Über die von vier Kugelbüscheln erzeugten Flächen-Sechseck-Waben

In this work, we consider four one-parameter families of spheres which do not belong to the same bundle with the additional condition that three of these families of spheres cut the spherez=0 belonging to the fourth family in a hexagonal three-web formed by the three families of curves defined by the equationsx ² +y ²=u ₁, x=u ₂, y=u ₃ x. Then, we determine these families of spheres in such a way that they form a hexagonal surface 4-web.     

Die Sechseck-n-Waben in der Ebene,welche von n Kreisbüscheln erzeugt werden

In this paper, the problem of determining the pencils of circles which form a hexagonal n-web in E₂, is completely solved. It is well-known that n pencils of circles orthogonal to a fixed circle form a hexagonal n-web. Therefore, the main problem is the determination of all circle pencils which form a hexagonal n-web and which do not cut a fixed circle orthogonally. In this connection the following results have been obtained: The number of hexagonal 4-webs is six, whereas the number of hexagonal 5-webs is two.Finally, after having proved that the number of hexagonal 6-webs is one, it is shown that, for n7, there exist no circle pencils forming a hexagonal n-web without being orthogonal to a fixed circle.     

Curvature of Poisson pencils in dimension three

A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese web is trivial. We suggest a simpler obstruction to flatness, which we call the curvature form of a Poisson pencil. This form can be defined in two ways: either via the Blaschke curvature form of the associated web, or via the Ricci tensor of a connection compatible with the pencil.We show that the curvature form of a Poisson pencil can be given by a simple explicit formula. This allows us to study flatness of linear pencils on three-dimensional Lie algebras, in particular those related to the argument translation method. Many of them appear to be non-flat.     

On the problem of classification of regular 4-webs formed by pencils of spheres

Shelehov’s theorem on bondary curves of a regular curvilinear 3-web is generalized to the case of an arbitrary regular codimension 1 (n + 1)-web; an example is given of a regular 4-web formed by pencils of spheres in the three-dimensional conformal space (W. Blaschke’s problem); it is proved that a spherical 4-web of the basic type cannot be regular.     

Classification of holomorphic spheres of constant curvature in complex Grassmann manifold G2,5

In this paper, we classify all nonsingular holomorphic spheres in complex Grassmann manifolds G_2,5 with the induced constant curvatures K=4, 2, 4/3,1 and 4/5 into some classes, up to unitary equivalence, in which none of the spheres are congruent; At the same time we also prove that there does not exist the nonsingular holomorphic sphere in G_2,5 with constant curvature 2/3.     

On the Structure of the Set of Zeros of Quaternionic Polynomials

We prove that any quaternionic polynomial (with the coefficients on the same side) has two types of zeroes: the zeroes are either isolated or spherical ones, i.e., those ones which form a whole sphere. What is more, the total quantity of the isolated zeroes and of the double number of the spheres does not outnumber the degree of the polynomial.     

The loop-product spectral sequence

The purpose of this paper is to prove that the shriek map associated to a finite codimensional sub-fiberwise embedding between Hilbert manifolds behaves properly in regard of the associated Serre Spectral sequences. We apply this result to evaluate the Chas–Sullivan loop product of the total space of a fibration. Then, we compute up to extension issues the loop homology of sphere bundle of spheres.     

On null Lagrangians

We consider multiple-integral variational problems where the Lagrangian function, defined on a frame bundle, is homogeneous. We construct, on the corresponding sphere bundle, a canonical Lagrangian form with the property that it is closed exactly when the Lagrangian is null. We also provide a straightforward characterization of null Lagrangians as sums of determinants of total derivatives. We describe the correspondence between Lagrangians on frame bundles and those on jet bundles: under this correspondence, the canonical Lagrangian form becomes the fundamental Lepage equivalent. We also use this correspondence to show that, for a single-determinant null Lagrangian, the fundamental Lepage equivalent and the Carathéodory form are identical.     

The gonality and the Clifford index of curves on an elliptic ruled surface

In this paper it is shown that the gonality of curves on an elliptic ruled surface is twice the degree of the restriction of the bundle map and the Clifford index of such curves is computed by pencils of minimal degree, under certain numerical conditions. It is also proved that any pencil computing the gonality and the Clifford index of curves is composed with the restriction of the bundle map under some stronger conditions. On the other hand, we found some counterexample to the constancy of gonality and Clifford index in a linear system.Received: 2 December 2003     

A family of quadratic forms associated to quadratic mappings of spheres

The general form of a real quadratic mapping of spheres can be determined by studying the diagonalization of each form in an associated family of quadratic forms. In particular, the eigenvalues provide a means for detecting maps which are of the Hopf type. When the eigenvalues are nonzero for every form in the family, the forms associated to $\text{?:}\text{S}{}^{\mathrm{n}}\text{→}\text{S}{}^{\mathrm{m}}$ give rise to a quadratic form on the tangent bundle of the unit sphere Sⁿ. If ? is of the Hopf type, nondegeneracy of each form occurs only when n=1,3,7,15.     

