Differential Geometry II - Urs Lang - 2017

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08.08.2017 10.30 - 11.00 Philip

L: What does Hopf-Rhinow state?

P: The following are equivalent: Completeness, Geodesic completeness, Geodesic completeness from a particular point

L: You are missing something: Heine-Borel. What is Heine-Borel?

P: Closed and bounded sets are compact

L: We also had another statement that follows from the other four. What is it?

P:I don’t know

L: It’s that for all p and q there is a minimizing geodesic. Does it imply the other four statements?

P: Obviously not, do you want a counterexample?

L: Yes.

P: The open disk in R^2

L: Is completeness independent of the metric?

P: No, consider the Poincare disk.

L: State and prove a theorem connecting curvature and topology

P: Synge theorem. [He helps on some of the details which I struggle with, he doesn’t 100% know the second variation formula by heart either.]

L: Can you equip S^2xS^2 with a positive metric?

P: [I try to use Synge to prove it, but obviously no statement can be made using Synge] Nothing follows from Synge about this statement.

L: Oh yes, right. It’s a conjecture though.

P: It would have worked to show that S^1xS^1 can’t have a positive metric, though.

L: Yes. Can you equip RP^2xRP^2 with a positive metric?

P: If yes, it follows that the two sheeted cover has trivial fundamental group by Synge and RP^2xRP^2 itself then has fundamental group Z/2Z, but it should actually be Z/2ZxZ/2Z.

L: What other topological consequences of curvature do you know?

P: The theorem that a negatively curved manifold has finitely generated fundamental group.

L: You are missing a condition.

P: Compactness.

L: Do you know another statement about the fundamental group from curvature?

P: For a manifold with Ricci curvature bounded from below, finitely generated subgroups of the fundamental group have polynomial growth of at most dim(M).

L: What exactly is the bound for the Ricci curvature?

P: [I state something confused and he helps me]

L: How would you prove it?

P: [Very rough outline of the proof, but he is satisfied]

L: Which volume comparison theorem did you use in the proof?

P: Bishop-Gromove, wait only Bishop

L: [Asks assistant about time, only 4 minutes] Tell me about Schur’s theorem.

P: Which one is that again?

L: The one about Einstein manifolds.

P: What are those again? Are they the ones with Ric=f*g?

L: Yes.

P: The theorem states that f is constant.

L: You are missing a condition. What happens if dim(M)=2?

P: The condition is probably that dim(M)>2.

L: Yes, indeed.

P: [I struggle to prove it and time saves me]