2. Smooth manifolds: A review
3
by taking the union of all charts that, together with any of the ones
originally chosen, satisfy condition (c). Hence, with a certain abuse
of language, we say that a smooth manifold is a set that satisfies
conditions (a)-(c), and the extension to the maximal atlas is done
without further comment.
Examples.
(1) The Euclidean space
Rn
is an n-dimensional manifold, covered
by only one chart U
Rn,
4: U
W1
the identity map.
(2) The sphere Sn = {x = (xux2,... xn+i) G R n + 1 : x\ + x\ +
-+£^+i = 1} in R
n + 1
is a manifold of dimension n. It can be covered
by two charts U+ = {x G
Sn:
xn+i -1 } with 0+: U+ - P
n
by
M*) = (l+ftTT''' •' i4^Tr) '
a n d
^ - ^ l ^
5
^
x
-+ i
X w i t h
0_(x) = ( i _ ^ i _ ^
x
)• The maps 0+ and 0_ are called
stereographic projections.
(3) The projective space RP
n
is the set of lines in R
n + 1
that pass
through 0 G R
n + 1
. More precisely, RP
n
is the quotient space of
R
n + 1
\ {0} by the equivalence relation
(xi,...,x
n +
i ) ~ (Axi,...,Ax
n +
i), A G R \ { 0 } .
The points of RP n will be denoted by [xi,... ,xn+i\. Define the
subsets Ui = {[xi,... ,x
n +
i] : Xi ^ 0} (i = 1,... ,n + 1) of RP n .
Then the maps 0;: Ui » Rn (z = 1,..., n + 1) given by
/([xi, . . . , X
n +
i ] ) = [xiX"
1
, . . . ,
Xi-iX^1, Xi+iX'1,
. . . , Xn+iX"
1
]
are 1-1 and onto. The projective space is covered by the charts
(^l,0l),---,(£4+l,0n+l).
(4) Any open subset U of a smooth manifold M is itself a smooth
manifold. The charts of U are the intersections of U with the charts
of M.
(5) If M and N are smooth manifolds, then the Cartesian product
M x N is also a smooth manifold of dimension equal to the sum of
the dimensions of M and N.
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