Description

Differential manifolds and Riemannian Geometry aims to cover the fundamental theory of Riemannian manifolds. This will include: an introduction to differentiable manifolds, Riemannian metrics and the associated Levi-Civita connection, and the definition of curvature. Geodesics, their length-minimizing properties, and Jacobi fields are also studied with the goal of establishing the Hopf-Rinow theorem. The course concludes by illustrating important connections between local properties of a manifold (geometry) and its global properties (topology). If there is time, we will briefly discuss differential operators on manifolds, such as the Laplacian. Attention will be paid to both rigour and developing the machinery to actually perform explicit calculations.

Prerequisites

Familiarity with differential geometry of surfaces in R³ (similar to Math-4230) or an introductory course in general relativity (similar to Math-4130) and an undergraduate background in linear algebra and analysis.

Textbooks

• Manuel Do Carmo, Riemann Geometry, Birkhaeuser (1992).
• John M Lee, Riemannian Manifolds, Springer Graduate Texts in Mathematics (online copies available in MUN library).