David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain Ω in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of t ...
A weak metric on a set is a function that satisfies the axioms of a metric except the symmetry and the separation axioms. The aim of this paper is to present some interesting weak metrics and to study some of their properties. In particular, we introduce a ...
During the years 1940–1970, Alexandrov and the “Leningrad School” have investigated the geometry of singular surfaces in depth. The theory developed by this school is about topological surfaces with an intrinsic metric for which we can define a notion of c ...
We define and study a natural weak metric on the Teichmüller space of the torus. A similar metric has been defined by W. Thurston on the Teichmüller space of higher genus surfaces and our definition is motivated by Thurston's definition. However, we shall ...
We prove a version of the Lp hodge decomposition for differential forms in Euclidean space and a generalization to the class of Lizorkin currents. We also compute the Lqp−cohomology of Rn. ...
In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called simple sin-gularities.Wefirstd ...
We prove the following version of Poincaré, duality for reduced L (q,p) -cohomology: For any 1 < q, p < a, the Lqp -cohomology of a Riemannian manifold is in duality with the interior Lp'q'-cohomology for 1/p + 1/p' = 1/q + 1/q' = 1. ...
We generalize the celebrated results of Bernhard Riemann and Gaston Darboux: we give necessary and sufficient conditions for a bilinear form to be flat. More precisely, we give explicit necessary and sufficient conditions for a tensor field of type (0, 2) ...
We study some relation between some geometrically defined classes of diffeomorphisms between manifolds and the Lq,p-cohomology of these manifolds. Some applications to vanishing and non vanishing results in Lq,p-cohomology are given. ...
We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture showing that a complete and essential conformal vector field on a non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski metric. ...
The Gauss-Bonnet Formula is a significant achievement in nineteenth century differential geometry for the case of surfaces and the twentieth century cumulative work of H. Hopf, W. Fenchel, C. B. Allendoerfer, A. Weil and S.S. Chern for higher-dimensional R ...
We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold (M,g) and the Lq,p-cohomology of that manifold. The Lq,p-cohomology of (M,g) is defined to be the quotient of the space of closed differential ...
We revisit the isoperimetric inequalities for finitely generated groups introduced and studied by N. Varopoulos, T. Coulhon and L. Saloff-Coste. Namely we show that a lower bound on the isoperimetric quotient of finite subsets in a finitely generated group ...
We prove a version of Myers-Steenrod's theorem for Finsler manifolds under the minimal regularity hypothesis. In particular we show that an isometry between C-k,C-alpha-smooth (or partially smooth) Finsler metrics, with k + alpha > 0, k is an element of N ...
The purpose of this paper is to give a self-contained proof that a complete manifold with more than one end never supports an L-q,L-p-Sobolev inequality (2
We give sufficient conditions for the naturallity of the exterior differential under Sobolev mappings. In other words we study the validity of the equation df∗α=f∗dα for a smooth form α and a Sobolev map f. ...
The aim of this note is to prove that any compact metric space can be made connected at a minimal cost, where the cost is taken to be the one-dimensional Hausdorff measure. ...
We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distance ...
Dans les années 1940-1970, Alexandrov et l'``'Ecole de Leningrad" ont développé une théorie très riche des surfaces singulières. Il s'agit de surfaces topologiques, munie d'une métrique intrinsèque pour laquelle on peut définir une notion de courbure, qui ...
