Maria Aparecida Soares Ruas

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• Professor Emeritus at University of São Paulo

In this paper we study the bi-Lipschitz triviality of deformations of an analytic function germ $f$ defined on a germ of an analytic variety $(X, 0)$ in $\mathbb C^n$. We introduce the notion of strongly rational $\mathscr R_X$-bi-Lipschitz trivial families and give an infinitesimal criterion which is a sufficient condition for the bi-Lipschitz tri...

In this work, we describe a prenormal form for the generators of the semigroup of a toric variety $X \subset \mathbb{C}^p$ with isolated singularity at the origin and smooth normalization. A complete description of the semigroup is given when $X$ is a variety of dimension $n$ in $\mathbb{C}^{2n}$. Moreover, for toric surfaces in $\mathbb{C}^4$, we...

We provide bi-Lipschitz invariants for finitely determined map germs f: (\K^n, 0) → (\K^p, 0), where \K = \R or \C. The aim of the paper is to provide partial answers to the following questions: “Does the bi-Lipschitz type of a map germ f: (\R^n, 0) → (\R^p, 0) determine the bi-Lipschitz type of the link of f and of the double point set of f? Reci...

We show that the Nash blowup of 2-generic determinantal varieties over fields of positive characteristic is non-singular. We prove this in two steps. Firstly, we explicitly describe the toric structure of such varieties. Secondly, we show that in this case the combinatorics of Nash blowups are free of characteristic. The result then follows from th...

We introduce the module of derivations $\Theta_{h,M}$ attached to a given analytic map $h:(\mathbb C^n,0)\to (\mathbb C^p,0)$ and a submodule $M\subseteq \mathcal O_n^p$ and analyse several exact sequences related to $\Theta_{h,M}$. Moreover, we obtain formulas for several numerical invariants associated to the pair $(h,M)$ and a given analytic map...

Given a germ of an analytic variety X and a germ of a holomorphic function f with a stratified isolated singularity with respect to the logarithmic stratification of X, we show that under certain conditions on the singularity type of the pair (f, X), the following relative analog of the well-known K. Saito’s theorem holds true: equality of the rela...

In this work we define some map-germs, called elementary joins, for the purpose of producing new ${\mathcal A}$-finite map-germs from them. In particular, we describe a general form of an ${\mathcal A}$-finite monomial map from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{p},0)$ for $p\geq 2n$ of any corank in terms of elementary join maps. Our main tools a...

We refine the affine classification of real nets of quadrics in order to obtain generic curvature loci of regular 3‐manifolds in R6$\mathbb {R}^6$ and singular corank one 3‐manifolds in R5$\mathbb {R}^5$. For this, we characterize the type of the curvature locus by the number and type of solutions of a system of equations given by four ternary cubi...

Given a germ of an analytic variety $X$ and a germ of a holomorphic function $f$ with a stratified isolated singularity with respect to the logarithmic stratification of $X$, we show that under certain conditions on the singularity type of the pair $(f,X)$, the following relative analog of the well known K. Saito's theorem holds true: equality of t...

This paper is a first step in order to extend Kummer's theory for line congruences to the case $\lbrace x, \xi \rbrace $, where $x: U \rightarrow \mathbb{R}^3$ is a smooth map and $\xi: U \rightarrow \mathbb{R}^3$ is a proper frontal. We show that if $\lbrace x, \xi \rbrace$ is a normal congruence, the equation of the principal surfaces is a multip...

In this paper, we give the generic classification of the singularities of 3-parameter line congruences in $\mathbb {R}^{4}$ . We also classify the generic singularities of normal and Blaschke (affine) normal congruences.

We refine the affine classification of real nets of quadrics in order to obtain generic curvature loci of regular $3$-manifolds in $\mathbb{R}^6$ and singular corank $1$ $3$-manifolds in $\mathbb{R}^5$. For this, we characterize the type of the curvature locus by the number and type of solutions of a system of equations given by 4 ternary cubics (w...

We relate the moduli space of analytic equivalent germs of reduced quasi-homogeneous functions at (C2,0) with their bi-Lipschitz equivalence classes. We show that any non-degenerate continuous family of (reduced) quasi-homogeneous (but not homogeneous) functions with constant Henry–Parusiński invariant is analytically trivial. Further, we show that...

