By Bryant D. E., Lindner C. C.
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Is a collection of ~nooth vector fields weakly controlled if (VFp) holds for all be a smooth map, stratified by A'. (VF~) (A, A'), and let We say that the stratified vector field ~he condition Tfo~ N. =~' o f Instead of (VFp) we also consider the following weaker condition on 51 at a stratum Y E A : if f (VFpf) We say that ~ maps Y into Y' E A then, as a germ at Y, Tp Y o ~ I f-lY ' = 0 . is controlled over ~' if ~ satisfies all conditions of type (VFf), (VF~) and (VFpf). The reader may verify himself that the vector fields in ( l .
A) y E C : this case is trivial. (b) fy ~ X' : we want to show s < ty, + so let us assume J = (eyI[0, t~))-IIE([0, J is an open subset of siC, ~)r~ f-l(e~y[0, [0, ty) t+ < s Y and look at the set s])l . 4) that 57 § J is compact. Since 0 E J we conclude that ty) + is compact, which is absurd. [0, Thus J = [0, ty), hence that tY§ as desired. s < We are left with (c) fy E X' Aa F~ ITI u-I f but by p y ~ C, (A' ' and hence A' ) ' ~' by py > 0: we argue as in (b), replacing [0, co), stratified by the origin and its complement, by the zero vector field.
Without more j . 27 ado we shall come to the point of partial stratifications. 1) Le_~t f : N - + P values is closed ~ if f be a smooth manpin~ whose set admits a partial stratification C @ of critical then f admits a Thom stratification. Proof Fist, note that we obtain a Whitney stratification augmenting @ P : (P - C) since by any stratum of (P - C), which is open in is open in @ . P ~' Z f-1(p _ C) , N . in ~' with ~ f(X) C U , f(Y) C V . or disjolut from (i) ~ , f by of N by taking with I claim first that To this end let From the very construction of X f-Iw ~ P , f'dW - $ the critical set of a Whitney stratification of P hence a smooth submanifold of And clearly we obtain a stratification and of it will automatically be Whitney regular over its strata to be sets of the form a stratum of P , ~' X, Y be strata in ~ ~ is ~ .