相关论文: A Short Study of Alexandroff Spaces
We find universal spaces for Alexandroff and finite spaces and explore some of its topological properties as well as their description as inverse limits of finite spaces and Alexandroff extensions. They can be used as a natural environment…
We construct examples of 3-dimensional compact aspherical Alexandrov spaces without boundary which are not topological manifolds.
Alexandrov spaces are defined via axioms similar to those given by Euclid. The Alexandrov axioms replace certain equalities with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded…
We construct the first non-trivial examples of compact non-isometric Alexandrov spaces which are isospectral with respect to the Laplacian and not isometric to Riemannian orbifolds. This construction generalizes independent earlier results…
Differentiable structure ensures that many of the basics of classical convex analysis extend naturally from Euclidean space to Riemannian manifolds. Without such structure, however, extensions are more challenging. Nonetheless, in…
Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang-Schroeder-Sturm. The purpose of this paper is to study…
In the present paper, we consider several valid notions of orientability of Alexandov spaces and prove that all such conditions are equivalent. Further, we give topological and geometric applications of the orientability. In particular, a…
In this paper we show that, in the definition of Alexandrov spaces with lower or upper curvature bound, the original conditions can be replaced with much weaker ones. For the purpose, we introduce `imaginary' comparison angles (and…
In this paper, we investigate Kondratiev spaces on domains of polyhedral type. In particular, we will be concerned with necessary and sufficient conditions for continuous and compact embeddings, and in addition we shall deal with pointwise…
Finite topological spaces became much more essential in topology, with the development of computer science. The task of this paper is to study and investigate some properties of such spaces with the existence of an ordered relation between…
In this paper we give a new proof for an almost isometry theorem in Alexandrov spaces with curvature bounded below.
We obtain sharp volume bounds on the boundaries of Alexandrov spaces with given lower curvature bound, dimension, and radius. We also completely classify the rigidity case and analyze almost rigidity. Our results are new even for smooth…
We consider an infinitesimal version of the Bishop-Gromov relative volume comparison condition as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under…
The well known equivalence between preorders and Alexandrov spaces is extended to an equivalence between arbitrary topological spaces and spatial fibrous preorders, a new notion to be introduced.
As a continuation of [MY], we determine the topologies of collapsing three-dimensional compact Alexandrov spaces with nonempty boundary.
We introduce a new geometric invariant called the obtuse constant of spaces with curvature bounded below. We first find relations between this invariant and the normalized volume. We also discuss the case of maximal obtuse constant equal to…
We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary, we obtain the classification of closed, $n$-dimensional,…
Inspired by a recent work of Grove-Petersen in [GP18], where the authors studied Alexandrov spaces with largest possible boundary. We study Alexandrov spaces with lower curvature bound 1 and with small boundary. When the radius of X is…
In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space…
The aim of this paper is to introduce the concept of Delta-Compact spaces along with some basic properties of it. Here, we try to establish the behavior of Delta-Compact spaces under the continuous mapping. Finally, we define another…