‎We prove that a remainder $Y$ of a non-locally compact‎ ‎rectifiable space $X$ is locally a $p$-space if and only if‎ ‎either $X$ is a Lindel"{o}f $p$-space or $X$ is $sigma$-compact‎, ‎which improves two results by Arhangel'skii‎. ‎We also show that if a non-locally compact‎ ‎rectifiable space $X$ that is locally paracompact has a remainder $Y$ which has locally a $G_{delta}$-diagonal‎, ‎then...

To any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that ...

Consider a bundle of circles passing through 0 in 4-dimensional space. It is said to be rectifiable if there is a germ of diffeomorphism at 0 that takes all circles from our bundle to straight lines. We will give a classification of all rectifiable bundles of circles containing sufficiently many circles in general position. This result is surprisingly different from those in dimensions 2 and 3 ...

The main motivation of this paper arises from the study of Carnot–Carathéodory spaces, where the class of 1-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree k, which are Hö...

Let Ω be a complex manifold and let V be a (holomorphic) subvariety of Ω of pure (complex) dimension k. Then integration over V defines a closed current of dimension 2k in Ω, denoted by [V ]. More generally, let {Vj} be a locally finite family of irreducible holomorphic subvarieties of Ω of pure dimension k and let {nj} be integers. Then T = ∑ nj [Vj ] is a 2kcurrent in Ω—these are the holomorp...

We prove that a metric measure space (X, d,m) satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space W1,2 is Hilbert is rectifiable. That is, a RCD∗(K,N)-space is rectifiable, and in particular for m-a.e. point the tangent cone is unique and euclidean of dimension at most N. The proof is based on a maximal function argument combined with an original Almost Splitting ...

Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M. There is a natural Riemannian metric on the total space B consistent with the metric on M . With respect to that metric, the volume of a rectifiable section σ : M → B is the mass of the image σ(M) as a rectifiable n-current in B. For any homology class of sections of B, there is a mass-minimizing Cartesian curr...