Density of stable maps is the common thread of this paper. We review Whitney's contribution to singularities of differentiable mappings and Thom-Mather theories on $C^{\infty}$ and $C^{0}$-stability. Infinitesimal and algebraic methods are presented in order to prove Theorem A and Theorem B on density of proper stable and topologically stable mappi...

We study 3-manifolds in \({\mathbb {R}}^5\) with corank 1 singularities. At the singular point we define the curvature locus using the first and second fundamental forms, which contains all the local second order geometrical information about the manifold. Also, we relate the geometry of these objects to the geometry of regular 3-manifolds in \({\m...

We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in $$C^{\infty }(N\times [0,1],P)$$ C ∞ ( N × [ 0 , 1 ] , P ) is dense if and only if the pair of dimensions $$(\dim N, \dim P)$$ ( dim N , dim P ) is in the extra-nice dimensions. This result is parallel to Mather’s characterization of the nice dim...

In this paper, we give the generic classification of the singularities of 3-parameter line congruences in $\mathbb{R}^4$. We also classify the generic singularities of Blaschke (affine) normal congruences.

In this work we define some map-germs, called elementary joins, for the purpose of producing new ${\mathcal A}$-finite map-germs from them. In particular, we describe a general form of an ${\mathcal A}$-finite monomial map from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{p},0)$ for $p\geq 2n$ of any corank in terms of elementary join maps. Our main tools a...

In this course we introduce the main tools to study the Lipschitz geometry of real and complex singular sets and mappings: the notions of semialgebraic sets and mappings and basic notions of Lipschitz geometry. The course then focuses on the real setting, presenting the outer Lipschitz classification of semialgebraic curves, the inner classificatio...

For given natural numbers d1,d2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_1,d_2$$\end{document} we describe the topology of a generic polynomial mapping F=(f,g):...

It was shown by Henry and Parusiński in 2003 that the bi‐Lipschitz right equivalence of function germs admits moduli. In this article, we introduce the notion of Lipschitz simple function germ and present the complete classification in the complex case. For this, we present several bi‐Lipschitz invariants associated to functions germs. In particula...

Denote by $H(d_1,d_2,d_3)$ the set of all homogeneous polynomial mappings $F=(f_1,f_2,f_3): \C^3\to\C^3$, such that $\deg f_i=d_i$. We show that if $\gcd(d_i,d_j)\leq 2$ for $1\leq i<j\leq 3$ and $\gcd(d_1,d_2,d_3)=1$, then there is a non-empty Zariski open subset $U\subset H(d_1,d_2,d_3)$ such that for every mapping $F\in U$ the map germ $(F,0)$...

We relate the moduli space of analytic equivalent germs of reduced quasi-homogeneous functions at $(\mathbb{C}^2,0)$ with their bi-Lipschitz equivalence classes. We show that any non-degenerate continuous family of (reduced) quasi-homogeneous functions with constant Henry-Parusi\'nski invariant is analytically trivial. Further we show that there ar...

We extend the notions of μ*-sequences and Tjurina numbers of functions to the framework of Bruce–Roberts numbers, that is, to pairs formed by the germ at 0 of a complex analytic variety X ⊆ ℂ ⁿ and a finitely ${\mathcal R}(X)$ -determined analytic function germ f : (ℂ ⁿ , 0) → (ℂ, 0). We analyze some fundamental properties of these numbers.

In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of itself. The parabolic set itself is detected in this way, and each arc is given a sign, which changes at the s...

We study 3-manifolds in $\mathbb{R}^5$ with corank $1$ singularities. At the singular point we define the curvature locus using the first and second fundamental forms, which contains all the local second order geometrical information about the manifold.

We study the geometry of surfaces in ℝ4 with corank 1 singularities. For such surfaces, the singularities are isolated and, at each point, we define the curvature parabola in the normal space. This curve codifies all the second-order information of the surface. Also, using this curve, we define asymptotic and binormal directions, the umbilic curvat...

