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.