Orals Questions - Ricardo Mendes, Sept 25, 2007

(the ones I remember)

AT- What spaces can a closed disc cover? What about CP^2?

AT- Calculate the homotopy groups of Maps(S^1, S^3) in terms of the homotopy groups of S^3.

AT- Calculate the cohomology ring of CP^2 using the Serre Spectral Sequence.

AT- Let M be a closed connected oriented manifold of dimension n. Show
that any (n-2) homology class is represented by a smooth codimension 2
submanifold.

DG- List the spaces of constant curvature that you know.

(This was a very open-ended question, and as we were talking about it they asked many other questions, including:

- When are two flat tori isometric?

- Why are the sphere and projective space the only constant positive curvature spaces of dimension n when n is even?

- State the Synge, Bonnet-Myers, and Preissman theorems.)

DG- Show that for any metric of positive curvature on S^2, any two
closed geodesics must intersect. Give an example of a metric on S^2
which has two disjoint closed geodesics.

DG- Show that any metric of non-negative curvature on the cylinder has a closed geodesic.

DG- Let H be a connected hypersurface in R^n, and assume that its Gauss
curvature (i.e., product of principal curvatures) is non-zero at every
point. Can it be positive/negative? What if H is compact? Show that
when H is compact it is homeomorphic to a sphere.

DG- State the version of the Gauss-Bonnet Theorem on a region bounded by curve.

DG- What is a symmetric space? What can you say about the sectional
curvatures of a symmetric space? Give examples of symmetric spaces with
non-negative curvature, and also with non-positive curvature.