Differential Geometry II - 2020

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

Preissmann

• State and prove thm 4.20

Gauss equations

• Def: 2nd FF \(h(X,Y)\) and \(h_N(X,Y)\) (Mind: How are the LC-connections \(D,\bar{D}\) and metrics \(g,\bar{g}\) of \(M\) and \(\bar{M}\) related)

• Proof: Show symmetry of (\h\)

• State Gauss equations

• State in terms of \(sec\) instead of \(R\)

Hadamard, Cartan

• State thm 4.12

• Claim in proof: \(sec\leq0\) implies no conjugate points. (This was an exercise in script)

• Prove claim.