arXiv subjects

Tadashi Fujioka

Publications and source records attributed to Tadashi Fujioka.

16 recordsLinked to original sources

arXiv2026

The disjoint disks property for Busemann $G$-spaces

We prove that every finite-dimensional Busemann $G$-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere $L=S(c,r)$, we show that every embedded arc contained in an exact distance level is a homotopical $Z_2$-set in $L$. It follows that $L$ has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for $L\times\mathbb R$, and a local avoidance argument at the center yields DDP for the ambient $G$-space. Since finite-dimensional Busemann $G$-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.

math.MG↗

arXiv2026

Busemann G-spaces with convex balls

We prove that any Busemann G-space such that every sufficiently small metric ball is convex is a topological manifold. The key ingredient in the proof is Ivanov's Helly theorem. The appendix contains a counterexample to a question of Berestovskii--Halverson--Repovš.

math.MG↗

arXiv2026

Finsler structure of Busemann G-spaces

We provide two sufficient conditions for a Busemann G-space to admit a differentiable DC atlas with a continuous Finsler metric, from the viewpoint of comparison geometry. These results generalize previous work on G-spaces with Riemannian curvature bounds, namely the Alexandrov and CAT conditions, to the Finsler setting.

math.DG↗

arXiv2026

Topological regularity of Busemann spaces of nonpositive curvature

We extend the topological results of Lytchak-Nagano and Lytchak-Nagano-Stadler for CAT(0) spaces to the setting of Busemann spaces of nonpositive curvature, i.e., BNPC spaces. We give a characterization of locally BNPC topological manifolds in terms of their links and show that the singular set of a locally BNPC homology manifold is discrete. We also prove that any (globally) BNPC topological 4-manifold is homeomorphic to Euclidean space. Applications include a topological stability theorem for locally BNPC G-spaces. Our arguments also apply to spaces admitting convex geodesic bicombings.

math.DG↗

arXiv2026

Busemann and MCP

We study the structure of Busemann spaces with measures satisfying the measure contraction property (MCP). The main results are rigidity theorems and structure theorems under the assumption of geodesic completeness or non-collapse. The appendix contains some observations on the tangent cones of geodesically complete Busemann spaces.

math.DG↗

arXiv2025

Alexandrov spaces are CS sets

We prove that the extremal stratification of an Alexandrov space introduced by Perelman-Petrunin is a CS stratification in the sense of Siebenmann. We also show that every space of directions of an Alexandrov space without proper extremal subsets is homeomorphic to a sphere. In the polyhedral case, the same holds for every iterated space of directions.

math.DG↗

arXiv2025

Lipschitz homotopy convergence of Alexandrov spaces II

We establish a quantitative version of the Lipschitz homotopy convergence introduced by Mitsuishi and Yamaguchi for a moduli space of compact Alexandrov spaces without collapsing. Along the way, we obtain a Lipschitz version of Petersen's homotopy stability theorem that is applicable to more general settings, including CAT spaces. We also show that the Lipschitz homotopies can be chosen to preserve the singular strata of Alexandrov spaces, i.e., extremal subsets.

math.DG↗

arXiv2024

Uniform boundedness on extremal subsets in Alexandrov spaces

In this paper, we study extremal subsets in Alexandrov spaces with dimension $n$, curvature $\geκ$, and diameter $\le D$. We show that the following three quantities are uniformly bounded above in terms of $n$, $κ$, and $D$: (1) the number of extremal subsets in an Alexandrov space; (2) the Betti numbers of an extremal subset; (3) the volume of an extremal subset. The proof is an application of essential coverings introduced by Yamaguchi.

math.DG↗

arXiv2023

Noncritical maps on geodesically complete spaces with curvature bounded above

We define and study the regularity of distance maps on geodesically complete spaces with curvature bounded above. We prove that such a regular map is locally a Hurewicz fibration. This regularity can be regarded as a dual concept of Perelman's regularity in the geometry of Alexandrov spaces with curvature bounded below. As a corollary we obtain a sphere theorem for geodesically complete CAT(1) spaces.

math.DG↗

arXiv2023

Euler characteristics of collapsing Alexandrov spaces

We prove that the Euler characteristic of a collapsing Alexandrov space (in particular, a Riemannian manifold) is equal to the sum of the products of the Euler characteristics with compact support of the strata of the limit space and the Euler characteristics of the fibers over the strata. This was conjectured by Semyon Alesker.

math.DG↗

arXiv2023

A lower bound for the curvature integral under an upper curvature bound

We prove that the integral of scalar curvature over a Riemannian manifold is uniformly bounded below in terms of its dimension, upper bounds on sectional curvature and volume, and a lower bound on injectivity radius. This is an analogue of an earlier result of Petrunin for Riemannian manifolds with sectional curvature bounded below.

math.DG↗

arXiv2023

Extremal subsets in geodesically complete spaces with curvature bounded above

We introduce the notion of an extremal subset in a geodesically complete space with curvature bounded above, i.e., a GCBA space. This is an analogue of an extremal subset in an Alexandrov space with curvature bounded below introduced by Perelman and Petrunin. We prove that under an additional assumption the set of topological singularities in a GCBA space forms an extremal subset. We also exhibit some structural properties of extremal subsets in GCBA spaces.

math.DG↗

arXiv2022

Application of good coverings to collapsing Alexandrov spaces

Let $M$ be an Alexandrov space collapsing to an Alexandrov space $X$ of lower dimension. Suppose $X$ has no proper extremal subsets and let $F$ denote a regular fiber. We slightly improve the result of Perelman to construct an infinitely long exact sequence of homotopy groups and a spectral sequence of cohomology groups for the pair $(M,X,F)$. The proof is an application of the good coverings of Alexandrov spaces introduced by Mitsuishi-Yamaguchi. We also extend this result to each primitive extremal subset of $X$.

math.DG↗

arXiv2021

Collapsing to Alexandrov spaces with isolated mild singularities

Let $M_j$ be a sequence of Riemannian manifolds with sectional curvature bound below collapsing to a compact Alexandrov space $X$ of dimension $k$. Suppose that all but finitely many points of $X$ are $(k,δ)$-strained and that the space of directions at each exceptional point contains $k+1$ directions making obtuse angles with each other. We prove that $M_j$ admits a structure of locally trivial fibration over $X$ for sufficiently large $j$. The same is true for collapsing sequences of Alexandrov spaces such that the infimum of the volume of the spaces of directions is sufficiently large relative to $δ$.

math.DG↗

arXiv2021

A fibration theorem for collapsing sequences of Alexandrov spaces

Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently large compared to the weakness of singularities of $X$, then $f_j$ is a locally trivial fibration. Moreover, we show some properties on the intrinsic metric and the volume of the fibers of $f_j$.

math.DG↗

arXiv2021

Regular points of extremal subsets in Alexandrov spaces

We define regular points of an extremal subset in an Alexandrov space and study their basic properties. We show that a neighborhood of a regular point in an extremal subset is almost isometric to an open subset in Euclidean space and that the set of regular points in an extremal subset has full measure and is dense in it. These results actually hold for strained points in an extremal subset. Applications include the volume convergence of extremal subsets under a noncollapsing convergence of Alexandrov spaces, and the existence of a cone fibration structure of a metric neighborhood of the regular part of an extremal subset. In an appendix, a deformation retraction of a metric neighborhood of a general extremal subset is constructed.

math.DG↗