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Raúl Oset Sinha

Publications and source records attributed to Raúl Oset Sinha.

18 recordsLinked to original sources
arXiv2025

A characterization of quasi-homogeneity in terms of liftable vector fields

We prove under certain conditions that any stable unfolding of a quasi-homogeneous map-germ with finite singularity type is substantial. We then prove that if an equidimensional map-germ is finitely determined, of corank 1, and either it admits a minimal stable unfolding or it is of multipliticy 3, then it admits a substantial unfolding if and only if it is quasi-homogeneous. Based on this we pose the following conjecture: a finitely determined map-germ is quasi-homogeneous if and only if it admits a substantial unfolding.

math.AG↗
arXiv2025

Bi-Lipschitz triviality of function-germs on singular varieties

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 triviality of deformations of $f$ on $(X,0).$ As a corollary it follows that when $X$ and $f$ are homogeneous of the same degree, all deformation of $f$ of the same or higher degrees are bi-Lipschitz trivial. We then prove a rigidity result for deformations of $f$ on $X$ when both are weighted homogeneous with respect to the same set of weights.

math.AG↗
arXiv2024

A note on 1-parameter stable unfoldings

We give two characterisations of when a map-germ admits a 1-parameter stable unfolding, one related to the $\mathscr K_e$-codimension and another related to the normal form of a versal unfolding. We then prove that there are infinitely many finitely determined map-germs of multiplicity 4 from $\mathbb K^3$ to $\mathbb K^3$ which do not admit a 1-parameter stable unfolding.

math.AG↗
arXiv2023

Augmentation of singularities: $μ/τ$-type conjectures and simplicity

For function germs $g:(\mathbb C^n,0)\to (\mathbb C,0)$ it is well known that $1\leq\frac{μ(g)}{τ(g)}$ and it has recently been proved by Liu that $\frac{μ(g)}{τ(g)}\leq n$. We give an upper bound for the codimension of map-germs $f:(\mathbb C^n,0)\to (\mathbb C^p,0)$ given as augmentations of other map-germs with which we prove the analog to the first inequality (known as Mond's conjecture) for augmentations $h:(\mathbb C^n,0)\to (\mathbb C^{n+1},0)$. Furthermore, we show that the quotient given by the image Milnor number and the codimension of any augmentation in the pair of dimensions $(n,n+1)$ is less than $\frac{1}{4}(n+1)^2$ and prove the analog to the second inequality for map-germs with $n=1$ and augmentations with $n=2,3$. We then prove a characterization of when a map-germ is an augmentation, finding a counterexample for the characterization given by Houston. Next, we give sufficient conditions for when the augmentation is independent of the choice of stable unfolding by studying different notions of equivalence of unfoldings. Moreover, these results allow us to give sufficient conditions for the simplicity of an augmentation, providing context to locate the moduli for non-simple augmentations.

math.AG↗
arXiv2022

Curvature loci of 3-manifolds

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 (which is a determinantal variety in some cases). We also study how singularities of the curvature locus of a regular 3-manifold can go to infinity when the manifold is projected orthogonally in a tangent direction.

math.DG↗
arXiv2022

Axial curvatures for corank 1 singular $n$-manifolds in $\mathbb R^{n+k}$

For singular $n$-manifolds in $\mathbb R^{n+k}$ with a corank 1 singular point at $p\in M^n_{\mbox{sing}}$ we define up to $l(n-1)$ different axial curvatures at $p$, where $l=\min\{n,k+1\}$. These curvatures are obtained using the curvature locus (the image by the second fundamental form of the unitary tangent vectors) and are therefore second order invariants. In fact, in the case $n=2$ they generalise all second order curvatures which have been defined for frontal type surfaces. We relate these curvatures with the principal curvatures in certain normal directions of an associated regular $(n-1)$-manifold contained in $M^n_{\mbox{sing}}$. We obtain many interesting geometrical interpretations in the cases $n=2,3$. For instance, for frontal type 3-manifolds with 2-dimensional singular set, the Gaussian curvature of the singular set can be expressed in terms of the axial curvatures. Similarly for the curvature of the singular set when it is 1-dimensional. Finally, we show that all the umbilic curvatures which have been defined for singular manifolds up to now can be seen as the absolute value of one of our axial curvatures.

math.DG↗
arXiv2020

Geometry of surfaces in $\mathbb R^5$ through projections and normal sections

We study the geometry of surfaces in $\mathbb R^5$ by relating it to the geometry of regular and singular surfaces in $\mathbb R^4$ obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which are not second order geometry for surfaces in $\mathbb R^5$ but are in $\mathbb R^4$. We also relate the umbilic curvatures of each type of surface and their contact with spheres. We then consider the surfaces as normal sections of 3-manifolds in $\mathbb R^6$ and again relate asymptotic directions and contact with spheres by defining an appropriate umbilic curvature for 3-manifolds.

