Mathematics

Differential Geometry

Module code: 858G1
Level 7 (Masters)
15 credits in spring teaching
Teaching method: Lecture
Assessment modes: Unseen examination, Coursework

On this module, we will cover:

  • Manifolds and differentiable structures
  • Lie derivatives
  • Parallel transport
  • Riemannian metrics and affine connections
  • Curvature tensor
  • Sectional curvature
  • Scalar curvature
  • Ricci curvature
  • Bianchi identities
  • Schur's lemma
  • Complete manifolds
  • Hopf-Rinow theorem
  • Hadmard's theorem
  • Geodescis and Jacobi fields
  • Bonnet-Meyer and Synge theorems
  • Laplace-Beltrami operator
  • Heat kernels and index theorem.

Module learning outcomes

  • Understand the concept of a Riemannian metric and its curvature tensor.
  • Understand the concepts of Sectional, Ricci and Scalar curvature and their relations and significances.
  • Understand the notion of a geodesis and Jacobi field and solve the corresponding equations for some basic cases.
  • Understand the Laplace-Beltrami operator on a Rimannian Manifolds and its basic properties.