Tits topology and fundamental groups of compact Riemannian manifolds

Let M be a compact Riemannian manifold without conjugate points. We generalize the Tits topology on the ideal boundary of the universal covering space of M. Then we show that if π₁(M) is amenable and is compact with respect to the Tits topology, then M is flat. This work was supported by Grant No.R01-2006-000-10047-0(2006) from the Basic Research Program of the Korea Science & Engineering Foundation.     

Five-Dimensional Bieberbach Groups with Holonomy Group Z2⊕Z2

In this paper we determine all five-dimensional compact flat Riemannian manifolds with holonomy group Z ₂Z ₂. The classification is achieved by classifying their fundamental groups up to isomorphism. The Betti numbers of all these manifolds are also computed.     

Multivariate rational data fitting: general data structure,maximal accuracy and object orientation

Sections 1 and 2 discuss the advantages of an object-oriented implementation combined with higher floating-point arithmetic, of the algorithms available for multivariate data fitting using rational functions. Section 1 will in particular explain what we mean by higher arithmetic. Section 2 will concentrate on the concepts of object orientation. In sections 3 and 4 we shall describe the generality of the data structure that can be dealt with: due to some new results virtually every data set is acceptable right now, with possible coalescence of coordinates or points. In order to solve the multivariate rational interpolation problem the data sets are fed to different algorithms depending on the structure of the interpolation points in then-variate space.This text is a preparatory publication for the development of a scientific expert system for multivariate rational interpolation. The issues addressed are relevant to the implementation of such a system.     

On hyper-Kähler manifolds of typeA ∞

In this paper we shall construct new families of 4m dimensional non-compact complete hyper-Kähler manifolds on whichm dimensional torus acts. In the 4 dimensional case our manifolds should be considered as hyper-Kähler manifolds which correspond to the extended Dynkin diagram of typeA .     

Standard pseudo-Hermitian structure on manifolds and seifert fibration

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function onM. AsM supports a contact form, there exists a characteristic vector field dual to the contact structure. If induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature or of nonpositive curvature . By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When 0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.Dedicated to Professor Sasao Seiya for his sixtieth birthday     

The Tits boundary of a 2-complex

We investigate the Tits boundary of -complexes that have only a finite number of isometry types of cells. In particular, we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the -complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two -complexes are quasi-isometric, then the cores of their Tits boundaries are bi-Lipschitz.

     

Geometry in Grassmannians and a generalization of the dilogarithm

In this paper we introduce some polyhedra in Grassman manifolds which we call Grassmannian simplices. We study two aspects of these polyhedra: their combinatorial structure (Section 2) and their relation to harmonic differential forms on the Grassmannian (Section 3). Using this we obtain results about some new differential forms, one of which is the classical dilogarithm (Section 1). The results here unite two threads of mathematics that were much studied in the 19th century. The analytic one, concerning the dilogarithm, goes back to Leibnitz (1696) and Euler (1779) and the geometric one, concerning Grassmannian simplices, can be traced to Binet (1811). In Section 4, we give some of this history along with some recent related results and open problems. In Section 0, we give as an introduction an account in geometric terms of the simplest cases.     

Differentiable mappings with an infinite number of critical points

In this paper we shall give some sufficient conditions in order that the so-called -category of a pair of differentiable manifolds be infinite.

     

On the asymptotic geometry of nonpositively curved graphmanifolds

In this paper we study the Tits geometry of a 3-dimensional graphmanifold of nonpositive curvature. In particular we give an optimal upper bound for the length of nonstandard components of the Tits metric. In the special case of a -metric we determine the whole length spectrum of the nonstandard components.

     

Real quadrics in C n , complex manifolds and convex polytopes

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C ⁿ which are invariant with respect to the natural action of the real torus (S ¹) ⁿ onto C ⁿ . The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.     

One-sided Littlewood-Paley theory

In this article we develop the theory of one-sided versions of the g function of Littlewood and Paley, the area function S of Lusin and the\(g_\lambda ^* \) that admit weighted norm estimates with weights belonging to the classes A _p ⁺ of Sawyer. In Sections 1 and 2 we give definitions and some lemmas that shall be needed. Section 3 is devoted to the study of the one-sided version of the functions g and S. In Section 4 we obtain a good λ estimate for the one-sided\(g_\lambda ^* \) function, and in Sections 5 and 6 we apply the results already obtained to fractional integrals and multiplier operators.     

Isomorphic homotopy groups of certain regular sets and their images

In this paper we study the relation between the topology of the set R(f) of regular points and the topology of its image f(R(f)), for some special maps acting between two manifolds M and N. The results are oriented towards negative examples for the inverse problem of deciding whether a given closed subset of the source manifold is a critical set.     

The central limit problem for mixing sequences of random variables

Summary In this paper the central limit problem is solved for sums of random variables having bounded variances and satisfying certain mixing conditions. In case of a stochastic process these mixing conditions essentially say that as time passes events concerning the future of the process are almost independent from the events in the past. It turns out that the class of limit laws for sums of mixing random variables is exactly the same as for the bounded variances case of independent random variables. We also shall give criteria for convergence to any specified law of this class of possible limit laws. Finally we shall derive the central limit theorem involving a kind of Lindeberg-Feller condition and as a corollary thereof a kind of Ljapounov theorem.     

Besov Spaces and a Trace Ideal

Let 2 be the operator ideal of all absolutely 2-summing operators and let (2)2,1(a) be the ideal of operators of 2-approximation type l2,1. In this paper we give some sufficient conditions for matrices and kernels related to Besov spaces to generate operators belonging to the ideal (2)2,1(a).     

On three-dimensional conformally flat quasi-Sasakian manifolds

LetM be a 3-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure function is defined. With the help of this function, we find necessary and sufficient conditions forM to be conformally flat. Next it is proved that ifM is additionally conformally flat with = const., then (a)M is locally a product ofR and a 2-dimensional Kählerian space of constant Gauss curvature (the cosymplectic case), or (b)M is of constant positive curvature (the non cosymplectic case; here the quasi-Sasakian structure is homothetic to a Sasakian structure). An example of a 3-dimensional quasi-Sasakian structure being conformally flat with nonconstant structure function is also described. For conformally flat quasi-Sasakian manifolds of higher dimensions see [O₁]     

The strong maximum principle revisited

In this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g.

(i)

two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4);

(ii)

the exterior Dirichlet boundary value problem (Section 5);

(iii)

the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7);

(iv)

Euler-Lagrange inequalities on a Riemannian manifold (Section 9);

(v)

comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10).

The case of p-regular elliptic inequalities is briefly considered in Section 11.     

Localization of MV-algebras and lu-groups

The aim of the present paper is to define the localization MV-algebra of an MV-algebra A with respect to a topology on A; also, following the categorical equivalence between the category of lu-groups and the category of MV-algebras, we define the analogous notion for lu-groups. In Section 5 we prove that the maximal MV-algebra of quotients (defined in [7]) and the MV-algebra of fractions relative to an -closed system (defined in [6]) are MV-algebras of localization.In the last part of this paper (Section 6) we prove analogous results for lu-groups.