Denote by H(d1, d2, d3) the set of all homogeneous polynomial mappings F = (f1, f2, f3) : C 3 → C 3 , such that deg fi = di. We show that if gcd(di, dj) ≤ 2 for 1 ≤ i < j ≤ 3 and gcd(d1, d2, d3) = 1, then there is a non-empty Zariski open subset U ⊂ H(d1, d2, d3) such that for every mapping F ∈ U the map germ (F, 0) is A-finitely determined. Moreov...

Denote by $H(d_1,d_2,d_3)$ the set of all homogeneous polynomial mappings $F=(f_1,f_2,f_3): \C^3\to\C^3$, such that $\deg f_i=d_i$. We show that if $\gcd(d_i,d_j)\leq 2$ for $1\leq i<j\leq 3$ and $\gcd(d_1,d_2,d_3)=1$, then there is a non-empty Zariski open subset $U\subset H(d_1,d_2,d_3)$ such that for every mapping $F\in U$ the map germ $(F,0)$ i...

For given natural numbers d1,d2 we describe the topology of a generic polynomial mappingF=(f,g):X→C^2, with deg f ≤ d1and deg g ≤d2. HereXis a complex plane or acomplex sphere.

We present new families of weighted homogeneous and Newton nondegenerate line singularities that satisfy the Zariski multiplicity conjecture.

In this work, we classify parametrized monomial surfaces f:(ℂ2,0)→(ℂ4,0) that are A-finitely determined. We study invariants that can be obtained in terms of invariants of a parametrized curve.

In this paper, we continue the study of the geometry of spacelike surfaces in de Sitter 5-space. We define invariants of the second fundamental form and consider their geometrical properties. We also investigate generic properties of the surfaces defined as solutions of the equations of asymptotic directions (AD) and lightlike binormal directions (...

A more general class than complete intersection singularities is the class of determinantal singularities. They are defined by the vanishing of all the minors of a certain size of a $m\times n$-matrix. In this note, we consider $\mathcal{G}$-finite determinacy of matrices defining a special class of determinantal varieties. They are called essentia...

This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern–Schwartz–MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (E...

We extend the notion of $\mu^*$-sequence and Tjurina number of functions to the framework of Bruce-Roberts numbers, that is, to pairs formed by the germ at $0$ of a complex analytic variety $X\subseteq \mathbb C^n$ and a finitely $\mathcal R(X)$-determined analytic function germ $f:(\mathbb C^n,0)\to (\mathbb C,0)$. We analyze some fundamental prop...

The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are bilipschitz equivalent. In this article we prove Lipschitz normal embeddedness of some algebraic subsets of the space...

For given natural numbers d1, d2 we describe the topology of a generic polynomial mapping F = (f, g) : X → C 2 , with deg f ≤ d1 and deg g ≤ d2. Here X is a complex plane or a complex sphere.

It is known that the bi-Lipschitz right classification of function germs admit moduli. In this article we introduce a notion called the Lipschitz simple function germs and present a full classification in the complex case. A surprising consequence of our result is that a function germ is Lipschitz modal if and only if it deforms to the smooth unimo...

We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in $C^{\infty}(N\times[0,1],P)$, also known as pseudo-isotopies, is dense if and only if the pair of dimensions $(\dim N, \dim P)$ is in the extra-nice dimensions. This result is parallel to Mather's characterization of the nice dimensions as the pa...

We study the geometry of surfaces in $\mathbb{R}^{4}$ with corank $1$ singularities. For such surfaces the singularities are isolated and at each point we define the curvature parabola in the normal space. This curve codifies all the second order information of the surface. Also, using this curve we define asymptotic and binormal directions, the um...

We construct singular varieties V G associated to a polynomial mapping G: ℂ ⁿ → ℂ ⁿ⁻¹ where n ≥ 2. Let G: ℂ ³ → ℂ ² be a local submersion, we prove that if the homology or the intersection homology with total perversity (with compact supports or closed supports) in dimension two of any variety V G is trivial then G is a fibration. In the case of a...

Mather proved that the smooth stability of smooth maps between manifolds is a generic condition if and only if the pair of dimensions of the manifolds are 'nice dimensions' while topologically stability is a generic condition in any pair of dimensions. And, by a result of du Plessis and Wall $C^1$-stability is also a generic condition precisely in...

