Lecture Notes for 120 - UCLA Department of Mathematics

CHAPTER 6
Intrinsic Calculations on Surfaces
The goal in this chapter is to show that many of the the calculations we do
on a surface can be done intrinsically. This means that they can be done without
reference to the normal vector and thus are only allowed to depend on the
first fundamental form. The highlights are the Gauss equations, Codazzi-Mainardi
equations, and the Gauss-Bonnet theorem.
6.1. Calculating Christoffel Symbols and The Gauss Curvature
We start by showing that the tangential components of the derivatives
@ 2 q
@w 1 @w 2
can be calculated intrinsically. For the Gauss formulas this amounts to showing
that the Christoffel symbols can be calculated from the first fundamental form. In
particular, this shows that they can be computed knowing only the first derivatives
of q (u, v) despite the fact that they are defined using the second derivatives!
Proposition 6.1.1. The Christoffel symbols satisfy
uuu = 1 @g uu
2 @u
uvu = 1 @g uu
2
@v = vuu
vvv = 1 @g vv
2 @v
uvv = 1 @g vv
2 @u = vuv
uuv = @g uv
@u
vvu = @g uv
@v
1
2
1
2
@g uu
@v
@g vv
@u
Proof. We prove only two of these as the proofs are all similar. First use the
product rule to see
uvu = @2 q
@u@v · @q
@u = ✓ @
@v
✓ ◆◆ @q
· @q
@u
133
@u = 1 @
2 @v
✓ @q
@u · @q ◆
= 1 @g uu
@u 2 @v