math.DG↗
arXiv2019

The axial curvature for corank 1 singular surfaces

For singular corank 1 surfaces in $\mathbb R^3$ we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola we define a new type of curvature called the axial curvature, which generalizes the singular curvature for frontal type singularities. We then study contact properties of the surface with respect to the plane orthogonal to the axial vector and show how they are related to the axial curvature. Finally, for certain fold type singularities, we relate the axial curvature with the Gaussian curvature of an appropriate blow up.

math.DG↗
arXiv2019

Relating second order geometry of manifolds through projections and normal sections

We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in $\mathbb{R}^6$ (resp. $\mathbb R^5$) with regular (resp. singular corank 1) surfaces in $\mathbb R^5$ (resp. $\mathbb R^4$). For example we show how to generate a Roman surface by a family of ellipses different to Steiner's way. Furthermore, we give necessary conditions for the 2-jet of the parametrisation of a singular 3-manifold to be in a certain orbit in terms of the topological types of the curvature loci of the singular surfaces obtained as normal sections. We also study the relations between the regular and singular cases through projections. We show there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular we define asymptotic directions for singular corank 1 3-manifolds in $\mathbb R^5$ and relate them to asymptotic directions of regular 3-manifolds in $\mathbb R^6$ and singular corank 1 surfaces in $\mathbb R^4$.

math.DG↗
arXiv2018

The flat geometry of the $I_{1}$ singularity: $(x,y)\mapsto(x,xy,y^{2},y^{3})$

We study the flat geometry of the least degenerate singularity of a singular surface in $\mathbb R^4$, the $I_{1}$ singularity parametrised by $(x,y)\mapsto(x,xy,y^{2},y^{3})$. This singularity appears generically when projecting a regular surface in $\mathbb R^5$ orthogonally to $\mathbb R^4$ along a tangent direction. We obtain a generic normal form for $I_1$ invariant under diffeomorphisms in the source and isometries in the target. We then consider the contact with hyperplanes by classifying submersions which preserve the image of $I_1$. The main tool is the study of the singularities of the height function.

math.DG↗
arXiv2018

The extra-nice dimensions

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 pairs $(n,p)$ for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have $\mathcal A_e$-codimension 1 hyperplane sections. They are also related to the simplicity of $\mathcal A_e$-codimension 2 germs. We give a sufficient condition for any $\mathscr A_e$-codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps.

math.CV↗
arXiv2018

The geometry of corank $1$ surfaces in $\mathbb{R}^{4}$

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 umbilic curvature and study the flat geometry of the surface. It is shown that we can associate to this singular surface a regular one in $\mathbb{R}^{4}$ and relate their geometry.

math.DG↗
arXiv2017

On the geometry of folded cuspidal edges

We study the geometry of cuspidal $S_k$ singularities in $\mathbb R^3$ obtained by folding generically a cuspidal edge. In particular we study the geometry of the cuspidal cross-cap $M$, i.e. the cuspidal $S_0$ singularity. We study geometrical invariants associated to $M$ and show that they determine it up to order 5. We then study the flat geometry (contact with planes) of a generic cuspidal cross-cap by classifying submersions which preserve it and relate the singularities of the resulting height functions with the geometric invariants.

math.DG↗
arXiv2017

A relation between the curvature ellipse and the curvature parabola

At each point in an immersed surface in $\mathbb R^4$ there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. More recently, at the singular point of a corank 1 singular surface in $\mathbb R^3$, a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in $\mathbb R^4$ to $\mathbb R^3$ in a tangent direction corank 1 singularities appear generically. The projection has a cross-cap singularity unless the direction of projection is asymptotic, where more degenerate singularities can appear. In this paper we relate the geometry of an immersed surface in $\mathbb R^4$ at a certain point to the geometry of the projection of the surface to $\mathbb R^3$ at the singular point. In particular we relate the curvature ellipse of the surface to the curvature parabola of its singular projection.

math.DG↗
arXiv2017

First order local invariants of planar caustics

We study first order local invariants of Vassiliev type for Lagrangian immersions with generic planar caustics. For this we produce some examples of 2-parameter families of Lagrangian maps and study their bifurcation diagrams.

math.AT↗
arXiv2016

On the flat geometry of the cuspidal edge

We study the geometry of the cuspidal edge $M$ in $\mathbb R^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$. We classify submersions on a model of $M$ by diffeomorphisms and recover the contact of $M$ with planes from that classification. The contact of $M$ with lines is measured by the singularities of orthogonal projections of $M$. We list the generic singularities of the projections and obtain the generic deformations of the apparent contour (profile) when the direction of projection varies locally in $S^2$. We also relate the singularities of the height functions and of the projections to some geometric invariants of the cuspidal edge.

math.DG↗
arXiv2016

Liftable vector fields, unfoldings and augmentations

We study liftable vector fields of smooth map-germs. We show how to obtain the module of liftable vector fields of any map-germ of finite singularity type from the module of liftable vector fields of a stable unfolding of it. As an application, we obtain the liftable vector fields for the family $H_k$ in Mond's list. We then show the relation between the liftable vector fields of a stable germ and its augmentations.

math.CV↗
arXiv2016

On the simplicity of multigerms

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 multigerms from $\mathbb C^3$ to $\mathbb C^3$.

math.CV↗