Volume 48, Issue 2 - Special Issue: GEOMETRIC SINGULARITY THEORY, EDITORS: Wojciech Domitrz (Warsaw University of Technology, Poland), Goo Ishikawa (Hokkaido University, Japan), Shyuichi Izumiya (Hokkaido University, Japan)

We study singularities of smooth mappings F̄ of ℝ 2n into symplectic space (ℝ 2n , ω̇) by their isotropic liftings to the corresponding symplectic tangent bundle (Tℝ 2n ,w). Using the notion of local solvability of lifting as a generalized Hamiltonian system, we introduce new symplectic invariants and explain their geometric meaning. We prove that a basic local algebra of singularity is a space of generating functions of solvable isotropic mappings over F̄ endowed with a natural Poisson structure. The global properties of this Poisson algebra of the singularity among the space of all generating functions of isotropic liftings are investigated. The solvability criterion of generalized Hamiltonian systems is a strong method for various geometric and algebraic investigations in a symplectic space. We illustrate this by explicit classification of solvable systems in codimension one.

We have already defined the evolutes and the involutes of fronts without inflection points. For regular curves or fronts, we can not define the evolutes at inflection points. On the other hand, the involutes can be defined at inflection points. In this case, the involute is not a front but a frontal at inflection points. We define evolutes of frontals under conditions. T he definition is a generalisation of both evolutes of regular curves and of fronts. By using relationship between evolutes and involutes of frontals, we give an existence condition of the evolute with inflection points. We also give properties of evolutes and involutes of frontals.

We study centre symmetry sets and equidistants for a I-parameter family of plane curves where, for a special member of the family, there exist two inflexions with parallel tangents. Some results can be obtained by reducing a generating family to normal forms, but others require direct calculation from the generating family.

Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown in (15), that, if the cone structure is regarded as a control system, then the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-Riemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the constrained Hamiltonian equations on normal and abnormal geodesics.

We consider a developable surface tangent to a surface along a curve on the surface. We call it an osculating developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface which characterize these singularities. As a by-product, we show that a curve is a contour generator with respect to an orthogonal projection or a central projection if and only if one of these invariants constantly equal to zero.

Let X n be an affine variety of dimension n and Y n be a quasi-projective variety of the same dimension. We prove that for a quasi-finite polynomial mapping ƒ : X n → Y n , every non-empty component of the set Y n \ ƒ (X n ) is closed and it has dimension greater or equal to n μ(ƒ), where μ(ƒ) is a geometric degree of ƒ. Moreover, we prove that generally, if ƒ : X n → Y n is any polynomial mapping, then either every non-empty component of the set is of dimension ≥ n μ(ƒ) or ƒ contracts a subvariety of dimension ≥ n μ(ƒ) + 1.

We study here the relative cohomology and the Gauss-Manin connections associated to an isolated singularity of a function on a manifold with boundary, i.e. with a fixed hyperplane section. We prove several relative analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange, concerning the properties of the Gauss-Manin connection as well as its relations with the Picard-Lefschetz monodromy and the asymptotics of integrals of holomorphic forms along the vanishing cycles. Finally, we give an application in isochore deformation theory, i.e. the deformation theory of boundary singularities with respect to a volume form. In particular, we prove the relative analog of J. Vey's isochore Morse lemma, J .-P. Fran~oise's generalisation on the local normal forms of volume forms with respect to the boundary singularity-preserving diffeomorphisms, as well as M. D. Garay's theorem on the isochore version of Mather's versa! unfolding theorem.

This is a survey article about the study of the links of some complex hypersurface singularities in ℂ 3 . We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group SU (2), Nil 3 or Sol 3 , respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of S 5 . Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in S 5 as codimension two contact submanifolds.

Let ƒ, g : (ℝ n , 0) --+ (ℝ, 0) be analytic functions. We will show that if ▽ƒ(0) = 0 and g ƒ∈ (ƒ) r+2 then I and g are C r -right equivalent, where (ƒ) denote ideal generated by ƒ and r ∈ ℕ.

We use the method of algebraic restrictions to classify symplectic U 7 , U 8 and U 9 singularities. We use discrete symplectic invariants to distinguish symplectic singularities of the curves. We also give the geometric description of symplectic classes