Bifurcation diagrams and caustics of simple quasi border singularities

Quasi boundary and quasi corner singularities of functions are discussed. They correspond to the classifications of Lagrangian projections with a boundary or a corner. The geometry of bifurcation diagrams and caustics of simple quasi boundary and quasi corner singularities in R³ and R⁴ are described.     

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.     

Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

An affine characterization of the Veronese surface

IfM ² is a nondegenerate surface in a 4-dimensional Riemannian manifold , then there is a natural affine metricg defined onM ². It is shown that this affine metricg is conformal to the induced Riemannian metric onM ² if and only ifM ² is a minimal submanifold of in the usual Riemannian sense. If the conformal factor is a constant, then the two metrics are said to be homothetic. It is shown that there does not exist a nondegenerate surface in Euclidean space ⁴ or hyperbolic spaceH ⁴ whose affine metric is homothetic to the induced Riemannian metric. Furthermore, ifM ² is a nondegenerate surface in the standard 4-sphereS ⁴ whose affine metric is homothetic to the induced Riemannian metric, thenM ² is a Veronese surface.T. Cecil was supported by NSF Grant No. DMS-9101961.     

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.     

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.     

Affine rotational surfaces with vanishing affine curvature

We give a classification of affine rotational surfaces in affine 3-space with vanishing affine Gauss-Kronecker curvature. Non-degenerated surfaces in three dimensional affine space with affine rotational symmetry have been studied by a number of authors (I.C. Lee. [3], P. Lehebel [4], P.A. Schirokow [10], B. Su [12], W. Süss [13]). In the present paper we study these surfaces with the additional property of vanishing affine Gauss-Kronecker curvature, that means the determinant of the affine shape operator is zero. We give a complete classification of these surfaces, which are the affine analogues to the cylinders and cones of rotation in euclidean geometry. These surfaces are examples of surfaces with diagonalizable rank one (affine) shape operator (cf. B. Opozda [8] and B. Opozda, T. Sasaki [7]). The affine normal images are curves.     

The mixed Lp affine surface areas for functions

In this paper, we propose a definition of a general mixed L_p Affine surface area, ?n ≠ p ∈ ?, for multiple functions. Our definition is di?erent from and is “dual” to the one in [11] by Caglar and Ye. In particular, our definition makes it possible to establish an integral formula for the general mixed L_p Affine surface area of multiple functions (see Theorem 3.1 for more precise statements). Properties of the newly introduced functional are proved such as affine invariance, and related affine isoperimetric inequalities are proved.     

Singularities of Lagrangian varieties and related topics

We study the classification problem for generic projections of Lagrangian submanifolds. A classification list for symmetric Lagrangian submanifolds is obtained and the generic evolutions of symmetric caustics are illustrated. We show how the singular Lagrangian varieties appear in the invariant theory of binary forms and we introduce the basic concepts of the desingularization procedure. Applications to differential geometry, geometrical optics, and mechanics are presented.     

Projectively flat affine surfaces

We study non-degenerate affine surfaces in A³ with a projectively flat induced connection. The curvature of the affine metric , the affine mean curvature H, and the Pick invariant J are related by . Depending on the rank of the span of the gradients of these functions, a local classification of three groups is given. The main result is the characterization of the projectively flat but not locally symmetric surfaces as a solution of a system of ODEs. In the final part, we classify projectively flat and locally symmetric affine translation surfaces.     

General affine surface areas

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for L? affine surface areas are established.     

An affine analogue of the Hartman-Nirenberg cylinder theorem

Let X be a smooth, complete, connected submanifold of dimension in a complex affine space , and r is the rank of its Gauss map . The authors prove that if and in the pencil of the second fundamental forms of X, there are two forms defining a regular pencil all eigenvalues of which are distinct, then the submanifold X is a cylinder with -dimensional plane generators erected over a smooth, complete, connected submanifold Y of rank r and dimension r. This result is an affine analogue of the Hartman-Nirenberg cylinder theorem proved for and r = 1. For and , there exist complete connected submanifolds that are not cylinders. Received: 20 October 2000 / Revised version: 18 April 2001 / Published online: 18 January 2002     

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 the fundamental theorem for non-degenerate complex affine hypersurface immersions

Two geometric versions of the fundamental theorem for non-degenerate complex affine hypersurface immersions are proved. We consider non-degenerate complex affine hypersurface immersions with complex transversal connection form (or equivalently, with holomorphic normalization) and prove that the conormal map is a holomorphic map. These considerations inspired the definitions of complex semi-compatible and complex semi-conjugate connections. This allows us to formulate the integrability conditions of the fundamental theorem, on one hand in terms of the induced connection, which has to be complex semi-compatible and dualH-projective flat, and on the other hand, in terms of its semi-conjugate connection, which has to beH-projective flat. Using this results, we formulate the conditions of the fundamental theorem in terms of anyH-projective flat complex affine connection.Research partially supported by Contract MM 413/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia St. Kl. Ohridcki.     

Weyl sums over integers with affine digit restrictions

For any given integer q?2, we consider sets N of non-negative integers that are defined by affine relations between their q-adic digits (for example, the set of non-negative integers such that the number of 1's equals twice the number of 0's in the binary representation). The main goal is to prove that the sequence (αn)_n∈N is uniformly distributed modulo 1 for all irrational numbers α. The proof is based on a saddle point analysis of certain generating functions that allows us to bound the corresponding Weyl sums.     

The coordinatization of affine planes by rings

With every unitary free module of rank 2 there is naturally associated a generalized affine plane (e.g. the lines are just the cosets of all nonzero 1-generated submodules). Here we solve the converse problem by coordinatizing a given generalized affine plane which satisfies certain versions of Desargues' postulate.     

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.     

The Lagrangian Gauss image of a compact surface in Minkowski 3-Space

A generic compact surfaceQ in Minkowski 3-Space is naturally stratified by the loci where the orthogonal line bundle is tangent to the next lower stratum,SP D ⁰ Q M³. To each component inD ⁰ we associate a light-like hypersurface and in turn a Lagrangian loop in the cotangent bundle of the circle. We then establish an inequality relating the Euler characteristic of the indefinite component ofQ with the total Gauß-Maslov index of the associated Lagrangian loops.