Instructor: Adam S. Sikora (Math Dept.)
Class meets on Tu,Th: TuTh 12:30PM - 1:50PM in Math Bldg. 235
Attendance is expected.
My office hour: TBA

Primary Text: Riemannian Geometry by Manfredo P. do Carmo

Course description: This is a first course in Riemannian geometry, a mathematical subject that has rich interactions with algebra, analysis, and topology. A Riemannian metric is a geometric structure on a smooth manifold that allows measuring angles and distances. After a brief review of smooth manifolds, the course will introduce Riemannian metrics, affine connections, geodesics, curvature, and isometries.

Tentative Topics:

1. Review of differentiable manifolds: smooth manifolds, tangent spaces, smooth vector fields, Lie brackets, flows, Lie derivatives, cotangent spaces, vector bundles, differential forms.
2. Riemannian metrics, local and global isometries
3. Affine connections, covariant derivatives, parallel transport, the Levi-Civita connection.
4. Riemannian curvature, sectional curvature, Ricci curvature, scalar curvature.
5. Geodesics, metric structure on Riemannian manifolds, exponential map,
6. Jacobi fields, Hopf-Rinow, Cartan-Hadamard, and Killing-Hopf theorems.
7. spaces of constant curvature, especially hyperbolic geometry.
8. Other topics of interest for students.

Grading: Your letter grade for the course will be based on biweekly homework assignments.
There will be no tests, no final exam.

1. Define and work fluently with the foundational objects of Riemannian geometry, including smooth manifolds, Riemannian metrics, Levi-Civita connections, geodesics, curvature tensors, and covariant derivatives.
2. Compute geometric quantities in concrete examples, such as Christoffel symbols, geodesics, sectional curvature, Ricci curvature, and scalar curvature.
3. Read, understand, and critically engage with advanced mathematical texts and research-level material in differential and Riemannian geometry.
4. Communicate sophisticated mathematical arguments clearly and rigorously.