Jeffrey S. Case
Associate Professor
The Pennsylvania State University, Department of Mathematics

Contents: Spring 2014
MAT 982, Junior Seminar, Riemannian Geometry
T 4:30-6:30pm, Fine 801
S 1-4pm, Fine 1001

Office: 1004 Fine Hall
Office Hours: TBA, or by appointment

Our plan is to discuss some comparison theorems in Riemannian geometry involving sectional curvature, Ricci curvature, and possibly scalar curvature, depending on student interest. We will use as references the following books; I may add a few articles to this list later.
  • Barrett O'Neill, Semi-Riemannian Geometry, with Applications to Relativity. This has a quick introduction to smooth manifolds which will be sufficient for our needs, and also provides a nice overview of Riemannian geometry.
  • Jeff Cheeger and David Ebin, Comparison Theorems in Riemannian Geometry. This covers many comparison theorems involving sectional curvature, and will most likely be the primary reference for this course.
  • Peter Petersen, Riemannian Geometry. This gives another nice overview of Riemannian geometry and covers many comparison theorems for both sectional curvature and Ricci curvature.
For further information, please visit Blackboard.

Lecture Plans

  1. Feb. 11 Cancelled. Make-up date to be scheduled later.
  2. Feb. 18 Quick introduction to smooth manifolds. Manifolds, curves, vector fields, tensors.
  3. Feb. 25 Finish introduction to smooth manifold and start discussing Riemannian manifolds. Riemannian metrics, the Levi-Civita connection, and geodesics.
  4. Mar. 8 (Danni) Geodesics and the exponential map. (Clayton) The Hopf-Rinow theorem. (Max) Jacobi fields and the second variation of the length functional.
  5. Mar. 25 (Max) Continue second variation of length functional. (Clayton) Index lemmas. (Danni) Myers' Theorem
  6. Apr. 5 (Max) Rauch Comparison Theorem, Cartan--Hadamard Theorem, Cartan--Ambrose--Hicks Theorem. (Clayton) Toponogov's Theorem. (Danni) Closed geodesics and the cut locus.
  7. Apr. 19 (Max) Morse theory, following Ch. 4 and Milnor's book. (Danni) Quarter-pinching sphere theorem. (Clayton) Soul theorem.
  8. Apr. 26 TBA

Previous Courses:
MAT 175, Fall 2013
MAT 104, Spring 2012