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Register a new userAffine functions on Alexandrov spaces
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We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
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The equivariant Gromov--Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by Lie homomorphisms. Additional lower bounds on curvature and volume strengthen this result to convergence by monomorphisms, so that symmetries can only increase on passing to the limit.
Let $(X,d)$ be an $n$-dimensional Alexandrov space whose Hausdorff measure $mathcal{H}^n$ satisfies a condition giving the metric measure space $(X,d,mathcal{H}^n)$ a notion of having nonnegative Ricci curvature. We examine the influence of large volume growth on these spaces and generalize some classical arguments from Riemannian geometry showing that when the volume growth is sufficiently large, then $(X,d,mathcal{H}^n)$ has finite topological type.
In this paper we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank-$n$ Carnot algebra is isomorphic to the exterior algebra of $mathbb{R}^n$. Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.
It is shown that a construction of Z. Zhang and Y. Xiao on open subsets of Ptolemaic spaces yields, when the subset has boundary containing at least two points, metrics that are Gromov hyperbolic with parameter $log 2$ and strongly hyperbolic with parameter $1$ with no further conditions on the open set. A class of examples is constructed on Hadamard manifolds showing these estimates of the parameters are sharp.
In this work we study the issue of geodesic extendibility on complete and locally compact metric length spaces. We focus on the geometric structure of the space $(Sigma (X),d_H)$ of compact balls endowed with the Hausdorff distance and give an explicit isometry between $(Sigma (X),d_H)$ and the closed half-space $ Xtimes mathbb{R}_{ge 0}$ endowed with a taxicab metric. Among the applications we establish a group isometry between $mbox{Iso} (X,d)$ and $mbox{Iso} (Sigma (X),d_H)$ when $(X,d)$ is a Hadamard space.
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