Whitney Umbrella Research Articles

Lagrange mappings of the first open Whitney umbrella

In this paper we give a classification of simple stable singularities of Lagrange projections of the first open Whitney umbrella, the simplest singularity of Lagrange varieties. Our classification extends the ADE-classification, due to Arnold, of simple stable singularities of Lagrange projections of smooth Lagrange submanifolds. We also prove a criterion of equivalence of stable Lagrange projections of open Whitney umbrellas which is analogous to Mather's fundamental theorem on stable map-germs.

Lscr ]0-equivalence of maps

We use the theory of the integral closure of an ideal to study the equivalence of map-germs under C0 coordinate changes in the target. We also derive a formula for the number of double points of a map germ from Cn → C2n in terms of the Segre number of dimension 0 of an ideal associated to the double point locus of f, and the number of Whitney umbrellas of the composition of f with a generic projection to C2n−1.

Lines of principal curvature around umbilics and Whitney umbrellas

In this paper is studied the configuration of lines of curvature near a Whitney umbrella which is the unique stable singularity for maps of surfaces into R 3. The pattern of such configuration is established and characterized in terms of the 3-jet of the map. The result is used to establish an expression for the Euler-Poincare characteristic in terms of the number of umbilics and umbrellas. 1. Introduction. The bending or curvature pattern of a smooth mapping a : M -» R3, where M is a compact oriented two dimensional manifold, will be represented here by singular points, Sa, at which the mapping has rank less than 2 and the bending can be re- garded to be infinite; the umbilic points, Ua, at which the bending is finite but equal in all directions: and by the family of lines of principal curvature T\,a and Ti,a defined on M (Ua U 4. They showed that Darbouxian umbilic points characterize those with local structurally stable configuration, under small C3 deformations of the surface. See also the work of Bruce and Fidal (B-F). A study of principal foliations near the set Sa of singular points, aiming to characterize their local stability, was carried out by Gutierrez and Sotomayor (GS2). To this end they gave two sufficient conditions, the first of which is the Whitney singularity condition for stability of mappings in the sense of Singularity Theory (G-G). However, in this paper, an erroneous

Sums of two squares in analytic rings

We study analytic singularities for which every positive semidefinite analytic function is a sum of two squares of analytic functions. This is a basic useful property of the plane, but difficult to check in other cases; in particular, what about z(2)=xy, z(2)=yx(2)-y(3), z(2)=x(3)+y(4) or z(2)=x(3)-xy(3)? In fact, the unique positive examples we can find are the Brieskorn singularity, the union of two planes in 3-space and the Whitney umbrella. Conversely we prove that a complete intersection with that property (other than the seven embedded surfaces already mentioned) must be a very simple deformation of the two latter, namely, z(2)=x(2)+(-1)(k)y(k), k≥3, or z(2)=yx(2)+(-1)(k)y(k), k≥4. In particular, except for the stems z(2)=x(2) and z(2)=yx(2), all singularities are real rational double points.

Functions on a crosscap

The study of the differential geometry of surfaces in 3-space has a long and celebrated history. Over the last 20 years a new approach using techniques from singularity theory has yielded some interesting results (see, for example [3, 5, 19] for surveys). Of course surfaces arise in a number of ways: they are often defined explicitly as the image of a mapping f: R2[rightward arrow]R3. Since the subject is differential geometry one normally asks that these defining mappings are smooth, that is infinitely differentiable, however it is not true, in any sense, that most such parametrisations will yield manifolds. For such mappings have self-intersections, and more significantly they may have crosscaps (also known as Whitney umbrellas). Moreover if we perturb the maps these singularities will persist; that is they are stable. (See [14, 20] for details.) Consequently when studying the differential geometry of surfaces in 3-space there are good reasons for studying surfaces with crosscaps. In this paper we carry out a classification of mappings from 3-space to lines, up to changes of co-ordinates in the source preserving a crosscap. We can apply our results to the geometry of generic crosscap points. In [10] we computed geometric normal forms for the crosscap and used them to study the dual of the crosscap. We shall see that the approach here yields more information than that obtained in [10], although the latter has the advantage of being more explicit (in particular various aspects of the geometry can be compared using the normal forms). We refer the reader to [16, 18] for background material concerning singularity theory.

