Differential Geometry I - Dietmar Salamon - 2017

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22.1.2018 09:30-10:00 Nicholas

The mighty Salamon will most likely manage to intimidate you at several points during the exam. But fear not, he can also smile, it just depends on what you’re saying. The exam is on the blackboard, with Salamon and some assistant sitting at the table.

He starts out with examples, something which I indeed didn’t really study. Thankfully, it was: give me a motion with pure sliding, this is fairly simple. He then wanted me to give the Phi too, at which point I started failing, and in response, Salamon looked as though he might yell at me or hit me. He did not.

Next, I had to explain when two manifolds with const sect curv k are isometric, with proof, and unlike the script, it is not enough to simply say “global CAH and csc theorem”.

Now the exam started to go really well, I got to talk about the exponential map, Hopf Rinow, which lemmata are used, prove curve shortening lemma. At this point he is smiling rather than looking angry. At some point he also wants an example of a MF where the exponential map is not surjective (IR^2 minus the origin), with explanation.

Then he asks me what I want to talk about, so I prove the isometries theorem, with mostly oral explanations and writing down the key formulas, and the diagram.

Lastly, he wants to know why the exponential map is smooth. This follows from Picard Lindelöf, geodesics are integral curves of the geodesic spray.

That was it, the exam ended two or three minutes early, and started without delay.

22.1.2018 10:30-11:00 Clemens

First I was asked whether I wanted to speak in German or English. I chose English.

S:Give an example of a manifold that is not complete. (R^2 without one point)

S:Give an example of a connected manifold where the exponential map is not surjective. (My idea was to take R^2 with some parts chopped out, but I got I bit confused with the distinction of the manifold M as a subset of R^2 and the tangent space at a point p in M (which is R^2))

Then he asked if there are conditions s.t. the exponential map is surjective and I said Hopf-Rinow Theorem. He wanted to see the proof of this. I draw a picture and explained the main steps of the proof. He then asked what we need in the proof and I said the curve-shortening lemma. That was enough, I didn't even have to give the exact formulation of the lemma.

S:Give an example of a symmetric manifold with non-constant sectional curvature.

I couldn't think of any easy examples, so I said the space of symmetric positive definite matrices P.

S:What dimension?

I:Hmm I don't know, doesn't matter..?

S:Well, it surely doesn't work for dimension one. (because we then have constant curvature 0) But two also doesn't work I think. Lets say dimension 3. The space of symm. pos. def. 3-matrices is the same as the hyperbolic 3 space H^3.

I say "okay" and make a really confused face.

S:What is the hyperbolic 3 space?

I give the defintion. He notices that I don't know much about the hyperbolic space, so he switches the topic.

Then he wanted me to do some stuff with locally symmetric manifolds, which I couldn't really do...

In general, he always started with examples and then went over to theorems that are related to the example or can be used in some way. If you're insecure he askes you about the definitions or more basic statements.

31.2.2018 14:00-14:30 Konstantin

Professor Salamon war sehr nett, jedoch immer darauf aus einen zu verwirren. Wenn er merkt, dass man irgendwo unsicher ist oder etwas falsch beantwortet hat, bohrt er gerne nach. Trotzdem war die Prüfung sehr angenehm, vor allem da er sehr sympathisch wirkt.

Fragen: 1. Beispiel wo die Exponential-Map nicht surjektiv ist. Dann wann ist es der Fall. -> Hopf Rinow mit beweis Idee.

Dann wieso ist die Exponential-Map glatt? Geodesic Spray und dann Vektorfluss.

2. Beispiel einer Symmetrischen Mannigfaltigkeit (S^2) Wie beweist man dies? Er wollte dass ich die Isometrie aufschreibe, habe diese aber nicht auswendig gewusst. Dies hat ihn sichtlich nicht erfreut. Dann wie man es sonst beweisen könnte, (constant sectional curvature und complete connected simply connected). Wieso ist S^2 simply connected?

Dann Guassmap und Sectional Curvature