Finite Determinacy of Smooth Map-Germs

Wall, C. T. C.

doi:10.1112/blms/13.6.481

Cited by 356 publications

(339 citation statements)

References 54 publications

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“…We refer to the survey article of Wall [15], for background singularity theory. A few further remarks concerning multigerms will be helpful.…”

Section: Multigerms: Preliminaries and Techniquesmentioning

confidence: 99%

On the classification and bifurcation of multigerms of maps from surfaces to 3-space

Kirk¹,

Hobbs²

2001

MATH. SCAND.

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The A -classification of multigerm singularities is discussed, based on the theory of complete transversals. An A -classification of r-multigerms from the plane to 3-space of A -codimension ≤ 6 − r is carried out and the bifurcation geometry of these singularities analysed. This work has applications to the study of two-dimensional spatial motions, giving local models for the singularities which appear on general trajectories of rigid body motions from the plane to 3-space. In addition, our classification is extensive enough to give the full list of simple multigerm singularities from the plane to 3-space.

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“…We refer to the survey article of Wall [15], for background singularity theory. A few further remarks concerning multigerms will be helpful.…”

Section: Multigerms: Preliminaries and Techniquesmentioning

confidence: 99%

On the classification and bifurcation of multigerms of maps from surfaces to 3-space

Kirk¹,

Hobbs²

2001

MATH. SCAND.

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“…The question is to determine whether a function germ is characterized (up to some equivalence relation) by its Taylor expansion at the origin. Many authors have given versatile criteria during the three last decades [Wa,Ku1,Ku2,Ku3,KuT,TW,Ma,Ta,Y,Ko1,Ko2,Wi].…”

Section: Introductionmentioning

confidence: 99%

Bi-Lipschitz Sufficiency of Jets

Valette

2009

J Geom Anal

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Abstract. We give some theorems of bi-Lipschitz or C 1 sufficiency of jets which are expressed by means of transversality with respect to some strata of a stratification satisfying the (L) condition of T. Mostowski. This enables us to prove that the number of metric types of intersection of smooth transversals to a stratum of an (a) regular stratification of a subanalytic set is finite. IntroductionIn this paper we study the problem of sufficiency of jets. This is a classical problem of singularity theory. The question is to determine whether a function germ is characterized (up to some equivalence relation) by its Taylor expansion at the origin. Many authors have given versatile criteria during the three last decades [Wa, Ku1, Ku2, Ku3, KuT, TW, Ma, Ta, Y, Ko1, Ko2, Wi].We give here several results about sufficiency of jets. We study the intersection of a stratified space with a transversal to a given stratum, the zero locus of maps and the maps themselves up to a homeomorphism. We focus on the bi-Lipschitz or C 1 equivalence in each case.Our criteria for sufficiency of jets are proved by studying stratifications satisfying the (L) condition of T. Mostowski. This condition has been introduced to obtain the biLipschitz triviality along the strata. Existence of stratifications satisfying this condition has been proved for complex analytic or real subanalytic sets [M, P].It is important to note that for determinacy of transversal (section 4) we will not assume that the given stratum is a stratum of an (L) regular stratification (which would be a rather strong assumption). We will just fix an (L) regular stratification which is compatible with our given stratum. Actually such a stratification always exists for a given subanalytic set X. Then, we will put transversality conditions with respect to this stratification generalizing the results of [TW].In the two strata case we will get some results of C 1 determinacy. It is interesting to note that these theorems will apply not only on manifolds but also on the more general class of spaces having an isolated singularity at the origin. Moreover the criteria obtained are better than the ones given in [Wa] or [Ta] in the sense that the order of determinacy is lower.We end this paper by two further applications of the results of section 3. The first one is a finiteness result which is also a generalization of a result of [TW]. We prove that the number of Lipschitz types of intersection of smooth direct transversals at a given point is finite when the stratification satisfies the Whitney (a) condition (Theorem 6.1.1). The second one is about Kuo and Trotman's blowing up. This is a transformation of stratified spaces introduced by T. C. Kuo and D. Trotman which is known to improve the regularity.In [TW] the authors provide explicit criteria using the (t) condition about stratifications. This condition has been introduced by R. Thom and is weaker than the classical ones (Whitney and Kuo-Verdier). It does not guarantee the topological triviality along the strata. It is prov...

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“…See [24] [31]. We denote by Diff + 0 (S 1 , S) the group of orientation preserving diffeomorphism-germs (S 1 , S) fixing S pointwise, and we treat f y0 up to Diff…”

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confidence: 99%

Global classification of curves on the symplectic plane

Ishikawa¹

2008

Real and Complex Singularities

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Finite Determinacy of Smooth Map-Germs

Cited by 356 publications

References 54 publications

On the classification and bifurcation of multigerms of maps from surfaces to 3-space

On the classification and bifurcation of multigerms of maps from surfaces to 3-space

Bi-Lipschitz Sufficiency of Jets

Global classification of curves on the symplectic plane

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