Singularities of the Minkowski set and affine equidistants for a curve and a surface

Generic singularities of envelopes of families of chords and bifurcations of affine equidistants defined by a pair of a curve and a surface in R³ are classified. The chords join pairs of points of the curve and the surface such that the tangent line to the curve is parallel to the tangent plane to the surface. The classification contains singularities of stable Lagrange and Legendre projections, boundary singularities and some less known classes appearing at the points of the surface and the curve themselves.     

Topological finite-determinacy of functions with non-isolated singularities

We introduce the concept of topological finite-determinacy for germs of analytic functions within a fixed ideal I, which provides a notion of topological finite-determinacy of functions with non-isolated singularities. We prove the following statement which generalizes classical results of Thom and Varchenko: let A be the complement in the ideal I of the space of germs whose topological type remains unchanged under a deformation within the ideal that only modifies sufficiently large order terms of the Taylor expansion. Then A has infinite codimension in I in a suitable sense. We also prove the existence of generic topological types of families of germs of I parametrized by an irreducible analytic set.     

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss–Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss–Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.     

Geometry of curves and surfaces through the contact map

We introduce a new approach to the study of affine equidistants and centre symmetry sets via a family of maps obtained by reflexion in the midpoints of chords of a submanifold of affine space. We apply this to surfaces in R³, previously studied by Giblin and Zakalyukin, and then apply the same ideas to surfaces in R⁴, elucidating some of the connexions between their geometry and the family of reflexion maps. We also point out some connexions with symplectic topology.     

Simple tangential family germs and perestroikas of their envelopes

A tangential family is a 1-parameter system of regular curves emanating tangentially from another regular curve. We classify simple tangential family germs up to A-equivalence. We describe perestroikas of envelopes of simple tangential family germs of small codimension under small deformations of the germ among tangential families.     

On associated quasi connections

The purpose of this paper is to give a necessary and sufficient condition on the existence of associated splittings (defined in this paper) and to consider some applications to associated quasi-connections on fibred manifolds and vector bundles, using the idea and extending Theorem 1 from [2]. In Section 1, a general condition on the existence of associated splittings is given. In Section 2, the basic constructions concerning q.c.s. used in the next Section are briefly described following [7]; they extend the q.c.s. of Wang [8, 1, 2]. In Section 3 there are proved two theorems on associated q.c.s. using essentially the main theorem from Section 1.     

A simple proof of removable singularities for coupled fermion fields

This paper presents a simple proof of a removable singularity theorem for coupled fermion fields on compact four-dimensional manifolds. New methods are employed and the hypotheses here are weak.     

Differential invariants of surfaces

The algebra of differential invariants of a suitably generic surface S⊂R³, under either the usual Euclidean or equi-affine group actions, is shown to be generated, through invariant differentiation, by a single differential invariant. For Euclidean surfaces, the generating invariant is the mean curvature, and, as a consequence, the Gauss curvature can be expressed as an explicit rational function of the invariant derivatives, with respect to the Frenet frame, of the mean curvature. For equi-affine surfaces, the generating invariant is the third order Pick invariant. The proofs are based on the new, equivariant approach to the method of moving frames.     

Tangent and normal bundles in almost complex geometry

We define and study pseudoholomorphic vector bundle structures, particular cases of which are tangent and normal bundle almost complex structures. As an application we deduce normal forms of almost complex structures along a pseudoholomorphic submanifold.In dimension four we relate these normal forms to the problem of pseudoholomorphic foliation of a neighborhood of a curve and the question of non-deformation and persistence of pseudoholomorphic tori.     

