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In-Depth Information
—
—
f
f
()
=
()
np
p
.
Now, by Theorem 9.10.3(2), g is a geodesic if and only if
≤
()
=
()
(
()
)
g
t
a
t
n
g
t
for some function a. It follows that
=≤∑
(
)
=¢ ∑
(
(
)
)
¢-
(
¢∑
(
)
¢
)
=- ¢∑
(
)
¢
ag
n
o
g
g
n
o
g
g
n
o
g
g
n
o
g
,
because g¢ •(
n
g) = 0. In other words, g is a geodesic if and only if
(
¢∑
(
)
¢
)(
)
=
gg
≤+
nn
o
g
o
g
0.
(9.73)
If
n
(
p
) = (n
1
(
p
),n
2
(
p
),n
3
(
p
)) and g(t) = (x
1
(t),x
2
(t),x
3
(t)), then solving equation (9.73)
reduces to solving the three second-order differential equations
2
3
∂
∂
n
x
xx x
dx
dt
dx
dt
dx
dt
j
j
i
Â
k
(
)
(
)
+
nxx x
,
,
,
,
=
0
,
i
=
1 2 3
, , .
(9.74)
i
123
123
2
k
jk
,
=
1
The theorem now follows from theorems about solutions to such equations. See
[Thor79] for more details. In fact it is shown there that there is a
maximal
open inter-
val containing 0 over which the unique geodesic g(t) is defined.
The form of the result in Theorem 9.10.12 leads to a natural question. Can one
extend the domain of the geodesic g in the theorem from (-e,e) to
all
of
R
?
Definition.
A surface
S
is said to be
geodesically complete
if every geodesic g : (a,b)
Æ
S
extends to a geodesic
:
R
Æ
S
.
g
Neither the open unit disk nor
R
2
9.10.13. Example.
-
0
is geodesically complete.
9.10.14. Theorem.
(The Hopf-Rinow Theorem)
(1) A surface is geodesically complete if and only if it is complete in the topolog-
ical sense.
(2) If a surface is geodesically complete, then any two points can be joined by
minimal-length geodesics.
(3) In a closed and compact surface there is a minimal-length geodesic between
any two points.
Proof.
See [Hick65] or [McCl97]. Part (2) implies part (3) because by Theorem 5.5.7
every compact metrizable space is complete.
Moving on to property (4) of straight lines, we first need to define what it means
for a vector field along a curve to consist of parallel lines. Let
S
be a surface in
R
3
and let
