We show that a group G acts properly and effectively on a locally compact and σ-compact metric space (X, d) if and only if there exists a compatible G-invariant Heine-Borel metric dp on X such that G is homeomorphic to a closed subgroup of the group of isometries Iso(X, dp).

In this paper, spaces with σ-locally countable weak-bases are characterized as the weakly open msss-images of metric spaces (or g-first countable spaces with σ-locally countable cs-networks). To find the internal characterizations of certain images of metric spaces is an interesting research topic on general topology. Recently, S. Xia introduced the concept of weakly open mappings, by using it,...

We derive the (g − 2)(g − 3)/2 linearly independent relations among the products of pairs in a basis of holomorphic abelian differentials in the case of compact non-hyperelliptic Riemann surfaces of genus g ≥ 4. By the Kodaira-Spencer map this leads to the modular invariant metric on the moduli space induced by the Siegel metric.

We derive the (g − 2)(g − 3)/2 linearly independent relations among the products of pairs in a basis of holomorphic abelian differentials in the case of compact non-hyperelliptic Riemann surfaces of genus g ≥ 4. By the Kodaira-Spencer map this leads to the modular invariant metric on the moduli space induced by the Siegel metric.

Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness theorem is proved for Einstein metrics on compact quotients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian version of the Miyaoka-Yau inequality. A smooth Riemannian manifold (M, g) is said [1] to be Einstein if its Ricci curvature is a constant multiple of g. Any i...

Definition. Let X be a metric space with metric d. (a) A collection {G α } α∈A of open sets is called an open cover of X if every x ∈ X belongs to at least one of the G α , α ∈ A. An open cover is finite if the index set A is finite. (b) X is compact if every open cover of X contains a finite subcover. Definition. Let X be a metric space with metric d and let A ⊂ X. We say that A is a compact s...

in this paper, we present some common fixed point theorems for two generalized nonexpansive multivalued mappings in cat(0) spaces as well as in uced banach spaces. moreover, we prove the existence of fixed points for generalized nonexpansive multivalued mappings in complete geodesic metric spaces with convex metric for which the asymptotic center of a bounded sequence in a bounded closed convex...

considering the increasing interest in fuzzy theory and possible applications,the concept of fuzzy metric space concept has been introduced by severalauthors from different perspectives. this paper interprets the theory in termsof metrics evaluated on fuzzy numbers and defines a strong hausdorff topology.we study interrelationships between this theory and other fuzzy theories suchas intuitionis...