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)
^ * ^
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