Algebraic Geometry

Lecture notes for a term-long course on Riemannian geometry which I took as a Masters student. The focus of the coursewas in familiarity working with Riemannian metrics, the Levi-Civita connection, geodesics and curvature, both in a local coordinate description and using coordinate-free expressions. Then we moved on to Riemannian submanifolds (and considering the second fundamental form), Jacobi fields, completeness, and theorems of Hopf-Rinow, Cartan-Hadamard, Bonnet-Myers, and Synge

Do Carmo's fantastic book on Riemannian Geometry is the primary reference here, though the books by Peterson and Lee can be consulted, as they also follow the standard material.

Course Schedule

1. Introduction to Riemannian Manifolds:

Definition of Riemannian manifolds, examples, and the Riemannian metric as a smoothly varying inner product on tangent spaces. The key takeaway: Understanding how the Riemannian metric allows one to measure lengths and angles on manifolds

2. Connections and Covariant Derivatives:

Introducing affine connections, the Levi Civita connection as a metric-compatible and torsion free connection, and the covariant derivative playing a role in differentiating vector fields along curves.The key takeaway: Connections enable the comparison of vectors in different tangent spaces

3. Geodesics and the Exponential Map:

Definition and properties of geodesics as curves with zero acceleration, the exponential map for relating tangent spaces to the manifold, and normal coordinates and their construction. The key takeaway: Geodesics generalise the idea of striaght lines to curved space

4. Curvature and the Riemannian Curvature Tensor:

Introducing the Riemann curvature tensor, sectional curvature and its geometric interpretation, and symmetries and properties of the curvature tensor. The key takeaway: The heirarchy of curvatures, and understanding its use in quantifying hte derivation of a manifold from being flat

5. Jacobi Fields and Variations of Geodesics:

Definition and analysis of Jacobi fields, second variation formula, and the conjugate points. The key takeaway: Jacobi fields as tool for describing the behaviour of nearby geodesics and the stability of geodesic flows

6. The Hopf-Rinow Theorem

Statement and proof of the Hopf-Rinow Theorem, equivalence of completeness, geodesic completeness, and the compactness of closed balls. The key takeaway: The connections between geometric properaties and global topological outcomes

7. Spaces of Constant Curvature:

The classification of manifold with constant sectional curvature, using Euclidean, spherical, and hyperbolic geometries as models, and the isometry groups. The key takeaway: Understanding how constant curvature shapes global geometry and symmetry of spaces

8. The Cartan-Hadamard Theorem:

Statement and proof of the Cartan-Hadamard theorem, implications for manifolds of non-positive curvature, and the universal covering spaces and their properties. The key takeaways: Appreciating how non-positive curvature influences the global topology, which ensures manifolds are diffeomorphic to Euclidean spaces

9. The Bonnet-Myers Theorem:

Statement and proof of the theorem, consequences for manifolds with positive Ricci curvature, and diameter bounds and compactness results. Key takeaway: Positive curvature imposes strong restrictions on the size and topology of manifolds

10. Synge's Theorem and Applications:

Statement and proof of the theorem, orientability and its relations to curvature, and applications to the topology of even-dimensional manifolds. Key takeaway: The deep interplay between curvature, orientability, and the fundamental group of manifolds