Mather proved that the smooth stability of smooth maps between manifolds is a generic condition if and only if the pair of dimensions of the manifolds are 'nice dimensions' while topologically stability is a generic condition in any pair of dimensions. And, by a result of du Plessis and Wall $C^1$-stability is also a generic condition precisely in...

A more general class than complete intersection singularities is the class of determinantal singularities. They are defined by the vanishing of all the minors of a certain size of a $m\times n$-matrix. In this note, we consider $\mathcal{G}$-finite determinacy of matrices defining a special class of determinantal varieties. They are called essentia...

We describe the topology of a general polynomial mapping $F=(f, g):X\to\Bbb C^2$, where $X$ is a complex plane or a complex sphere.

The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are bilipschitz equivalent. In this article we prove Lipschitz normal embeddedness of some algebraic subsets of the space...

For a generic embedding of a smooth closed surface M into R⁴, the subset of R⁴ which is the affine λ−equidistant of M appears as the discriminant set of a stable mapping M × M → R⁴, hence their stable singularities are Ak, k = 2, 3, 4, and C2,2±. In this paper, we characterize these stable singularities of λ−equidistants in terms of the bi-local ex...

This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern--Schwartz--MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity...

In this paper, a systematic method is given to construct all liftable vector fields over an analytic multigerm $f: (\mathbb{K}^n, S)\to (\mathbb{K}^p,0)$ of corank at most one admitting a one-parameter stable unfolding.

In this work, we study the Whitney equisingularity of families of singular surfaces in $\mathbb{C}^3$ parametrized by $\mathcal{A}$-finitely determined map germs. We show that when $f:(\mathbb{C}^2,0)\rightarrow(\mathbb{C}^3,0)$ has corank $1$ and is finitely determined, then every $1$-parameter unfolding $F$ of $f$ which is topologically equisingu...

In this work, we study families of singular surfaces in $\mathbb{C}^3$ parametrized by $\mathcal{A}$-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding $F$ of a finitely determined map germ $f:(\mathbb{C}^2,0)\rightarrow(\mathbb{C}^3,0)$. We investigate the following conjecture: topolog...

The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are bilipschitz equivalent. In this article we prove that the model determinantal singularity, that is the space of $m\ti...

Let f (t,z) = f0(z) +tg(z) be a holomorphic function defined in a neighbourhood of the origin in C× Cn. It is well known that if the one-parameter deformation family {ft}defined by the function f is a m-constant family of isolated singularities, then {ft} is topologically trivial—a result of A. Parusiński. It is also known that Parusiński’s result...

Consider a family $\{f_t\}$ of complex polynomial functions with line singularities, and assume that $f_0$ is weighted homogeneous. We show that if the $1$-st L\^e number and the $1$-st polar number of $f_t$ at $\mathbf{0}$ are independent of $t$, then the family $\{f_t\}$ is equimultiple. In particular, $\{f_t\}$ is equimultiple if the $1$-st pola...

The study of Essentially Isolated Determinantal Singularites or EIDS was initiated by Ebeling and Gusein-Zade. They are non-smoothable as determinantal singularities, and in general have non-isolated singularities. Their singularities are generic in a deleted neighborhood of the origin, hence their description as "essentially isolated". In this pap...

The study of Essentially Isolated Determinantal Singularites or EIDS was initiated by Ebeling and Gusein-Zade. They are non-smoothable as determinantal singularities, and in general have non-isolated singularities. Their singularities are generic in a deleted neighborhood of the origin, hence their description as "essentially isolated". In this pap...

We prove several results regarding the simplicity of germs and multigerms obtained via the operations of augmentation, simultaneous augmentation and concatenation and generalised concatenation. We also give some results in the case where one of the branches is a non stable primitive germ. Using our results we obtain a list which includes all simple...

The de Sitter space is known as a Lorentz space with positive constant curvature in the Minkowski space. A surface with a Riemannian metric is called a spacelike surface. In this work we investigate properties of the second order geometry of spacelike surfaces in de Sitter space S51 by using the action of GL(2;ℝ)×SO(1; 2) on the system of conics de...

For a generic embedding of a smooth closed surface $M$ into $\mathbb R^4$, the subset of $\mathbb R^4$ which is the affine $\lambda$-equidistant of $M$ appears as the discriminant set of a stable mapping $M \times M \to \mathbb R^4$, hence their stable singularities are $A_k, \, k=2, 3, 4,$ and $C_{2,2}^{\pm}$. In this paper, we characterize these...

