Math 497C
Oct 7, 2004
1
Curves and Surfaces
Fall 2004, PSU
Lecture Notes 9
2.2
Definition of Gaussian Curvature
Let
M
⊂
R
3
be a regular embedded surface, as we defined in the previous
lecture, and let
p
∈
M
. By the
tangent space
of
M
at
p
, denoted by
T
p
M
,
we mean the set of all vectors
v
in
R
3
such that for each vector
v
there exists
a smooth curve
γ
: (
−
,
)
→
M
with
γ
(0) =
p
and
γ
(0) =
v
.
Exercise 1.
Let
H
⊂
R
3
be a plane. Show that, for all
p
∈
H
,
T
p
H
is the
plane parallel to
H
which passes through the origin.
Exercise 2.
Prove that, for all
p
∈
M
,
T
p
M
is a 2dimensional vector sub
space of
R
3
(
Hint:
Let (
U, X
) be a proper regular patch centered at
p
, i.e.,
X
(0
,
0) =
p
. Recall that
dX
(0
,
0)
is a linear map and has rank 2. Thus it
suﬃces to show that
T
p
M
=
dX
(0
,
0)
(
R
2
)).
Exercise 3.
Prove that
D
1
X
(0
,
0) and
D
2
X
(0
,
0) form a basis for
T
p
M
(
Hint:
Show that
D
1
X
(0
,
0) =
dX
(0
,
0)
(1
,
0) and
D
2
X
(0
,
0) =
dX
(0
,
0)
(0
,
1)).
By a
local gauss map
of
M
centered at
p
we mean a pair (
V, n
) where
V
is an open neighborhood of
p
in
M
and
n
:
V
→
S
2
is a continuous mapping
such that
n
(
p
) is orthogonal to
T
p
M
for all
p
∈
M
. For a more explicit
formulation, let (
U, X
) be a proper regular patch centered at
p
, and define
N
:
U
→
S
2
by
N
(
u
1
, u
2
) :=
D
1
X
(
u
1
, u
2
)
×
D
2
X
(
u
1
, u
2
)
D
1
X
(
u
1
, u
2
)
×
D
2
X
(
u
1
, u
2
)
.
Set
V
:=
X
(
U
), and recall that, since (
U, X
) is proper,
V
is open in
M
. Now
define
n
:
V
→
S
2
by
n
(
p
) :=
N
◦
X
−
1
(
p
)
.
1
Last revised: October 8, 2004
1
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Exercise 4.
Check that (
V, n
) is indeed a local gauss map.
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 Math, Geometry

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