Normal forms of the whitney umbrella with respect to the contact group preserving a cone

Normal forms of the whitney umbrella with respect to the contact group preserving a cone

Period-doubling for two-dimensional non-invertible maps: Renormalization group analysis and quantitative universality

Two types of universal behavior associated with period-doubling in two-dimensional non-invertible maps are investigated. The first, called C-type, may occur as a generic two-parameter phenomenon when the map possesses a fold singularity. The second kind of universal behavior, called FQ-type, is associated with the singularity arising from projecting Whitney's umbrella. Typically, this kind of period-doubling scaling behavior appears as a phenomenon of codimension three, i.e. at some critical point in the three-dimensional parameter space. Numerical results are presented to illustrate the self-similarity of the parameter space topography near the critical points C and FQ. The possibility of observing these new types of critical behavior in physical systems is discussed.

Symplectic and Lagrange stabilities of open Whitney umbrellas

In this paper we formulate and prove the criteria for the Lagrange stability of isotropic map-germs of corank at most one. They include an analogue of Mather’s characterisation of the stability of differentiable mappings. Also they give an generalisation of Arnol’d’s criterion for the Lagrange stability of Lagrange immersions. Meanwhile, there naturally arise some special singularities: the open Whitney umbrellas, which are characterised as symplectically stable isotropic map-germs of corank at most one. Unless otherwise stated, any manifolds and mappings are assumed to be of class C∞.

Universality and scaling in non-invertible two-dimensional maps

Two new types of period-doubling universality are discussed for non-invertible two-dimensional maps. The first type may appear when the map possesses the simplest fold singularity. The second type is associated with a singularity arising from projecting Whitney's umbrella.

Immersed 𝑛-manifolds in 𝐑²ⁿ and the double points of their generic projections into 𝐑²ⁿ⁻¹

We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities—the Whitney umbrellas—of an n n -manifold into R 2 n − 1 \mathbf {R}^{2n-1} , which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed n n -manifold in R 2 n \mathbf {R}^{2n} . We also study generic projections of an embedded n n -manifold in R 2 n \mathbf {R}^{2n} into R 2 n − 1 \mathbf {R}^{2n-1} and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in R 4 \mathbf {R}^{4} . The problem of lifting a map into R 2 n − 1 \mathbf {R}^{2n-1} to an embedding into R 2 n \mathbf {R}^{2n} is also studied.

Thom polynomials for open Whitney umbrellas of isotropic mappings

A smooth mapping $f:L^n → (M^{2n},ω)$ of a smooth n-dimensional manifold L into a smooth 2n-dimensional symplectic manifold (M,ω) is called isotropic if f*ω vanishes. In the last ten years, the local theory of singularities of isotropic mappings has been

Homoclinic points near degenerate homoclinics

We consider periodic pertubations of a family of autonomous vector fields which has a degenerate homoclinic orbit. We show that the set of parameters for which homoclinic points exist is foliated by Whitney umbrellas.

Parametrization of a Singular Lagrangian Variety

We give stabilization and parametrization theorems for a class of singular varieties in the space of polynomials of one variable and generalize the results of Arnol’d and Givental’. The class contains the open swallowtails and the open Whitney umbrella. The parametrization is associated with the singularity of a stable mapping (in the sense of Thom and Mather) of kernel rank one.

Bifurcation of degenerate homoclinics

We analyze the continuation and bifurcation of homoclinic orbits near a given degenerate homoclinic orbit. We show that the existence of such degenerate homoclinic orbit is a codimension three phenomenon, and that generically the set of parametervalues at which a nearby homoclinic exists forms a codimension one surface which shows a singularity of Whitney umbrella type at the critical parametervalue. The line of self-intersecting points of such surface corresponds to systems which have two nearby homoclinics.

Parametrization of a singular Lagrangian variety

We give stabilization and parametrization theorems for a class of singular varieties in the space of polynomials of one variable and generalize the results of Arnol’d and Givental’. The class contains the open swallowtails and the open Whitney umbrella. The parametrization is associated with the singularity of a stable mapping (in the sense of Thom and Mather) of kernel rank one.

The local model of an isotropic map-germ arising from one-dimensional symplectic reduction

AbstractIn this paper, we classify generic isotropic map-germs arising from the symplectic reduction process relative to a hypersurface (i.e. one-dimensional reduction), up to symplectic equivalence in the C category. These models are open Whitney umbrellas of arbitrary dimension and their suspensions. These singularities appear in the generalized Cauchy problem for HamiltonJacobi equations.

Lagrangian imbeddings of surfaces and unfolded Whitney umbrella

Lagrangian imbeddings of surfaces and unfolded Whitney umbrella

Lines of principal curvature for mappings with Whitney umbrella singularities

Lines of principal curvature for mappings with Whitney umbrella singularities