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