Differential Geometry Questions

1. Given the surface of revolution pictured at right, what are all the geodesics on this surface?
   Things to think about:
   
   • What are all the simply closed geodesics?
   • Which geodesics “go to infinity”?
   • What is the asymptotic behavior of geodesics, especially as they go down?
   • Are there any geodesics that are bounded other than the simply closed ones?
2. Suppose you know all Ricci curvatures on a 3-manifold. What can you say about the sectional curvatures? Does this hold for 4-manifolds?
3. Is the cut locus of a complete, connected surface connected? (There are two cases you should treat separately: compact and non-compact) Is there a surface with S¹ as its cut locus? Can you put a metric on a surface of genus g such that the cut locus of a point in this metric is a circle?
4. What are all the isometries of the hyperbolic plane. Determine explicitly which isometries are conjugate to each other.

Logic and Finite Model Theory Questions

1. Give a complete description of the first order theory of bi-infinite chains (i.e. the simple graph induced by < on ℤ). Is connectivity definable over simple graphs? Can you modify this definability argument to the finite case?
2. Prove the 0-1 law for first order logic.