Differential Geometry II - 2018

Please sign with your name and the date on which you had your exam. If you use this wiki, contribute to it as well or terrible things will happen to you: like me kicking you with my fists.

Sebastian, 13.08.18, 10:30-11:00

• No non vanishing vector field on \(S^2\) (with 2 proofs: Hairy Ball and Euler Characteristic)
• Compute Euler characteristic of \(S^2\) (explicitly with the Vector field from the Problem Sheets)
• Show that every smooth map \(\mathbb{CP}^2 \to \mathbb{CP}^2\) has a fixed point. (Lefschetz Trace formula, using the ring structure of \(\mathbb{CP}^2\))
• Sketch the proof of Lefschetz Trace Formula (Stated the three main steps orally, that was enough).
• Show that \(H^\star(\mathbb{CP^2}) \cong \mathbb{R}[\omega]/\omega^3\) (the Poincaré dual of \(\mathbb{CP}^1\) is the Euler class of \(H\), and \(\omega^2 \neq 0\)
• State and prove Moser Isotopy (I sketched the proof with some problems using Cartan's Formula).

Salamon is very friendly. Whenever I was unsure of at detail he would ask more specific questions until I was sure.

Leon, 13.08.18, 17:00-17:30

• Cp1 self intersection number (two proofs)
• Diagonal in S2 x S2, self intersection number via integral lattice
• fixpoint for mapping in CP2, via
• L(F)=1+d+d^2, d integer
• Defn Thom-Form
• Defn Euler class
• Statement of "Euler Class and Integration"
• Statement of P.D.
• give explicit construction for PD(Q) for a submanifold
• proof P.Q=int tau_p tau_q via the explicit construction
• proof Gauss-Bonnet
• why is deg(nu)=2 times euler characteristic
• behaviour of Gauss-Bonnet under parity of m.

Andreas, 14.08.18, 15:50-16:30

• Compute deRham-cohomology of RP^2
• Existence of a fixpoint of self-maps of RP^2 (Lefschetz-Fixpoint-Theorem)
• Trace formula for the Lefschetz number (Thm. 6.4.8), compute Lefschetz number in the setting above.
• Sketch of the proof, if M is compact, oriented, boundaryless.
• Short outline: How to proceed in the general case.
• Statement of Poincare-Duality
• Intersection numbers and Poincare-Duality (Thm. 6.4.7)
• Definition of a Thom form, proof of Existence (as in the lecture)
• Construction of Poincare Dual of Q via Thom form tau_epsilon, proof of duality (as in the lecture)
• Short sketch of proof of Thm. 7.2.18, setting and geometric picture (as in the lecture)

(Localization of the wedge-product around intersections)

• What is your favourite theorem? Gauß-Bonnet. Setting, Statement, sketch of proof, in particular why deg(Gauß-map) = half Euler characteristic.

Prof. Salamon is very relaxed and he lets you talk as long as your statements are correct. Interrupts you shortly, if you have a typo on the blackboard or if you start to move to far away from the initial question / setting. Handwaving instead of technical details is ok, as long as you mention the central steps.