Smooth tropical surfaces with infinitely many tropical lines

We study the tropical lines contained in smooth tropical surfaces in ℝ³. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however, there exist smooth tropical surfaces of any degree with infinitely many tropical lines.     

Konstant gedrallte Kongruenzflächen im einfach isotropen Raum

Let S be a congruence of straight lines in an isotropic space of degree one. i.e. a three dimensional real affine space with the metric ds²=dx²+dy². The purpose of this paper is to discuss ruled surfaces of the congruence S for which the parameter of distribution has a constant value δ (δ — surfaces). We study the curves of the middle surface and the focal surfaces (if there exist) of S which are base curves of δ — surfaces.Moreover we consider two congruences S and S′ which are “M-equivalent” and we investigate a correspondence between δ — surfaces of S and S′.     

Line congruences as surfaces in the space of lines

The space of lines in R³ can be viewed as a four dimensional homogeneous space of the group of Euclidean motions, E(3). Line congruences arise in the classical method of transforming one surface to another by lines. These transformations are particularly interesting if some geometric property of the original surface is preserved. Line congruences, then, are two parameter families of lines and can be studied as surfaces in the space of lines. In this paper, we use the method of moving frames to study line congruences. We calculate the first order invariants of line congruences for which there are two real focal surfaces, and give the geometric meaning of these invariants. We look specifically at the case where the two first order invariants are constant and give a simple proof of Bäcklund's Theorem which relates to the transformation of one pseudospherical surface, a surface of constant negative Gaussian curvature, to another. These transformations are of interest since pseudospherical surfaces correspond to solutions to the sine-Gordon equation. We also give a proof of Bianchi's permutability theorem for pseudospherical surfaces in this context. Finally, we use the results of these theorems to generate some pseudospherical surfaces. All of these concepts and results are understood in terms of the structure equations of the line congruence.     

The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions

We prove existence of Schoen's and other triply periodic minimal surfaces via conjugate (polygonal) Plateau problems. The simpler of these minimal surfaces can be deformed into constant mean curvature surfaces by solving analogous Plateau problems in S³. The required contours in S³ are obtained by working with the great circle orbits of Hopf S¹-actions in the same way as with families of parallel lines in ³. Annular Plateau problems give new embedded minimal surfaces in S³. For many of the minimal surfaces in ³ global Weierstraß representations are derived.Dedicated to Wilhelm Klingenberg     

Singularities of the affine chord envelope for two surfaces in four-space

We study local singularities of the envelope of the set of straight lines (chords) joining those points of two surfaces embedded in ?⁴ which are not in general position to each other.     

On surfaces inP 6 with no trisecant lines

Examples of surfaces inP ⁶ with no trisecant lines are constructed. A partial classification recovering them is given and conjectured to be the complete one. Dedicated to the memory of F. Serrano     

Skew loops and quadric surfaces

A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in R ³ with a point of positive curvature and no skew loops are the quadrics. In particular: Ellipsoids are the only closed surfaces without skew loops. Our efforts also yield results about skew loops on cylinders and positively curved surfaces. Received: January 7, 2002 RID="*" ID="*"The first author was partially supported by the NSF grant DMS-0204190.     

Joachimsthal surfaces inS 3

A representation of the Joachimsthal surfaces (having a family of curvature lines that lie in totally geodesic 2-spheres) in the sphereS ³ is obtained. It is proved that, if a surface of constant mean curvature inS ³ has one family of curvature lines lying in totally geodesic 2-spheres, then it is a surface of rotation. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 221–229, February, 2000.     

A geometric interpretation of stationary-wave type solutions of the sine-Gordon equation

Using the connection of solutions of the sine-Gordon equationz _xy =sinz with the construction of a grid of asymptotic lines on surfaces of constant negative curvature inE ³ we show that those surfaces and only those that have curvature –1 and any two asymptotic lines of a single family of which are congruent correspond to solutions of stationary-wave typez=z(x+by).Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 81–87.     

