Special curves on ruled surfaces in Galilean and pseudo-Galilean spaces

We develop the theory of ruled surfaces in the pseudo-Galilean space G ₃ ¹. Furthermore, we prove the Bonnet theorem and investigate the osculating quadrics for ruled surfaces in G ₃ ¹as well as in the Galilean space G ₃. This revised version was published online in June 2006 with corrections to the Cover Date.     

The local classifications of the ruled surfaces of normals and binormals of a regular space curve

The ruled surfaces of normals and binormals of a space curve is locally classified under the left-right action according to the types of the curve. In order to do this some useful results are obtained on the relationship of the powers of terms in the Taylor series of an invertible function and its inverse.     

The degree of the variety of rational ruled surfaces and Gromov-Witten invariants

We compute the degree of the variety parametrizing rational ruled surfaces of degree in by relating the problem to Gromov-Witten invariants and Quantum cohomology.

     

Analytic sphere eversion using ruled surfaces

Sphere eversions have been described so far by either pictures with minimal topological complexity, numerical evolution or complex equations. We write down relatively simple explicit formulas for the whole eversion, both analytic and topologically simpler, including also Boy surface (real projective plane), using a family of ruled surfaces. We show their usefulness in visualizing the process using commonly available modeling software.     

Pseudo‐spherical normal Darboux images of curves on a lightlike surface

We introduce the notions of the pseudospherical normal Darboux images for the curve on a lightlike surface in Minkowski 3‐space and study these Darboux images by using technics of the singularity theory. Furthermore, we give a relation between these Darboux images and Darboux frame from the viewpoint of Legendrian dualities.     

Parameter spaces for curves on surfaces and enumeration of rational curves

Let S be a smooth, minimal rational surface. The geometry of the Severi variety parametrising irreducible, rational curves in a given linear system on S is studied. The results obtained are applied to enumerative geometry, in combination with ideas from Quantum Cohomology. Formulas enumerating rational curves are found, some of which generalised Kontsevich's formula for plane curves.     

Degeneracy of holomorphic curves in surfaces

Let X be a complex projective algebraic manifold of dimension 2 and let D1,…,Du be distinct irreducible divisors on X such that no three of them share a common point. Let f: C→X\(U1≤i≤uDi) be a holomorphic map. Assume that u≥4 and that there exist positive integers n1,…,nu, c such that ninj(Di.Dj) = c for all pairs i, j. Then f is algebraically degenerate, i.e. its image is contained in an algebraic curve on X.     

Nonexistence of certain complete minimal surfaces with planar ends

In this paper we prove that two lines bounding an immersed minimal surface in a slab in R ³ homeomorphic to a compact Riemann surface minus two disks and a finite number of points must be parallel. This theorem is extended to a higher dimensional minimal hypersurface. Also it is proved that if the Gauss map of a complete embedded minimal surface of finite total curvature at a planar end has order two, then the intersection of the surface with the plane asymptotic to the planar end cannot admit a one-to-one orthogonal projection onto any line in the plane. Received: November 26, 1998     

Isoperimetric curves on hyperbolic surfaces

Least-perimeter enclosures of prescribed area on hyperbolic surfaces are characterized.

     

Some special surfaces in the pseudo-Galilean space

We describe some special surfaces in pseudo-Galilean spaces such as helical surfaces, ruled screw surfaces, surfaces of revolution and in particular tori of revolution. We define special surfaces and find their main properties.      

Del Pezzo surfaces with nonrational singularities

Normal algebraic surfacesX with the property rk(Div(X)⊗ℚ/≡)=1, numerically ample canonical classes, and nonrational singularities are classified. It is proved, in particular, that any such surfaceX is a contraction of an exceptional section of a (possibly singular) relatively minimal ruled surface with a nonrational base. Moreover, f is uniquely determined by the surfaceX. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 451–467, September, 1997. Translated by O. V. Sipacheva     

Anti de Sitter horospherical flat timelike surfaces

We investigate a special timelike surfaces in Anti de Sitter 3-space.We call such a timelike surface an Anti de Sitter horospherical flat surface which belongs to a class of surfaces given by one parameter families of Anti de Sitter horocycle.We give a generic classification of singularities and study the geometric properties of such surfaces from the viewpoint of Legendrian singularity theory.     

On Finite Morphisms of Ruled Surfaces

We prove that a surjective morphism :XX between ruled surfaces is finite and it descends to a finite morphism :CC between base curves of X and X. When is restricted to the fibres of X, it has a constant degree, say a, and then deg=a deg. In addition, we have several properties on the inverse image of a minimal section and a fibre of X as well as on the direct images. We also investigate precisely the case when both C and C are elliptic and X is the fibre product C ×_C X especially.     

Judging or setting weight steady-state of rational Bezier curves and surfaces

Many works have investigated the problem of reparameterizing rational B~zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.     

On orbit closures of spherical subgroups in flag varieties

Let be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on with finitely many orbits, and let V be an H-orbit closure in . Expanding the cohomology class of V in the basis of Schubert classes defines a union V₀ of Schubert varieties in with positive multiplicities. If G is simply-laced, we show that these multiplicities are equal to the same power of 2. For arbitrary G, we show that V₀ is connected in codimension 1. If moreover all multiplicities are 1, we show that the singularities of V are rational and we construct a flat degeneration of V to V₀ in . Thus, for any effective line bundle L on , the restriction map is surjective, and for all . Received: April 17, 2000     

A ruled residue theorem for function fields of elliptic curves

It is shown that a valuation of residue characteristic different from 2 and 3 on a field E has at most one extension to the function field of an elliptic curve over E, for which the residue field extension is transcendental but not ruled. The cases where such an extension is present are characterised.     

管状曲面在曲线论中的应用

研究了围绕曲线的管状曲面上的曲率线,渐近线与测地线,给出它们的方程,揭示了这些曲线与Bertrand曲线或Mannheim曲线之间的关系,采用新的方法给出一条曲线是Bertrand曲线或Mannheim曲线的充要条件的另一种证明以及Mannheim侣线的曲率与挠率之间的关系．