We show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers ar...

We study the essentially isolated determinantal singularities (EIDS), defined by Ebeling and Gusein-Zade [S. M. Guseĭn-Zade and W. Èbeling, On the indices of 1-forms on determinantal singularities, Tr. Mat. Inst. Steklova 267 (2009) 119–131], as a generalization of isolated singularity. We prove in dimension 3, a minimality theorem for the Milnor n...

We construct a singular variety ${\mathcal{V}}_G$ associated to a polynomial mapping $G : \C^{n} \to \C^{n - 1}$ where $n \geq 2$. We prove that in the case $G : \C^{3} \to \C^{2}$, if $G$ is a local submersion but is not a fibration, then the homology and the intersection homology with total perversity (with compact supports or closed supports) in...

We give an exact formula for the number of cusps of a general polynomial mapping $F=(f, g):\Bbb C^2\to\Bbb C^2.$ Namely, if deg $f=d_1$ and deg $g=d_2$ and $F$ is general, then $F$ has exactly $d_1^2+d_2^2+3d_1d_2-6d_1-6d_2+7$ simple cusps. Moreover, if $d_1>1$ or $d_2>1$, then the set $C(F)$ of critical points of $F$ is a smooth connected curve, w...

In this article we obtain a formula relating inflections, bitangencies and the Milnor number of a plane curve germ. Moreover, we present an extension of the formula obtained by the first author and Luis Fernando Mello for a class of plane curves with singularities.

We generalise the operations of augmentation and concatenations defined in Cooper et al. (Compos Math 131(2):121–160, 2002) in order to obtain multigerms of analytic (or smooth) maps \((\mathbb {K}^n,S)\rightarrow (\mathbb {K}^p,0)\) with \(\mathbb {K}=\mathbb {C}\) or \(\mathbb {R}\) from monogerms and some special multigerms. We then prove that a...

We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.

Let f : X -> Y be a dominant polynomial mapping of affine varieties. For generic y in Y we have Sing(f^{-1}(y)) = f^{-1}(y) \cap Sing(X): As an application we show that symmetry defect hypersurfaces for two generic members of the irreducible algebraic family of n-dimensional smooth irreducible subvarieties in general position in C^{2n} are homeomor...

Let X, Y ⊂ κm(κ = R,C) be smooth manifolds. We investigate the central symmetry of the configuration of X and Y. For p ∈ κm we introduce a number μ(p) of pairs of points x ∈ X and y ∈ Y such that p is the center of the interval xy. We show that if X, Y (including the case X = Y ) are algebraic manifolds in a general position, then there is a closed...

We show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers ar...

We provide a positive answer to Zariski's conjecture for families of singular surfaces in $\mathbb C^3,$ under the condition that the family has a smooth normalisation. As a corollary of the result, we obtain a surprising characterization of the Whitney equisingularity of one parameter families of $\mathcal A$ finitely determined map-germs $f_t: (\...

In this paper, two sufficient conditions are provided for given two \(\mathcal K \)-equivalent map-germs to be bi-Lipschitz \(\mathcal A \)-equivalent. These are Lipschitz analogues of the known results on \(C^r\) \(\mathcal A \)-equivalence \((0\le r\le \infty )\) for given two \(\mathcal K \)-equivalent map-germs. As a corollary of one of our res...

Using standard methods for studying singularities of projections and of contacts, we classify the stable singularities of affine $\lambda$-equidistants of $n$-dimensional closed submanifolds of $\mathbb R^q$, for $q\leq 2n$, whenever $(2n,q)$ is a pair of nice dimensions.

In this paper we investigate the relation betwen the Nash modification and the Bi-Lipschtiz equivalent germs in the cases of two germs and for a family of hypersurfaces with isolated singularities.

Let (V,0) be the germ of an analytic variety in \$\mathbb{C}^n\$ and f an analytic function germ defined on V. For functions with isolated singularity on V, Bruce and Roberts introduced a generalization of the Milnor number of f, which we call Bruce–Roberts number, μ BR (V,f). Like the Milnor number of f, this number shows some properties of f and...

