By Jean-Pierre Eckmann, Peter Wittwer
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Ql' ... , Qt: 'ife shall make suring that it is a submersion. n x 'S Ql ' ••. , Qt. p • If the graph is not already transverse it seems reasonable that we might be able to force it to be so by -------+----------- 52 IFf x 0 bodily translating it. point. The picture above illustrates the idea when Therefore we take (x, s) f(x) + s. P Q is a is transverse to s All that remains to be shown ••• , Qt· is close enough to s then f s is as close as we please to s more formally, we have to check that given a fundamental neighbourhood V of f we can find an s I 0 for which f s lies in V.
At N C lRn For the purpose of illustration take the special case when an open set. a smooth In other words iN. By a TN -+ N. and 7T : N x lRn -+ is N is pro- jection on the first factor, so that a smooth vector field is a smooth mapping e N -+ smooth mapping. N x lRn of the form Suppose e1, x ••• , en -+ (x, e(x)) are the components of vector field aoes no more than associate with each point ... , 24 e : with x E e: N -+ lRn a then our N the vector Thus, for instance, in the case when n =2 we can sketch a vector field by the simple device of drawing at each point X E N an arrow starting at that point, and in the direction given by the Here are some examples, which the reader is advised to check for himself: e indeed a good exercise is to invent simple formulae for and sketch the corresponding vector fields.
Its virtue lies rather in the fact that it is an easily understood prototype of transversality theorems of greater complexity and application. Our object here was simply to lay bare the underlying idea behind the proofs of such theorems. ~4. Thom's Transversality Theorem Our next transversality theorem is rather more useful than that of the preceding section. Indeed it will suffice for all the applications we shall require in this book. Its statement does not possess quite the same immediate intuitive appeal as our previous result: however, in the next section we shall discuss a simple application which should clarify the situation.