# C3.11 Riemannian Geometry - Material for the year 2020-2021

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Differentiable Manifolds is required. An understanding of covering spaces will be strongly recommended.

### Assessment type:

- 2-hour written examination paper

Riemannian Geometry is the study of curved spaces and provides an important tool with diverse applications from group theory to general relativity. The surprising power of Riemannian Geometry is that we can use local information to derive global results.

This course will study the key notions in Riemannian Geometry: geodesics and curvature. Building on the theory of surfaces in R^3 in the Geometry of Surfaces course, we will describe the notion of Riemannian submanifolds, and study Jacobi fields, which exhibit the interaction between geodesics and curvature.

We will prove the Hopf--Rinow theorem, which shows that various notions of completeness are equivalent on Riemannian manifolds, and classify the spaces with constant curvature.

The highlight of the course will be to see how curvature influences topology. We will see this by proving the Cartan--Hadamard theorem, Bonnet--Myers theorem and Synge's theorem.

The candidate will have great familiarity working with Riemannian metrics, the Levi-Civita connection, geodesics and curvature, both in a local coordinate description and using coordinate-free expressions. The candidate will gain understanding of Riemannian submanifolds, Jacobi fields, completeness, and be able to prove and apply fundamental results in the subject, including the theorems of Hopf--Rinow, Cartan--Hadamard, Bonnet--Myers and Synge.

Riemannian manifolds: basic examples of Riemannian metrics, Levi-Civita connection.

Geodesics: definition, first variation formula, exponential map, minimizing properties of geodesics.

Curvature: Riemann curvature tensor, sectional curvature, Ricci curvature, scalar curvature.

Riemannian submanifolds: examples, second fundamental form, Gauss--Codazzi equations.

Jacobi fields: Jacobi equation, conjugate points.

Completeness: Hopf--Rinow and Cartan--Hadamard theorems

Constant curvature: classification of complete manifolds with constant curvature.

Second variation and applications: second variation formula, Bonnet--Myers and Synge's theorems.

- M.P. do Carmo, Riemannian Geometry, (Birkhauser, 1992).
- J.M. Lee, Riemannian Manifolds: An Introduction to Curvature, (Springer, 1997).
- S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, (Springer, 1987).
- W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edition, (Academic Press, 1986).