Related papers: Real and bi-Lipschitz versions of the Theorem of N…

In this paper, we prove metric analogues, in any dimension and in any co-dimension, of the famous Theorem of Mumford on smoothness of normal surfaces and the beautiful Theorem of Ramanujam that gives a topological characterization of…

We show that a function $f : X \to \mathbb R$ defined on a closed uniformly polynomially cuspidal set $X$ in $\mathbb R^n$ is real analytic if and only if $f$ is smooth and all its composites with germs of polynomial curves in $X$ are real…

The classical Theorem of Mumford states that a topologically regular complex algebraic surface in $\mathbb{C}^3$ with an isolated singular point is smooth. We proof that any Lipschitz regular complex algebraic set is smooth. No restriction…

In 1957 Cartan proved his celebrated Theorem B and deduced that if $\Omega\subset{\mathbb R}^n$ is an open set and $X$ is a coherent real analytic subset of $\Omega$, then $X$ has the analytic extension property: Each real analytic function…

In this paper it is shown that for locally trivial complex analytic morphisms between some reduced spaces the Relative Riemann-Hilbert Theorem still holds up to torsion, i.e. tame flat relative connections on torsion-free sheaves are in…

Let $X$ be a real analytic orbifold. Then each stratum of $X$ is a subanalytic subset of $X$. We show that $X$ has a unique subanalytic triangulation compatible with the strata of $X$. We also show that every ${\rm C}^r$-orbifold, $1\leq…

We begin by showing that every real analytic orbifold has a real analytic Riemannian metric. It follows that every reduced real analytic orbifold can be expressed as a quotient of a real analytic manifold by a real analytic almost free…

We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily subanalytic) between two subanalytic sets, then their tangent cones are bi-Lipschitz homeomorphic. As a consequence of this result, we show that any Lipschitz…

We prove a relative version over $\mathbb{Q}$ of Nash-Tognoli theorem, that is: Let $M$ be a compact smooth manifold with closed smooth submanifolds $M_1,\dots,M_\ell$ in general position, then there exists a nonsingular real algebraic set…

In this paper, we present several definitive characterizations of the $C^1$ smoothness of definable sets in terms of their tangent cones and some other metric properties. In particular, we recover some of the beautiful characterizations…

Dahmen and Schmeding have obtained the result that although the smooth Lie group $G$ of real analytic diffeomorphisms $\mathbb S^{\,1.}\to\mathbb S^{\,1.}$ has a compatible analytic manifold structure, it does not make $G$ a real analytic…

Let $\mathbb{Q}$ be the field of rational numbers and let $X$ be a subset of $\mathbb{R}^n$. We say that $X$ is $\mathbb{Q}$-algebraic if it is the common zero set in $\mathbb{R}^n$ of a family of polynomials in…

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a…

The paper is devoted to metric properties of singularities. We investigate the relations among topology, metric properties and smoothness. In particular, we present some higher dimensional analogous of Mumford's theorem on smoothness of…

On a real ($\mathbb F=\mathbb R$) or complex ($\mathbb F=\mathbb C$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {\it tracks} $X$ if $[Y,X]=fX$ for some continuous function…

In this paper, we extend a theorem of To\"en and Vaqui\'e to the non-Archimedean and formal settings. More precisely, we prove that a smooth and proper rigid analytic variety is algebraizable if and only if its category of perfect complexes…

This article is devoted to studying multiplicity and regularity of real analytic sets. We present an equivalence for real analytic sets, named blow-spherical equivalence, which generalizes differential equivalence and subanalytic…

The purpose of this paper is to generalize in a geometric setting theorems of Severi, Brown and Bochner about analytic continuation of real analytic functions which are holomorphic or harmonic with respect to one of its variables. We prove…

In this paper, we prove the following version of the famous Bernstein's theorem: Let $X\subset \mathbb R^{n+k}$ be a closed and connected set with Hausdorff dimension $n$. Assume that $X$ satisfies the monotonicity formula at $p\in X$.…

A Lie algebroid on a variety X/k is an extension \alpha: g_X \to T_X of the tangent sheaf both as O_X-module and Lie algebra over the base field, with the obvious compatibilities; and given a Lie algebroid one has its associated ring of…