Math 497C
Mar 3, 2004
1
Curves and Surfaces
Fall 2004, PSU
Lecture Notes 11
2.4 Intrinsic Metric and Isometries of Surfaces
Let
M
⊂
R
3
be a regular embedded surface and
p
,
q
∈
M
, then we deFne
dist
M
(
p, q
) := inf
{
Length[
γ
]

γ
:[0
,
1]
→
M, γ
(0) =
p, γ
(1) =
q
}
.
Exercise 1.
Show that (
M,
dist
M
) is a metric space.
Lemma 2.
Show that if
M
is a
C
1
surface, and
X
⊂
M
is compact, then
for every
²>
0
, there exists
δ>
0
such that
¯
¯
dist
M
(
p, q
)
−k
p
−
q
k
¯
¯
≤
²
k
p
−
q
k
for all
p
,
q
∈
X
, with
dist
M
(
p, q
)
≤
δ
.
Proof.
DeFne
F
:
M
×
M
→
R
by
F
(
p, q
) := dist
M
(
p, q
)
/
k
p
−
q
k
,i
f
p
6
=
q
and
F
(
p, q
) := 1 otherwise. We claim that
F
is continuous. To see this let
p
i
be a sequnce of points of
M
which converge to a point
p
∈
M.
We may
assume that
p
i
are contained in a Monge patch of
M
centered at
p
given by
X
(
u
1
,u
2
)=(
u
1
2
,h
(
u
1
2
))
.
Let
x
i
and
y
i
be the
x
and
y
coorindates of
p
i
.I
f
p
i
is suﬃciently close to
p
=(0
,
0), then, since
∇
h
(0
,
0) = 0, we can make sure that
k∇
h
(
tx
i
,ty
i
)
k
2
≤
²,
for all
t
∈
[0
,
1] and
0. Let
γ
,
1]
→
R
3
be the curve given by
γ
(
t
tx
i
i
(
tx
i
i
))
.
1
Last revised: November 8, 2004
1
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View Full DocumentThen, since
γ
is a curve on
M
,
dist
M
(
p, p
i
)
≤
Length[
γ
]
=
Z
1
0
q
x
2
i
+
y
2
i
+
h∇
h
(
tx
i
,ty
i
)
,
(
x
i
,y
i
)
i
2
dt
≤
Z
1
0
q
x
2
i
+
y
2
i
+
²
(
x
2
i
+
y
2
i
)
2
dt
≤
√
1+
²
q
x
2
i
+
y
2
i
≤
(1 +
²
)
k
p
−
p
i
k
So, for any
²>
0wehave
1
≤
dist
M
(
p, p
i
)
k
p
−
p
i
k
≤
²
provided that
p
i
is suﬃciently close to
p
. We conclude then that
F
is con
tinuous. So
U
:=
F
−
1
([1
,
²
]) is an open subset of
M
×
M
which contains
the diagonal ∆
M
:=
{
(
p, p
)

p
∈
M
}
. Since ∆
X
⊂
∆
M
is compact, we may
then choose
δ
so small that
V
δ
(∆
X
)
⊂
U
, where
V
δ
(∆
X
) denotes the open
neighborhood of ∆
X
in
M
×
M
which consists of all pairs of points (
p, q
)
with dist
M
(
p, q
)
≤
δ
.
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 Math, Geometry, Trigraph, Det, det gij, det X1, det lij

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