Difference between revisions of "Differential Geometry II - 2018"

Revision as of 06:32, 14 August 2018

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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.