关于Finsler子流形的平均曲率

通过使用由射影球丛诱导的体积元来研究Finsler子流形几何,推导了体积泛函的第一变分公式。给出了Finsler子流形的平均曲率形式和第二基本形式的定义,该定义在Riemannian情形下与通常的概念一致．此外,通过推导射影球丛纤维上的散度公式。给出了平均曲率形式的一种非常简洁的等价表示,并得到一些关于Minkowski空间中Finsler子流形的有趣的结果．     

On the Geometry of the 7-Sphere

A sphere of dimension 4n+3 admits three Sasakian structures and it is natural to ask if a submanifold can be an integral submanifold for more than one of the contact structures. In the 7-sphere it is possible to have curves which are Legendre curves for all three contact structures and there are 2 and 3-dimensional submanifolds which are integral submanifolds of two of the contact structures. One of the results here is that if a 3-dimensional submanifold is an integral submanifold of one of the Sasakian structures and invariant with respect to another, it is an integral submanifold of the remaining structure and is a principal circle bundle over a holmophic Legendre curve in complex projective 3-space.     

The Strong Thirteen Spheres Problem

The thirteen spheres problem asks if 13 equal-size non-overlapping spheres in three dimensions can simultaneously touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schütte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen-sphere problem (or the Tammes problem for 13 points), which calls for finding the maximum radius of and an arrangement for 13 equal-size non-overlapping spheres touching the unit sphere. In this paper, we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on an enumeration of irreducible graphs.     

Perturbation and robust stability of autonomous singular linear matrix difference equations

In the perturbation theory of linear matrix difference equations, it is well known that the theory of finite and infinite elementary divisors of regular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them to disappear. In this paper, the perturbation theory of complex Weierstrass canonical form for regular matrix pencils is investigated. By using matrix pencil theory and the Weierstrass canonical form of the pencil we obtain bounds for the finite elementary divisors of a perturbed pencil. Moreover we study robust stability of a class of linear matrix difference equations (of first and higher order) whose coefficients are square constant matrices.     

Generation of lattices of points for bivariate interpolation

Principal lattices in the plane are distributions of points particularly simple to use Lagrange, Newton or Aitken–Neville interpolation formulae. Principal lattices were generalized by Lee and Phillips, introducing three-pencil lattices, that is, points which are the intersection of three lines, each one belonging to a different pencil. In this contribution, a semicubical parabola is used to construct lattices of points with similar properties. For the construction of new lattices we use cubic pencils of lines and an addition of lines on them. AMS subject classification 41A05, 65D05, 41A63Research partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

Incompleteness Results in Kripke Bundle Semantics

Kripke bundle and C-set semantics are known as semantics which generalize standard Kripke semantics. In [4] and in [1, 2] it is shown that Kripke bundle and C-set semantics are stronger than standard Kripke semantics. Also it is true that C-set semantics for superintuitionistic logics is stronger than Kripke bundle semantics ([6]). Modal predicate logic Q-S4.1 is not Kripke bundle complete ([3] - it is also yielded as a corollary to Theorem 6.1(a) of the present paper). This is shown by using difference of Kripke bundle semantics and C-set semantics. In this paper, by using the same idea we show that incompleteness results in Kripke bundle semantics which are extended versions of [2].     

Minimal volume invariants,topological sphere theorems and biorthogonal curvature on 4‐manifolds

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4‐manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded.     

Centroaffine First Order Invariants of Surfaces in IR4

An investigation of the centroaffine geometry of surfaces in IR⁴ leads to the centroaffine first order invariants: the vector bundle valued second fundamental form, the affine semiconformal structure, the h³-semiconformal structure and the centroaffine metric. A classification of surfaces by their semiconformal structures according to signature and rank is given. This involves the study of the orbits of two pencils of symmetric bilinear forms on IR² under a change of basis. Combined with previous results ([Nomizu-Sasaki 93]) a complete classification of the zero-degenerate surfaces is obtained and examples of the other surface types are constructed.     

局部系数的示类

<正> 1935年 Stiefel-Whitney 发现了一个以正真正交群为构造群的球纤维丛(?)={B,K,Y=S~(m-1),O_m},K 是有限多面体,有一些不变类,通常称为是 Stiefel-Whitney 示性类(参阅[1]的第三章).(?)是以 Stiefel 流形 Y~q=V_(m,m-q)=O_m/O_q 为纤维丛的与(?)相配的纤维丛,q+1维的 Stiefel-Whitney 示性类记为 W~(q+1)(?),     

Uniqueness of the “nonsingular core” for Hermitian and other matrix pencils

Every pencil of hermitian matrices is conjunctive with a pencil of the form L ⊕ M, where L (the "minimal-indices" part) has no elementary divisors and M (the "nonsingular core") is a nonsingular pencil. Here it is shown that the conjunctivity type ofM is determined by that of L ⊕ M. The same method of proof applies to many other types of pencils, e.g. to congruence of pencils based on (1) a pair of symmetric matrices, (2) a pair of alternating matrices, or (3) a symmetric and an alternating matrix.