Projective analog of Egorov transformation

We prove the following assertion, which is a projective analog of the well-known Egorov theorem on surfaces in the Euclidean space: a family of lines v = const on a surface S in P ³ is a basis for Egorov transformation if and only if the surface bands defined on S by these lines belong to bilinear systems of plane elements. There exist a whole set of Egorov transformations that depend on one function of v with this family of lines as the basis of the correspondence.     

Weingarten surfaces and nonlinear partial differential equations

The sine-Gordon equation has been known for a long time as the equation satisfied by the angle between the two asymptotic lines on a surface inR ³ with constant Gauss curvature –1. In this paper, we consider the following question: Does any other soliton equation have a similar geometric interpretation? A method for finding all the equations that have such an interpretation using Weingarten surfaces inR ³ is given. It is proved that the sine-Gordon equation is the only partial differential equation describing a class of Weingarten surfaces inR ³ and having a geometricso(3)-scattering system. Moreover, it is shown that the elliptic Liouville equation and the elliptic sinh-Gordon equation are the only partial differential equations describing classes of Weingarten surfaces inR ³ and having geometricso(3,C)-scattering systems.     

Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P4

We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in P ⁴ such congruences are necessarily linear. Based on the results of Castelnuovo, the classification of three-component systems is obtained, revealing a close relationship of the problem with projective geometry of the Veronesé variety V ²⊂P ⁵ and the theory of associativity equations of two-dimensional topological field theory. Received: 15 August 2001     

Mappings of Ruled Surfaces in Simply Isotropic Space <Emphasis Type="Italic">I</Emphasis><Subscript>3</Subscript><Superscript>1</Superscript> that Preserve the Generators

In this paper we study some mappings of skew ruled surfaces in simply isotropic space which preserve the generators. We study isometries, conformal mappings and mappings which preserve the area. Furthermore, we study mappings of surfaces in I ₃ ¹ which preserve the asymptotic lines.Received December 18, 2001; in revised form July 12, 2002 Published online April 4, 2003     

Helix splines as an example of affine Tchebycheffian splines

The present paper summarizes the theory of affine Tchebycheffian splines and presents an interesting affine Tchebycheffian free-form scheme, the “helix scheme”. The curve scheme provides exact representations of straight lines, circles and helix curves in an arc length parameterization. The corresponding tensor product surfaces contain helicoidal surfaces, surfaces of revolution and patches on all types of quadrics. We also show an application to the construction of planarC ² motions interpolating a given set of positions. Because the spline curve segments are calculated using a subdivision algorithm, many algorithms, which are of fundamental importance in the B-spline technique, can be applied to helix splines as well. This paper should demonstrate how to create an affine free-form scheme fitting to certain special applications.     

Projections of generic surfaces of revolution

We study some geometric properties of a generic surface of revolution in ³, like its order of contact with lines and its projection onto planes. The projections of generic surfaces of revolution from any centre of projection are classified up to diffeomorphisms of the apparent contour.     

Surfaces of order three with a peak. II

In [1], we presented a theory of surfaces of order three in real projective three-spaceP ³. In the present paper, we prove that a surfaceF of order three with a peak contains one, two or three lines and there are four types of suchF based upon the configuration of these lines. We describe eachF by determining the existence and the distribution of elliptic, parabolic and hyperbolic points; that is, points ofF not lying on any line contained inF.     

Classification of incidence scrolls (I)

The aim of this paper is to obtain a classification of scrolls of genus 0 and 1, which are defined by a one-dimensional family of lines meeting a certain set of linear spaces in p ⁿ . These ruled surfaces will be called incidence scrolls and such a set will be the base of the incidence scroll. Unless otherwise stated, we assume that the base spaces are in general position. Received: 1 December 2000     

Calabi–Yau Threefolds and Moduli of Abelian Surfaces I

We describe birational models and decide the rationality/unirationality of moduli spaces A _d (and A _d ^lev ) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero.     

A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.