Differential Topology - Computer Graphics and Geometric Modeling - page 549

n

j i : UR Æ

i

by

c
c

c
c

c
c

c

) = Ê
Ë

ˆ
¯

1 i - 1 i + 1 n + 1

(

[ ]

j i cc , ,..., c ,..., , ,..., .

(8.46)

12 n + 1

c

i i i i

Clearly, the sets U i cover P n and the maps j i are well defined, one-to-one, and onto.

Note that

- 1

( ) = [

]

j i x ,..., x x ,..., x

1 1

, , x ,..., x .

1 n 1 i - i n

The map j i will be called the ith standard projection of P n onto R n . The Definition.

map j i -1 will be called the ith standard imbedding of R n in P n .

This extends the terminology introduced in Section 3.5 where we called j n+1
the standard projection of P n onto R n and j n+1 -1 the standard imbedding of R n in

P n .

Next, if

UUU =«

ij i j

and

- 1 :

( Æ ()

jjjj = o U j U ,

ji j i ij j ij

i

then

x
x

x
xx

1 x

x

x
x

) = Ê
Ë

ˆ
¯

1 i - 1 i + 1 n

j

j ji

(

x ,..., x ,..., , , ,...,

1 n

j j j j

is a C • map. This shows that P n is an n-dimensional C • manifold.

Definition. Consider n-dimensional projective space P n as the quotient space of S n
where we have identified antipodal points. Let [ p ] ΠP n denote the equivalence class
of p ΠS n . Define the canonical line bundle g n = ( E ,p, P n ) over P n as follows:

(1) E = {([ p ],t p ) Œ P n ¥ R n+1 | t Œ R }.

(2) p([ p ],t p ) = [ p ].
(3) For local coordinate charts, we choose any open set U ˜ Ã S n that does not
contain any antipodal points and let U =p( U ˜ ) Ã P n . Define a homeomorphism

¥Æ ()

-1

j : UR p U

U

by

() = ( ) j U x ,t x t , q ,

where q Œ U ˜ is the unique point so that x = [ q ].