Graphics Reference
In-Depth Information
(
)
j
i
:
U
Æ-
11
,
i
and
1
(
)
=
US
U
=
,
j
j
j
j
xy
,
x
,
1
+
1
{
}
(
)
γ
1
(
)
=
=
xy
,
S
x
0
,
xy
,
y
,
2
2
1
US
U
=
,
(
xy
,
)
=
x
,
-
3
3
{
}
1
(
)
Σ
(
)
=
=
xy
,
S
x
0
,
xy
,
y
,
4
4
induce a C
•
structure on the unit circle
S
1
(Exercise 8.8.1(a)). Note that the maps j
i
are just the inverses of the maps F
i
in Example 8.3.2.
To find a C
•
structure for the unit sphere
S
n
, n ≥ 1.
8.8.4. Example.
Solution.
Rather than generalizing the coordinate neighborhoods defined in
Example 8.8.3, we describe another collection {(
U
+
,j
+
),(
U
-
,j
-
)}:
n
{}
n
US e
=
,
j
j
:
U R
Æ
,
+
n
+
1
+
+
=-
{
n
}
n
US
e
,
:
U R
Æ
,
-
n
+
1
-
-
where j
+
and j
-
are the stereographic projections from
e
n+1
and -
e
n+1
, respectively,
that is, j
+
= p
n
and j
-
= p
n
r, where r is the reflection of
R
n+1
about the plane x
n+1
=
0. These coordinate neighborhoods induce a C
•
structure on
S
n
(Exercise 8.8.1(b)).
o
Let
M
n
and
N
k
be C
r
manifolds with C
r
structure coordinate neighborhoods
{(
U
i
,j
i
)} and {(
V
j
,y
j
)}, respectively. Define maps
¥Æ ¥ =
+
n
k
n
k
h
ij
:
UVRRR
i
j
by
(
)
=
(
()
()
)
h
u, v
j
u
,
y
v
.
ij
i
j
It is easy to check that the coordinate neighborhoods {(
U
i
¥
V
j
,h
ij
)} induce a C
r
struc-
ture on
M
¥
N
unless both manifolds have boundary, in which case one has to make
some modifications to get legitimate coordinate neighborhoods for the points of ∂
M
¥∂
N
. We will not present the details here, but they are straightforward.
Definition.
The C
r
structure induced by the coordinate neighborhoods {(
U
i
¥
V
j
,h
ij
)}
is called the
product C
r
structure
. It makes
M
¥
N
into an (n + k)-dimensional C
r
man-
ifold called the
product C
r
manifold
.
Next, we define the concept of a differentiable map.
Definition.
Let
M
n
and
N
k
be C
r
manifolds with coordinate neighborhoods {(
U
i
,j
i
)}
and {(
V
j
,y
j
)}, respectively. A map f :
M
n
Æ
N
k
is said to be of
class C
r
or a
C
r
map
if
-
1
:
()
Æ
k
yjj
oo
f
UR
j
i
i
i