Lecture Notes for 120 - UCLA Department of Mathematics

4.3. THE ABSTRACT FRAMEWORK 86
in order to get a basis. While N does depend on the parametrizations we note that
as it is normal to a plane in R 3 there are in fact only two choices ±N, just as with
planar curves.
This means that we shall consider frames ⇥ @q
@u
such frames
@ ⇥ @q @q
@w
@u @v
N ⇤ h
i
=
@ 2 q @ 2 q @N
@w@u @w@v @w
@q
@v
= ⇥ @q
@u
N ⇤ and derivatives of
@q
@v
N ⇤ [D w ]
where w can be either u or v.
The entries of D w are divided up into parts. The principal 2⇥2 block consisting
of what appears in the first two rows and columns, which depends only on tangential
information. This block corresponds to the [ w] that we defined in the plane using
general coordinates. The remaining parts, consisting of the third row and third
column, depends on normal information. Since N is a unit vector the 33 entry
actually vanishes:
0= @ |N|2
@w
@N
=2N ·
@w
showing that @N
@w
lies in the tangent space and hence does not have a normal component.
As before we have
In particular,
or
@ 2 ⇥ @q
@w 1 @w
@u 2
apple @Dv
[D u ][D v ]+
@u
apple @Dv
@u
@q
@v
N ⇤ @ 2 ⇥
=
@q
@w 2 @w
@u 1
apple @Du
@v
@q
@v
apple @Du
=[D v ][D u ]+
@v
+[D u ][D v ] [D v ][D u ]=0
As we shall see, other interesting features emerge when we try to restrict attention
to the tangential and normal parts of these matrices.
Elie Cartan developed an approach that uses orthonormal bases. Thus he chose
an orthonormal frame E 1 ,E 2 ,E 3 along part of the surface with the property that
E 3 = N is normal to the surface, and consequently E 1 ,E 2 form an orthonormal
basis for the tangent space. The goal is again to take derivatives. For that purpose
we can still use parameters
N ⇤
@ ⇥ ⇤ ⇥
E1 E 2 E 3 = @E1
@w
@w
@E 2
@w
@E 3
@w
⇤
=
⇥
E1 E 2 E 3
⇤
[Dw ]
The first observation is that [D w ] is skew-symmetric since we used an orthonormal
basis:
2 3
⇥ ⇤ t ⇥ ⇤ 1 0 0
E1 E 2 E 3 E1 E 2 E 3 = 4 0 1 0 5
0 0 1