Math 621 Syllabus

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


Smooth manifolds (the equivalent of Math 620).


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


  • Lee, Riemannian Geometry
  • do Carmo, Riemannian Geometry