Question

Which types of questions should I expect in oral examination in Differntial geometry ?

Topics covered

local curve theory
local area theory

Plane curves

Which type of questions should I expect and give me as much as examples as possible?

Answer 1

1. What can you tell me about Gauss-Bonnet? What is the Gaussian curvature of a surface? Draw surfaces of positive and negative curvatures. Does it depend on the particular way the surface is embedded in space? What is the Euler characteristic of the torus? Looking at the torus embedded in space as a donut, where are regions of positive and negative curvature?

2. Curvature tensor. Gauss theorem.

3. Is there a metric on the sphere with negative curvature? (Ans: no, by Gauss-Bonnet)

4. Describe the tensor product bundle over a manifold?

5. Describe affine connection?

6. Can a compact orientable surface with negative curvature cover another such surface?

Ans. In two dimensions, the only simply connected compact orientable surface is the sphere, which must have positive curvature somewhere by Gauss-Bonnet.

the Cartan-Hadamard theorem: the universal cover of a negatively curved manifold is non-compact, and thus in order for the space itself to be compact it must have non-trivial fundamental group.