- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 07:12:33
8
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=LEM&vol=57&iss=1&update_since=2024-03-29
L’Enseignement Mathématique
Enseign. Math.
LEM
0013-8584
2309-4672
General
10.4171/LEM
http://www.ems-ph.org/doi/10.4171/LEM
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© Fondation L’Enseignement Mathématique
57
2011
1
A note on the Chas-Sullivan product
François
Laudenbach
Université de Nantes, NANTES, FRANCE
We give a finite-dimensional approach to the Chas-Sullivan product on the free loop space of a manifold, which is not necessarily orientable or compact.
General
3
21
10.4171/LEM/57-1-1
http://www.ems-ph.org/doi/10.4171/LEM/57-1-1
From spaces of polygons to spaces of polyhedra following Bavard, Ghys and Thurston
François
Fillastre
Université de Cergy-Pontoise, CERGY-PONTOISE, FRANCE
Following work of W.P.~Thurston, C.~Bavard and É. Ghys constructed particular hyperbolic polyhedra from spaces of deformations of Euclidean polygons. We present this construction as a straightforward consequence of the theory of mixed-volumes. The gluing of these polyhedra can be isometrically embedded into complex hyperbolic cone-manifolds constructed by Thurston from spaces of deformations of Euclidean polyhedra. It is then possible to deduce the metric structure of the spaces of polygons embedded in complex hyperbolic orbifolds discovered by P. Deligne and G.D. Mostow.
General
23
56
10.4171/LEM/57-1-2
http://www.ems-ph.org/doi/10.4171/LEM/57-1-2
A Frobenius theorem for Cartan geometries, with applications
Karin
Melnick
University of Maryland, COLLEGE PARK, UNITED STATES
We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing the configuration of orbits for local Killing fields in a compact real-analytic Cartan geometry, and an open-dense theorem in the smooth case, which says that if there is a dense orbit, then there is an open, dense, locally homogeneous subset. Combining the Frobenius theorem with the embedding theorem of Bader, Frances, and the author gives a representation theorem that relates the fundamental group of the manifold with the automorphism group.
General
57
89
10.4171/LEM/57-1-3
http://www.ems-ph.org/doi/10.4171/LEM/57-1-3
Quadratic Form made a Perfect Power by a New Composition Theorem on Arbitrary Quadratic Forms
Ajay
Choudhry
Old J.N.U. Campus, NEW DELHI, INDIA
This paper deals with the diophantine equation $Q(x_1,\,x_2,\,\ldots,\,x_m)= y^n$, where $m$ and $n$ are arbitrary positive integers and $Q(x_1,\,x_2,\,\ldots,\,x_m)$ is an arbitrary quadratic form in the $m$ variables $x_1,\,x_2,\,\ldots,\,x_m$. While solutions of special cases of this equation have been published earlier, the general equation of this type has not been solved till now. To solve this equation, we first show that, given an arbitrary quadratic form $Q(x_1,\,x_2,\,\ldots,\,x_m)$ in $m$ variables, there exists a {\it composition formula} $Q(u_i)\,Q^2(v_i)=Q(w_i)$ where $u_i$ and $v_i$ ($i= 1,\,2,\,\ldots,\,m$) are arbitrary variables and the $w_i$ ($i=1,\,2,\,\ldots,\,m$) are cubic forms in the variables $u_i$ and $v_i$ ($i=1,\,2,\,\ldots,\,m$). This is a new composition formula, different from the standard composition formulae of the type $Q(u_i)Q(v_i)=Q(w_i)$ which are known for certain classes of quadratic forms. As the equation $Q(x_i)=y^n$ is not always solvable, we prove a theorem giving a necessary and sufficient condition for its solvability. We use the aforementioned composition formula to obtain parametric solutions of the equation $Q(x_i)=y^n$, and also give some numerical examples.
General
91
102
10.4171/LEM/57-1-4
http://www.ems-ph.org/doi/10.4171/LEM/57-1-4
A short geometric proof of a conjecture of Fulton
Nicolas
Ressayre
Université Claude Bernard Lyon 1, Villeurbanne Cedex, FRANCE
We give a new geometric proof of a conjecture of Fulton about the Littlewood-Richardson coefficients. This conjecture was first proved by Knutson, Tao and Woodward using the Honeycomb theory. A geometric proof was given by Belkale. Our proof is based on the geometry of the Horn cones.
General
103
115
10.4171/LEM/57-1-5
http://www.ems-ph.org/doi/10.4171/LEM/57-1-5
Sur l'ergodicité du flot géodésique en courbure négative ou nulle
Yves
Coudène
Université de Bretagne Occidentale, BREST, FRANCE
Cet article est consacré à la dynamique du flot géodésique sur les variétés à courbure négative ou nulle. Après avoir détaillé quelques résultats de dynamique topologique, on étudie les propriétés ergodiques du flot géodésique sur les variétés de rang un, de trois points de vue différents : d'abord relativement à la mesure riemannienne, ensuite par une approche entropique, enfin par des techniques de généricité.
General
117
153
10.4171/LEM/57-1-6
http://www.ems-ph.org/doi/10.4171/LEM/57-1-6
The stable rank of arithmetic orders in division algebras – an elementary approach
Joachim
Schwermer
Universität Wien, WIEN, AUSTRIA
Ognjen
Vukadin
Universität Wien, WIEN, AUSTRIA
arithmetic orders, skew fields
A well-known theorem of Bass implies that $2$ defines a stable range for an arithmetic order in a finite-dimensional semisimple algebra over an algebraic number field. The purpose of this note is to provide an independent and elementary proof of this fact for arithmetic orders contained in a finite-dimensional division algebra over an algebraic number field.
$K$-theory
Number theory
General
155
163
10.4171/LEM/57-1-7
http://www.ems-ph.org/doi/10.4171/LEM/57-1-7
Sur la coupe des vêtements. Variation autour d'un thème de Tchebychev
Étienne
Ghys
École Normale Supérieure de Lyon, LYON CEDEX 07, FRANCE
General
165
208
10.4171/LEM/57-1-8
http://www.ems-ph.org/doi/10.4171/LEM/57-1-8