Differential Geometry II - 2020
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Niko, 6.8.2020
- Levi-Civita Connection:
Def: torsion-free and compatible with metric Koszul-Formula Uniqueness of L-C-Connection (Mentioned case of Lie-group with bi-invariant Riem. metric and left-invariant vector fields, but I didn't remember and we just skipped that)
- Rauch comparison theorem
Def: Jacobi Field (and geometric interpretation) Thm 3.18: Statement
- Gauss equations
Def: 2nd FF \(h(X,Y)\) and \(h_N(X,Y)\) (Mind: what are relations of LC-connections and metric in \(M\) and \(\bar{M}\) Proof: Show symmetry of (\h\) State Gauss equations State in terms of \(sec\) instead of \(R\)
- Hadamard, Cartan
State theorem 4.12 Claim in proof: \(sec\leq0\) implies no conjugate points. (This was an exercise in script) Prove claim.
