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
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