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

• 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 theorem 4.12

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

Prove claim.