Graphics Reference
In-Depth Information
(3) Given any
Z
ΠVect(
M
) that agrees with
X
on h and any
T
ΠVect(
M
) that
agrees with h¢(t) on h, then
D
dt
X
()
=
(
)
(
( )
)
t
Z
h
t
.
T
(A vector field
A
ΠVect(
M
)
agrees
with a vector field
B
ΠVect(
M
,h) on h if
A
(h(t)) =
B
(t).)
One can show that every connection defines a unique covariant derivative along
curves. This is an easy consequence of the axioms. Specifically, let
n
∂
∂
Â
a
-
1
[
]
Æ
n
X
=
and
g
=
F
o
h
:
a b
,
R
,
i
u
i
i
=
1
where a
i
: [a,b] Æ
R
. Then
n
n
È
Í
˘
˙
D
dt
X
da
dt
d
dt
g
∂
Â
k
Â
i
i
k
=
+
G
a
.
j
∂
u
k
k
=
1
ij
,
=
1
Definition.
A vector field
X
(t) along a curve h is called a
parallel vector field
along h
if D
X
/dt =
0
.
One can show just like in Section 9.10 that a vector at the start point of a curve
defines a unique parallel vector field along the curve.
Definition.
A connection on
M
is said to be
compatible with the metric
on
M
if any pair of parallel vector fields along a curve h(t) have a constant inner
product, that is, if
X
(t) and
Y
(t) are parallel vector fields along h, then <
X
(t),
Y
(t)> is
constant.
If one has a connection compatible with the metric, then the covariant derivative
along a curve satisfies
d
dt
D
dt
X
YX
Y
D
dt
<
XY
,
>=<
,
>+<
,
>
.
Definition.
A connection —
X
Y
on
M
is said to be
symmetric
or
torsion-free
if it satisfies
[
]
—-—
YXXY
,
,
X
where [
X
,
Y
] is the vector field defined by
[
]
()
-
()
XY
,
f
=
X Y
f
Y X
f
p
p
p
for
p
Œ
M
and f ΠC
•
(
M
) called the
Lie bracket
of
X
and
Y
.
