The geometry of the moduli space of CP2 instantons

Summary The moduli space of self-dual connections (instantons) of instanton numberk=1 overCP ² is topologically a cone onCP ². The complement ^* of the vertex of this cone carries a natural Riemannian metric (the L ² metric). We find an explicit formula for this metric and deduce certain consequences. In particular, (i) the boundary is a totally geodesic submanifold of the completion, (ii) is positively curved at , and (iii) is negatively curved in certain directions near the vertex of the cone.Partially supported by N.S.F. grant DMS-8405661     

On the geometry of Urysohn's universal metric space

In recent years, much interest was devoted to the Urysohn space U and its isometry group; this paper is a contribution to this field of research. We mostly concern ourselves with the properties of isometries of U, showing for instance that any Polish metric space is isometric to the set of fixed points of some isometry φ. We conclude the paper by studying a question of Urysohn, proving that compact homogeneity is the strongest homogeneity property possible in U.     

The omega limit sets of subsets in a metric space

In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the -limit set (Y) of Y is the limit point of the sequence {(Cl Y) [i, )}_i=1 in 2^X and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor.This paper was completed while the author was visiting the Academy of Mathematics and Systems, Chinese Academy of Sciences. The author was supported by the National Science Foundation of China (Grant No. 10461003).     

The fundamental group of a compact metric space

We give a forcing free proof of a conjecture of Mycielski that the fundamental group of a connected locally connected compact metric space is either finitely generated or has the power of the continuum.

     

Nuclearity of a nonnegative definite integral kernel on a separable metric space

The aim of this paper is to characterize the nuclearity of an integral operator, defined by a continuous non-negative definite square integrable kernel on a separable metric space, in terms of the integrability of the trace of the kernel function. Nuclearity here plays a role forU-statistics.     

The link of the germ of a semi-algebraic metric space

In this paper we investigate the metric properties of semi-algebraic germs. More precisely we introduce a counterpart to the notion of link for semi-algebraic metric spaces, which is often used to study the topology. We prove that it totally determines the metric type of the germ. We give a nice consequence for semi-algebraically bi-Lipschitz homeomorphic semi-algebraic germs.

     

Diameter preserving surjections in the geometry of matrices

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping φ:Γ→Γ^′ between graphs from this class is shown to be an isomorphism provided that the following holds: Any two points of Γ are at a distance equal to the diameter of Γ if, and only if, their images are at a distance equal to the diameter of Γ^′. This result is then applied to the graphs arising from the adjacency relations of spaces of rectangular matrices, spaces of Hermitian matrices, and Grassmann spaces (projective spaces of rectangular matrices).     

Minimum of a functional in a metric space and fixed points

The existence of minimizers is examined for a function defined on a metric space. Theorems are proved that assert the existence of minimizers, and examples of the functions for which these theorems are valid are given. Then, these theorems are applied to proving theorems on fixed points of univalent and multivalued mappings of metric spaces. Finally, coincident points of two mappings are examined.     

Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry

Let be a proper partial geometry pg(s,t,2), and let G be an abelian group of automorphisms of acting regularly on the points of . Then either t≡2±od s+1 or is a pg(5,5,2) isomorphic to the partial geometry of van Lint and Schrijver (Combinatorica 1 (1981), 63–73). This result is a new step towards the classification of partial geometries with an abelian Singer group and further provides an interesting characterization of the geometry of van Lint and Schrijver.The author is Postdoctoral Fellow of the Fund for Scientific Research Flanders (FWO-Vlaanderen).     

An example of a nuclear space in infinite dimensional holomorphy

LetU be an open subset of a complex locally convex spaceE, andH(U) the space of holomorphic functions fromU toC. If the dualE′ ofE is nuclear with respect to the topology generated by the absolutely convex compact subsets ofE, then it is shown thatH(U) endowed with the compact open topology is a nuclear space. In particular, ifE is the strong dual of a Fréchet nuclear space, thenH(U) is a Fréchet nuclear space.     

An example of a nuclear space in infinite dimensional holomorphy

LetU be an open subset of a complex locally convex spaceE, andH(U) the space of holomorphic functions fromU toC. If the dualE′ ofE is nuclear with respect to the topology generated by the absolutely convex compact subsets ofE, then it is shown thatH(U) endowed with the compact open topology is a nuclear space. In particular, ifE is the strong dual of a Fréchet nuclear space, thenH(U) is a Fréchet nuclear space.     

Normal almost contact metric structures of hyperbolic type of the first kind on hypersurfaces of a space of constant holomorphic curvature of hyperbolic type

We consider normal integrable Sasakian almost contact metric structures of hyperbolic type of the first kind on hypersurfaces of a space of constant holomorphic curvature of hyperbolic type, in particular, on hypersurfaces of a flat A-space of hyperbolic type.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 22–32, January–March, 2005.Translated by V. Mackeviius     

The contraction principle for mappings on a metric space with a graph

We give some generalizations of the Banach Contraction Principle to mappings on a metric space endowed with a graph. This extends and subsumes many recent results of other authors which were obtained for mappings on a partially ordered metric space. As an application, we present a theorem on the convergence of successive approximations for some linear operators on a Banach space. In particular, the last result easily yields the Kelisky-Rivlin theorem on iterates of the Bernstein operators on the space .