Lecture Notes for 120 - UCLA Department of Mathematics

6.4. THE GAUSS AND CODAZZI EQUATIONS 144
we obtain after writing out the entries in the matrices
2
3
2
u u
6
7
4
5 + w 1u
4 v
w 1u
=
2
6
4
@
u w 2 u @ u w 2 v @L u w 2
@w 1 @w 1 @w 1
@
v w 2 u @ v w 2 v @L v w 2
@w 1 @w 1 @w 1
@L w2 u @L w2 v
@w 1 @w 1
0
@
u w 1 u @ u w 1 v @L u w 1
@w 2 @w 2 @w 2
@
v w 1 u @ v w 1 v @L v w 1
@w 2 @w 2 @w 2
@L w1 u @L w1 v
@w 2 @w 2
0
3
2
7
5 + 4
w 1v L u w 1
v
w 1v L v w 1
L w1u L w1v 0
u
w 2u
v
w 2u
u
w 2v L u w 2
v
w 2v L v w 2
L w2u L w2v 0
3 2
u
w 2u
v
w 2u
u
w 2v L u w 2
v
w 2v L v w 2
5 4
L w2u L w2v 0
3 2
u
w 1u
v
w 1u
u
w 1v L u w 1
v
w 1v L v w 1
5 4
L w1u L w1v 0
If we restrict attention to the the general terms of the entries in the first two columns
and rows using w 3 ,w 4 as indices instead of u, v we end up with
2 3
2 3
@ w4
w 2w 3
+ ⇥ u
⇤ w 2w 3
w 4 w 4
w
@w 1u w 1v L w4 4 v 5
w 1 w 2w 3
= @ w4
w 1w 3
+ ⇥ u
⇤ w 1w 3
w 4 w 4
w
1 @w 2u w 2v L w4 4 v 5
w 2 w 1w 3
L 2 w2w 3
L w1w 3
which can further be rearranged by isolating
@ w4
w 2w 3
@w 1
@ w4
w 1w 3
@w 2
+ ⇥ w 4
w 1u
w 4
w 1v
⇤ apple u
w2w 3
v
w 2w 3
sononeside:
⇥
w 4
w 2u
w 4
w 2v
⇤ apple u
w1w 3
v
w 1w 3
3
5
3
5
= L w4
w 1
L w2w 3
These are called the Gauss Equations.
The Riemann curvature tensor is defined as the left hand side of the Gauss
equations
R w4
w 1w 2w 3
= @ w4
w 2w 3
@w 1
@ w4
w 1w 3
@w 2
+ ⇥ w 4
w 1u
w 4
w 1v
⇤ apple u
w2w 3
v
w 2w 3
⇥
w 4
w 2u
w 4
w 2v
⇤ apple u
w1w 3
v
w 1w 3
It is clearly an object that can be calculated directly from the first fundamental
form, although it is certainly not always easy to do so. But there are some symmetries
among the indices that show that there is essentially only one nontrivial
curvature on a surface. On the face of it each index has two possibilities so there
are potentially 16 different quantities! Here are some fairly obvious symmetries
R w4
w 1w 2w 3
= R w4
w 2w 1w 3
,
In particular there are at least 8 curvatures that vanish
and up to a sign only 4 left to calculate
R w4
www 3
=0
R u uvu = R u vuu,
R v uvu = R v vuu,
R u uvv = R u vuv,
R v uvv = R v vuv
AslightlylessobviousformulaistheBianchi identity
R w4
w 1w 2w 3
+ R w4
w 3w 1w 2
+ R w4
w 2w 3w 1
=0
It too follows from the above definition, but with more calculations. Unfortunately
it doesn’t reduce our job of computing curvatures. The final reduction comes about
by constructing
R w1w 2w 3w 4
= R u w 1w 2w 3
g uw4 + R v w 1w 2w 3
g vw4
L w4
w 2
L w1w 3
,