Introductory description

The core of this course will be an introduction to Riemannian geometry - the study of Riemannian metrics on abstract manifolds. This is a classical subject, but is required knowledge for research in diverse areas of modern mathematics.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Review of basic notions on smooth manifolds; tensor fields.
• Riemannian metrics.
• Affine connections; Levi-Civita connection; parallel transport.
• Geodesics; exponential map; minimising properties of geodesics.
• The curvature tensor; sectional, Ricci and scalar curvatures.
• Training in making calculations: switching covariant derivatives; Bochner/Weitzenböck formula.
• Jacobi fields; geometric interpretation of curvature; second variation of length.
• Classical theorems in Riemannian Geometry: Bonnet-Myers, Hopf-Rinow and Cartan-Hadamard.

Learning outcomes

By the end of the module, students should be able to:

• See outline syllabus

Indicative reading list

Lee, J. M.: Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics, 176. Springer-Verlag, 1997.
Gallot, S., Hulin, D., Lafontaine, J.: Riemannian geometry. Springer. 2nd edition (1993)
Jost, J.: Riemannian Geometry and Geometric Analysis 5th edition. Springer-Verlag, 2008
Petersen, P.: Riemannian Geometry Graduate Texts in Mathematics, 171. Springer-Verlag, 1998
Kobayashi, S., Nomizu, K.: Foundations of differential geometry.
do Carmo, M: Riemannian geometry. Birkhäuser, Boston, MA, 1992.

Subject specific skills

See outline syllabus.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.