10 CHAPTER L Manifolds and Vector Fields d d(z) - I . f dg* \k+liq is invertible, so u is a diffeomorphism U\ — » U2 for suitable open neighbor- hoods of 0 in W1. Consider g = f o u~l : U2 - K9. Then we have 5(21, ...,zn) = (zi,... ,zk,gk+1(z),.. .,gq{z)), \ k 0 \ 0zJ Jk+ljnJ rank(dg(z)) = rank {d{f o u - 1 ) ^ ) ) = rank ( # ( t r ^ . c f o r 1 ^ ) ) = rank(#(s)) = k. dgl Therefore, TT^{Z) = 0 for & + 1 i a and fc + 1 j n: gi(z\...,zn)=gi(z\... 9 zk 9 0,...,0) fork + liq. Let v : U3 - • R9, where [73 = {y G R9 : (y 1 ,..., yk, 0,..., 0) G U2 C R n }, be given by / A ^7 \ 2/ *+l _ n H U „ l y" gk+\y\...,yk,0,...,0) ( \ yk+1-gk+1(y) V yq-9q(y) J \ yq-g'1(y\...,yk,0,...,0) J where y = (y1,..., yq, 0,..., 0) € Rn if q n and y = (y 1 ,..., yn) if q n. We have v(0) = 0, and is invertible thus v : F — R9 is a chart for a suitable neighborhood of 0. Now let U := / " W - l l / i . Then vo feu'1 = vog : Rn D u(U) -»• v(V) C R« looks as follows: cfr x i X " / x1 \ X ?fc+1(z) V 9q(x) J ( k+li ^ W - ^ W V ^(x)-^(x) / . = (xl\ xk 0 w • Corollary. Let f : M -^ N be C°° with Txf of constant rank k for all x G M. Then for each b G f(M) the set /_1(&) C M is a submanifold of M of dimension dimM — k. •

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