Criteria for Morin singularities for maps
                    into lower dimensions, and applications

Saji, Kentaro

doi:10.1090/conm/675/13599

Cited by 5 publications

(3 citation statements)

References 26 publications

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“…Note that K. Saji showed a different approach to this idea from the viewpoint of normal forms of singularities in his talk at Kobe, 2017 (similar method for singularities of surfaces are shown in [22,23,28]). Also, his papers about criteria for singularities suggest another approach [26,27].…”

Section: Introductionmentioning

confidence: 99%

One‐parameter families of Legendre curves and plane line congruences

Kabata

Takahashi

2022

Mathematische Nachrichten

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Families of curves in the Euclidean plane naturally contain singular curves, where the frame of classical differential geometry does not work well. We introduce the notions of one-parameter family of Legendre curves in the Euclidean plane, congruent equivalence and curvature. Especially, a one-parameter family of Legendre curves can contain singular curves, and is determined by the curvature up to congruence. We also give properties of one-parameter families of Legendre curves. As applications, we give a relation between one-parameter families of Legendre curves and Legendre surfaces. Moreover, we study plane line congruences (one-parameter families of lines in plane) in terms of the curvatures as one-parameter families of Legendre curves.

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Section: Introductionmentioning

confidence: 99%

One‐parameter families of Legendre curves and plane line congruences

Kabata

Takahashi

2022

Mathematische Nachrichten

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“…Criteria for types of Morin singularities of mappins from R m to R n (in case m = n = 2 they are folds and cusps) were presented in [9]. Moreover some results concerning the algebraic sum of cusps are contained in [2], [8], and in [3] in the complex case.…”

Section: Introductionmentioning

confidence: 99%

Criteria for singularities for mappings from two-manifold to the plane. The number and signs of cusps

Krzyżanowska

Nowel

2017

Kodai Math. J.

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Let M ⊂ R n+2 be a two-dimensional complete intersection. We show how to check whether a mapping f : M −→ R 2 is 1-generic with only folds and cusps as singularities. In this case we give an effective method to count the number of positive and negative cusps of a polynomial f , using the signatures of some quadratic forms. PreliminariesLet M, N be smooth manifolds such that m = dim M and n = dim N. Take p ∈ M. For smooth mappings f, g : M −→ N such that f (p) = g(p) = q, we say that f has first order contact with g at p if Df (p) = Dg(p), as mappingsq) denotes a set of equivalence classes of mappings f : M −→ N, where f (p) = q, having the same first order contact at p. Let J 1 (M, N) = (p,q)∈M ×N J 1 (M, N) (p,q)denote the 1-jet bundle of smooth mappings from M to N.With any smooth f : M −→ N we can associate a canonical mapping j 1 f : M −→ J 1 (M, N). Take σ ∈ J 1 (M, N), represented by f . Then by corank σ we denote the corank Df (p). Put S r = {σ ∈ J 1 (M, N) | corank σ = r}. According to [4, II, Theorem 5.4], S r is a submanifold of J 1 (M, N), with codim S r = r(|m − n| + r). Put S r (f ) = {x ∈ M | corank Df (p) = r} = (j 1 f ) −1 (S r ).Definition 2.1. We say that f : M −→ N is 1-generic if j 1 f ⋔ S r , for all r.According to [4, II, Theorem 4.4], if j 1 f ⋔ S r then either S r (f ) = ∅ or S r (f ) is a submanifold of M, with codim S r (f ) = codim S r .In the remaining we will need the following useful fact.

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“…Therefore, Morin singularities are fundamental and frequently arise as singularities of maps from one manifold to another, as observed by K. Saji in [12]. Morin singularities have been studied by many authors in different contexts as [7,1,4,10,11], and more recently [6,15,18,5,2,12,14,13,9]. In particular, papers of J.M.…”

Section: Introductionmentioning

confidence: 99%

Morin singularities of coframes and frames

Ruiz¹

2016

Preprint

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Inspired by the properties of an n-frame of gradients (∇f 1 , . . . , ∇fn) of a Morin map f : M → R n , with dim M ≥ n, we introduce the notion of Morin singularities in the context of singular n-coframes and singular n-frames. We also study the singularities of generic 1-forms associated to a Morin n-coframe, in order to generalize a result of T. Fukuda [4, Theorem 1], which establishes a modulo 2 congruence between the Euler characteristic of a compact manifold M and the Euler characteristics of the singular sets of a Morin map defined on M , to the case of Morin n-coframes and Morin n-frames.

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Criteria for Morin singularities for maps into lower dimensions, and applications

Abstract: We give criteria for Morin singularities for germs of maps into lower dimensions. As an application, we study the bifurcation of Lefschetz singularities.

Cited by 5 publications

References 26 publications

One‐parameter families of Legendre curves and plane line congruences

One‐parameter families of Legendre curves and plane line congruences

Criteria for singularities for mappings from two-manifold to the plane. The number and signs of cusps

Morin singularities of coframes and frames

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