Topics courses – syllabi
FQ25 Math 445-1 ("Differential Geometry"): Prof. Jared Wunsch
This class will be an introduction to Riemannian geometry. It will be at the second-year graduate level: the assumption is that you will have completed the first-year Geometry and Topology sequence. In particular, you should be familiar with the definitions and basic properties of smooth manifolds, the tangent and cotangent bundles, vector fields, and flows. I'll give some quick review of the basic concepts as we go, but it will be too fast to assimilate if you have not seen this material before. The class will essentially begin with "what is a Riemannian metric on a smooth manifold?" and go from there. The first year course deals almost exclusively with the topology of manifolds, which by definition all look locally the same; the point is that now we introduce extra structure (a metric) and obtain a rich class of objects that have interesting local, as well as global, invariants (e.g., curvature). This is the big difference between doing topology and doing geometry.
A very loose list of topics we'll cover (as time permits)
- Riemannian metrics, and examples, esp. symmetric spaces
- Connections; Levi-Civita connection
- Geodesics
- Exponential map
- Tubular neighborhoods
- Hopf-Rinow theorem
- Curvature
- Submanifolds; second fundamental form.
- Jacobi fields, conjugate points
- Comparison theorems
(This is probably too ambitious, but we'll try!)
The text wil be John Lee's "Introduction to Riemannian Manifolds," although this is a long book and we can only cover parts of it. You should have free access to this through NU's SpringerLink subscription.
Some (modest amount of) written work and an oral presentation will be required for students enrolled in the course. I will hand out one or more problem sets in the course of the quarter. Every student is expected to complete six problems and tell me which they have done. Additionally, each student will be responsible for carefully writing up two problems from the list and uploading their solutions to Canvas, where all students will be asked to read and (politely) critique the writeups. After any necessary corrections, each student will give a short talk on each of the problems they have completed in one or more in-class problems sessions in the latter half of the quarter.
[Post-qualifying-exam students working on a thesis problem can be exempted from this, and should talk to me.]
Class will meet at 2pm MWF in Lunt 101.