1. Differentiable Manifolds 9 If / : M -• N and g : N - P are smooth, then we have T(p of) = Tgo Tf. This is a direct consequence of (g o /)* = /* o #*, and it is the global version of the chain rule. Furthermore we have T(IdM) — IdTM- If / G C°°(M), then T / : TM - ^ T R ^ l x R . We define the differential of / by df := pr2 oT/ : TM - R. Let t denote the identity function on R. Then (Tf.Xx)(t) = Xx(t o / ) - **(/) , so we have df(Xx) = Xx(f). 1.12. Submanifolds. A subset N of a manifold M is called a submanifold if for each x € N there is a chart (C7, u) of M such that u(U D N) = u{U) H (Rk x 0), where R fe xO-IR fc x Rn~k = Rn. Then clearly N is itself a manifold with (U Pi AT, u\(U fl AT)) as charts, where (£/, n) runs through all submanifold charts as above. 1.13. Let / : Rn — R9 be smooth. A point x G R9 is called a regular value of / if the rank of / (more exactly: the rank of its derivative) is q at each point y of f~1(x). In this case, f~1(x) is a submanifold of Rn of dimension n — q (or empty). This is an immediate consequence of the implicit function theorem, as follows: Let x = 0 G Rq. Permute the coordinates (x1,. • • , xn) on Rn such that the Jacobi matrix has the left hand part invertible. Then u := (/,pr n _ 9 ) : Rn -+ R9 x Rn~q has invertible differential at y, so (U,u) is a chart at any y G / - 1 (0) , and we have / oiT" 1 ^ 1 ,..., zn) = (z1,..., ^ ) , so u(/ - 1 (0)) - u(E7) n (0 x Rn~q) as required. Constant rank theorem ([41, I 10.3.1]). Let f : W -• Rq be a smooth mapping, where W is an open subset ofRn. If the derivative df(x) has constant rank k for each x G W, then for each a EW there are charts (U, u) of W centered at a and (V, v) of Rq centered at f(a) such that v o / o u~l : u(U) — v(V) has the following form: (xi,..., xn) *-+ (xi,... , x/c, 0,..., 0). So f~1(b) is a submanifold of W of dimension n — k for each b G f(W). Proof. We will use the inverse function theorem several times. The deriva- tive df(a) has rank k n,q without loss we may assume that the upper left (k x fc)-submatrix of df(a) is invertible. Moreover, let a = 0 and f(a) = 0. We consider u: W - Rn, u(x \ ... ,x n ) := (f1^), • • •, fk(x),xk+1,... ,x n ). Then ( (df^\lik (df^\lik \ {dzi'ljk \dz*'k+ljn\ 0 IRn-fc I

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