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Friedrich Schur (1856-1932) a accompli d’importantes recherches sur les fondements de la géométrie à la même époque que Hilbert. Elles ont trouvé leur aboutissement dans un livre publié en 1909 et intitulé, comme celui de Hilbert, Grundlagen der Geometrie. La construction axiomatique exposée par Schur est originale et différente de celle de Hilbert. Elle trouve son origine dans les travaux de Pasch et Peano. Elle prend comme point de départ la géométrie projective et accorde une place centrale à...
On montre que la géométrie de Hilbert d’un domaine convexe de est à géométrie locale bornée c-à-d que pour un rayon fixé, toutes les boules sont bilipschitz à un domaine de euclidien. On en déduit que si la géométrie de Hilbert est hyperbolique au sens de Gromov, alors le bas de son spectre est strictement positif. On donne un contre-exemple en dimension trois qui montre que la réciproque n’est pas vraie pour les géométries de Hilbert non planes.
We consider the problem of isometric embedding of metric spaces into Banach spaces, and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a...
Let be a nonempty open set in a metric space with . Define
where is the distance from to the boundary of . For every , is a metric. We study the sharp Lipschitz constants for the metric under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.
In this paper we construct, for each aspherical oriented -manifold , a -dimensional class in the -homology of whose norm combined with the Gromov simplicial volume of gives a characterization of those nonzero degree maps from to which are homotopic to a covering map. As an application we characterize those degree one maps which are homotopic to a homeomorphism in term of isometries between the bounded cohomology groups of and .
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