This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.

#### Prerequisites

Smooth manifolds (the equivalent of Math 620).#### Syllabus

- Riemannian metrics and examples including the 3 model spaces.
- Connections/covariant derivatives on vector bundles; curvature of connections; parallel translation.
- Levi–Civita connection on a Riemannian manifold; geodesics as straightest curves; the Riemannian exponential map; normal coordinates and neighborhoods.
- Geodesics as (locally) shortest paths; metric properties of Riemannian manifolds; completeness and the Hopf–Rinow theorem.
- Symmetries of the curvature tensor; Ricci and scalar curvature.
- Riemannian submanifolds, intrinsic versus extrinsic geometry; surfaces in Euclidean 3-space; Gauss’s Theorem Egregium; the Gauss–Bonnet theorem.
- The second variation formula, Jacobi fields, conjugate points and elementary comparison geometry.
- The Bonnet–Myers theorems and the Cartan–Hadamard theorem.

#### References

- Lee,
*Riemannian Geometry* - do Carmo,
*Riemannian Geometry*