Emily presented about the tangent spherical image of a curve. This is a bit like a curves version of the Gauss map.
Jesse presented Shifrin 2.2#13.
Then I talked about what will come next: using the tangential components of the second derivatives \(x{uu}\), etc. This leads us to introduce the _Christoffel symbols and find a way to compute them.
Please read up on section 2.3 of Shifrin and start exercises there.