Graphics Reference
In-Depth Information
In fact, in expanded form we have
()
+
()
—=
U
w
X U
w
X U
.
.
X
1
12
2
13
3
()
()
—= -
U
w
X U
+
w
X U
X
2
12
1
23
3
()
-
()
—= -
U
w
X U
w
X U
.
X
3
13
1
23
2
where w
ij
(
X
)(
p
) is an abbreviation for w
ij
(
p
)(
X
(
p
)).
Proof.
The fact that the equations can be expressed as shown using only the forms
w
12
, w
13
, and w
23
follows from the fact that w
ij
=-w
ji
for all i and j, which also implies
that w
ii
= 0.
Looking at the equations in Theorem 9.16.6 should remind the reader of the
Serret-Frenet formulas. In fact, Exercise 9.16.1 shows that Theorem 9.16.6 is a gen-
eralization of these formulas. The Serret-Frenet formulas have no w
13
terms because
of the special nature of the Frenet frames. Of course, the comparison might seem
problematic because the Frenet frames are only defined along a curve and our con-
nection forms in equations (9.94) were assumed to be defined at all points of
R
3
.
However, one can show that any frame field on a curve can always be extended to a
tubular neighborhood of the curve so that equations (9.94) would have to hold as
long as the vector field
X
is tangent to the curve because of the way that the direc-
tional derivative is defined (Lemma 9.16.1). Note that the derivatives in the Serret-
Frenet formulas involve differentiating in a tangential direction.
9.16.7. Example.
Let g(s) be a curve in
R
2
parameterized by arc-length. As usual,
let T(s) =g¢(s) and let N(s) be the unit normal vector at g(s) so that (T(s),N(s)) induces
the standard orientation on
R
2
. Consider a frame field (
U
1
,
U
2
,
E
3
) defined in a neigh-
borhood of the curve so that
(
()
)
=
()
(
()
)
=
()
.
U
g
sTs d
U
g
sNs
1
2
Let
X
=
U
1
. Using the definition of the w
ij
and the notation of Theorem 9.16.6
d
ds
Ts
(
()
)
(
( )
)
=—
(
()
)
∑
(
()
)
=
()
∑
()
=
()
∑
()
=
wg
sTs
(
)
U
g
s
U
g
s
Ns
k
Ns
Ns
k
.
12
Ts
1
2
S
S
Furthermore, since the covariant derivative of
U
1
with respect to S(s) lies in
R
2
, we have
(
()
)
(
( )
)
=—
(
()
)
∑
(
()
)
=
wg
sTs
(
)
U
g
s
U
g
s
0
.
13
Ts
1
3
The form w
23
is zero for a similar reason. We see that using our special frame field
and vector field
X
the covariant derivative equations (9.94) capture all of the
geometry of the curve, namely, its curvature. In Exercise 9.16.1 you are asked to show
an analogous fact for space curves.
It should be noted that all of our results above are local and rather than talking
about vector fields and frame fields on
R
3
we could have developed the same results
