1. Differentiable Manifolds 11

1.14. Products. Let M and N be smooth manifolds described by smooth

atlases {Ua^ua)oceA and (Vp,vp)p£B, respectively. Then the family (Ua x

Vp,uaxvp : Ua x Vp —

W71 xRn)(a,p)eAxB ls a

smooth atlas for the cartesian

product M x N. Clearly the projections

are also smooth. The product (M x iV,prl5pr2) has the following universal

property:

For any smooth manifold P and smooth mappings / : P — » M and

g : P — A T the mapping

(/,?) :P^MxN, (f,g)(x) = (f(x),g(x)),

is the unique smooth mapping with prx o(/, g) = f and pr2 o(/, 5) = #.

From the construction of the tangent bundle in (1.9) it is immediately clear

that

TM

(

T ( p r i )

T(M x N)

T(pr2 )

) TiV

is again a product, so that T(M x N) = TM x TN in a canonical way.

Clearly we can form products of finitely many manifolds.

1.15. Theorem. Let M be a connected manifold and suppose that f : M —

M is smooth with f o f = f. Then the image f(M) of f is a submanifold

ofM.

This result can also be expressed as: 'smooth retracts' of manifolds are

manifolds. If we do not suppose that M is connected, then f(M) will not

be a pure manifold in general; it will have different dimensions in different

connected components.

Proof. We claim that there is an open neighborhood U of f(M) in M such

that the rank of Tyf is constant for y G U. Then by theorem (1.13) the

result follows.

For x e f{M) we have Txf o Txf = Txf; thus imTxf = kev(Id - Txf) and

mnkTxf + rank(/d - Txf) = dimM. Since rankT

x

/ and rank(/d - Txf)

cannot fall locally, rankT^/ is locally constant for x G /(M), and since

f(M) is connected, rankT

x

/ = r for all x G f(M).

But then for each x G f(M) there is an open neighborhood Ux in M with

rank Tyf r for all y G Ux. On the other hand

rankT

y

/ = rankT

y

(/ o / ) = rankT

/ ( y )

/ o Tyf TtmkTf{y)f = r

since f(y) G f(M).

So the neighborhood we need is given by U =

UXG/(M)

^ * ^