spaces.
  • Local and global properties of geodesics, first variation formula, Levi-Civita connection, parallel transport, discussion of examples (e.g. hyperbolic space, matrix groups).
  • Geodesics on surfaces of revolution, Clairaut's theorem.

Term 2

  • Different curvature notions and computations thereof: Riemmanian curvature tensor, sectional curvature, Ricci curvature, scalar curvature.
  • Spaces of constant curvature, the Cartan-Ambrose-Hicks theorem.
  • Integration on Riemannian manifolds, volume calculations in spaces of constant curvature.
  • The second variation formula and Bonnet-Myers theorem as a global comparison result.
  • Applications (e.g., the n-torus does not admit a metric of positive curvature; the global 2-dimensional Gau-Bonnet theorem from differential geometry implies same statement for 2-torus).

Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.

Reading List

Please see the Library Catalogue for the MATH4171 reading list.

Examination Information

For information about use of calculators and dictionaries in exams please see the Examination Information page in the Degree Programme Handbook.