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Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet

Bernard Bonnard, Monique Chyba (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Consider a sub-riemannian geometry(U,D,g) where U is a neighborhood of 0 in R3, D is a Martinet type distribution identified to ker ω, ω being the 1-form: ω = d z - y 2 2 d x , q=(x,y,z) and g is a metric on D which can be taken in the normal form: g = a ( q ) d x 2 + c ( q ) d y 2 , a=1+yF(q), c=1+G(q), G | x = y = 0 = 0 . In a previous article we analyze the flat case: a=c=1; we describe the conjugate and cut loci, the sphere and the wave front. The objectif of this article is to provide a geometric and computational framework to analyze the general case....

Metric Entropy of Homogeneous Spaces

Stanisław Szarek (1998)

Banach Center Publications

For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the metric entropy, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous...

Metric of special 2F-flat Riemannian spaces

Raad J. K. al Lami (2005)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In this paper we find the metric in an explicit shape of special 2 F -flat Riemannian spaces V n , i.e. spaces, which are 2 F -planar mapped on flat spaces. In this case it is supposed, that F is the cubic structure: F 3 = I .

Metric Perspectives of the Ricci Flow Applied to Disjoint Unions

Sajjad Lakzian, Michael Munn (2014)

Analysis and Geometry in Metric Spaces

In this paper we consider compact, Riemannian manifolds M1, M2 each equipped with a oneparameter family of metrics g1(t), g2(t) satisfying the Ricci flow equation. Adopting the characterization of super-solutions to the Ricci flow developed by McCann-Topping, we define a super Ricci flow for a family of distance metrics defined on the disjoint union M1 ⊔ M2. In particular, we show such a super Ricci flow property holds provided the distance function between points in M1 and M2 is itself a super...

Metric Ricci Curvature and Flow for PL Manifolds

Emil Saucan (2013)

Actes des rencontres du CIRM

We summarize here the main ideas and results of our papers [28], [14], as presented at the 2013 CIRM Meeting on Discrete curvature and we augment these by bringing up an application of one of our main results, namely to solving a problem regarding cube complexes.

Metric trees in the Gromov--Hausdorff space

Yoshito Ishiki (2023)

Commentationes Mathematicae Universitatis Carolinae

Using the wedge sum of metric spaces, for all compact metrizable spaces, we construct a topological embedding of the compact metrizable space into the set of all metric trees in the Gromov--Hausdorff space with finite prescribed values. As its application, we show that the set of all metric trees is path-connected and all its nonempty open subsets have infinite topological dimension.

Metrics in the sphere of a C*-module

Esteban Andruchow, Alejandro Varela (2007)

Open Mathematics

Given a unital C*-algebra 𝒜 and a right C*-module 𝒳 over 𝒜 , we consider the problem of finding short smooth curves in the sphere 𝒮 𝒳 = x ∈ 𝒳 : 〈x, x〉 = 1. Curves in 𝒮 𝒳 are measured considering the Finsler metric which consists of the norm of 𝒳 at each tangent space of 𝒮 𝒳 . The initial value problem is solved, for the case when 𝒜 is a von Neumann algebra and 𝒳 is selfdual: for any element x 0 ∈ 𝒮 𝒳 and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ ℒ 𝒜 ( 𝒳 ) , Z* = −Z and ∥Z∥ ≤ π, such...

Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields

Frédéric Campana, Henri Guenancia, Mihai Păun (2013)

Annales scientifiques de l'École Normale Supérieure

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor of a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields.

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