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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