Asymptotic curves are well defined on a smooth surface M in the Euclidean space R5 and are solutions of a binary quintic differential equation. We propose in this paper a definition of the lines of principal curvature on M that uses covariants of binary quintic forms as well as the metric structure on M.

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a L\^e-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplici...

We study horo-tight immersions of manifolds into hyperbolic spaces. The main result gives several characterizations of horo-tightness of spheres, answering a question proposed by Cecil and Ryan. For instance, we prove that a sphere is horo-tight if and only if it is tight in the hyperbolic sense. For codimension bigger than one, it follows that hor...

In this paper we investigate the classification of mappings up to \({\mathcal{K}}\)-equivalence. We give several results of this type. We study semialgebraic deformations up to semialgebraic C 0\({\mathcal{K}}\)-equivalence and bi-Lipschitz \({\mathcal{K}}\)-equivalence. We give an algebraic criterion for bi-Lipschitz \({\mathcal{K}}\)-triviality i...

We prove a new Morse-Sard type theorem for the asymptotic critical values of semi-algebraic mappings and a new fibration theorem at infinity for $C^2$ mappings. We show the equivalence of three different types of regularity conditions which have been used in the literature in order to control the asymptotic behaviour of mappings. The central role o...

The main goal of this work is to show that if two weighted homogeneous (but not homogeneous) function-germs $(\C^2,0)\to(\C,0)$ are bi-Lipschitz equivalent, in the sense that these function-germs can be included in a strongly bi-Lipschitz trivial family of weighted homogeneous function-germs, then they are analytically equivalent.

We study holomorphic function germs under equivalence relations that preserve an analytic variety. We show that two quasihomogeneous polynomials, not necessarily with isolated singularities, having isomorphic relative Milnor algebras are relative right equivalent. Under the condition that the module of vector fields tangent to the variety is finite...

A theory of bifurcation equivalence for forced symmetry breaking bifurcation problems is developed. We classify $(O(2),1)$ problems of corank 2 of low codimension and discuss examples of bifurcation problems leading to such symmetry breaking.

We recall Mather's Lemma providing effective necessary and sufficient conditions for a connected submanifold to be contained in an orbit. In Theorem 5.2 we show that two quasihomogeneous polynomials $f$ and $g$ having isomorphic relative Milnor algebras $M_v(f)$ and $M_v(g)$ are $R_v$-equivalent. In Theorem 5.4 we prove that two complex-analytic hy...

In this paper we introduce the concept of the index of an implicit differential equation $F(x,y,p)=0,$ where $F$ is a smooth function, $p=\frac{dy}{dx}$, $F_{p}=0$ and $F_{pp}=0$ at an isolated singular point. We also apply the results to study the geometry of surfaces in $\mathbb{R}^{5}$.

In this work we determine relations between the local Euler obstruction of an analytic map f and the Chern obstruction of a convenient collection of 1-forms associated to f. We give applications to singularity theory.

We use an inequality due to Bochnak and Lojasiewicz, which follows from the Curve Selection Lemma of real algebraic geometry in order to prove that, given a C r function \({f\colon U\subset{{\mathbb R}^m}\to{\mathbb R}}\), we have $$\lim\limits_{\substack{y\to x \\ y\in \text{crit}(f)}} \frac{|f(y)-f(x)|}{|y-x|^r}=0, \hbox{for all} \;x\in \text{cri...

In this work we determine relations between the local Euler obstruction of an analytic map f and the Chern obstruction of a convenient collection of 1-forms associated to f . We give applications to singularity theory.

We study asymptotic curves on generically immersed surfaces in R 5 . We characterise asymptotic directions via the contact of the surface with flat objects (k-planes, k = 1–4), give the equation of the asymptotic curves in terms of the coefficients of the second fundamental form and study their generic local configurations.

We study the horospherical geometry of submanifolds in hyperbolic space. The main result is a formula for the total absolute horospherical curvature of M, which implies, for the horospherical geometry, the analogues of classical inequalities of the Euclidean Geometry. We prove the horospherical Chern-Lashof inequality for surfaces in 3-space and th...

All -simple corank-1 germs from to , where n ≠ 4, have an M-deformation, that is a deformation in which the maximal numbers of isolated stable singular points are simultaneously present in the image.