HORIZONTAL DISTRIBUTIONS ON SPACES OF FIBERS

Abstract

Geometric methods for systems of partial differential equations and multiple integral problems in the calculus of variations lead naturally to differentiable manifolds that resemble fiber bundles but do not possess a structure group; in terms of local coordinates, π:B→M_n|(xⁱ, q^α)→(xⁱ), dim(B) = N + n, dim(M_n) = n. The standard notions of horizontal distributions, horizontal and vertical subspaces of T(B), T(B) = V(B) ⊕ H(B), horizontal lifts of curves in M_n, and horizontal and vertical dual subspaces with Λ¹(B) = V*(B) ⊕ H*(B) are shown to be well defined in B. The absence of a structure group is compensated for by an analysis based on the homogeneous ideals V and H that are generated by the canonical bases of V*(B) and H*(B), respectively. The differential system constructed from the generators of the horizontal ideal is shown to lead to a unique system of connection 1-forms and torsion 2-forms under the requirements that they have vacuous intersections with the horizontal ideal. The horizontal ideal is shown to be completely integrable if and only if the torsion 2-forms vanish throughout B, in which case the curvature 2-forms are congruent to zero mod H, and the curvature 2-forms are shown to have a vacuous intersection with H if and only if the horizontal distribution is affine. The paper concludes with a study of the mapping properties of the connection, torsion and curvature. These are significantly more general than those of a fiber bundle since the absence of a structure group allows mappings of the form 'xⁱ = φⁱ(x,q), 'q^α = φ^α (x,q).     

Graded Poisson structures on the algebra of differential forms

We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior derivative is a Hamiltonian graded vector field.     

Symplectic Yang–Mills theory,Ricci tensor,and connections

A Yang–Mills theory in a purely symplectic framework is developed. The corresponding Euler–Lagrange equations are derived and first integrals are given. We relate the results to the work of Bourgeois and Cahen on preferred symplectic connections.     

Great circular surfaces in the three-sphere

In this paper, we consider a special class of the surfaces in the 3-sphere defined by one-parameter families of great circles. We give a generic classification of singularities of such surfaces and investigate the geometric meanings from the view point of spherical geometry.     

Double structures and jets

We show how the double vector bundle structure of the manifold of double velocities, with its submanifolds of holonomic and semiholonomic double velocities, is mirrored by a structure of holonomic and semiholonomic subgroups in the principal prolongation of the first jet group. We use the actions of these groups to construct holonomic and semiholonomic submanifolds in the manifold of double contact elements, and show that these give rise to affine bundles where a semiholonomic element has well-defined holonomic and curvature components.     

Maurer–Cartan forms and the structure of Lie pseudo-groups

This paper begins a series devoted to developing a general and practical theory of moving frames for infinite-dimensional Lie pseudo-groups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for a pseudo-group. Our approach is completely explicit and avoids reliance on the theory of exterior differential systems and prolongation. The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the pseudo-group on submanifolds. The third paper [61] will apply Gr?bner basis methods to prove a fundamental theorem on the freeness of pseudo-group actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera for generating systems of differential invariants and also their syzygies. Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles, and solving equivalence and symmetry problems arising in geometry and physics.     

Nonholonomic problems and the theory of distributions

This article constitutes an appendix to the book by P. A. Griffiths, Exterior differential systems and the calculus of variations. Birkhäuser, 1983. It especially focusses on the distinction between holonomic and nonholonomic mechanical and variational problems, and indicates how rich and interesting the phenomena are in the nonholonomic case.Appendix to the Russian translation of [G]. The letter G followed by a number will refer to the corresponding reference in [G]; the quoted references, as well as [G] itself, can be found at the end of the list of references below.     

Rota’s umbral calculus and recursions

Umbral Calculus can provide exact solutions to a wide range of linear recursions. We summarize the relevant theory and give a variety of examples from combinatorics in one, two and three variables.Dedicated to the Memory of Gian-Carlo Rota     

Focal points and support functions in affine differential geometry

The notions of focal point and support function are considered for a nondegenerate hypersurfaceM ⁿ in affine spaceR ⁿ⁺¹ equipped with an equiaffine transversal field. IfM ⁿ is locally strictly convex, these two concepts are related via an Index theorem concerning the critical points of the support functions onM ⁿ . This is used to obtain characterizations of spheres and ellipsoids in terms of the critical point behavior of certain classes of affine support functions.Research supported by NSF Grant No. DMS-9101961.