Rapport asymptotique de courbure, courbure positive et non effondrement

Deruelle, Alix

Séminaire de théorie spectrale et géométrie, Volume 30 (2011-2012), pp. 47-75.

We focus on a geometric invariant associated to any noncompact Riemannian manifold : the asymptotic curvature ratio introduced by Gromov. We study how it interacts with the topology of the underlying manifold with other geometric constraints such as positive asymptotic volume ratio, nonnegative (Ricci) curvature and finiteness of the fundamental group (at infinity).

On s’intéresse ici à un invariant géométrique associé à toute variété riemannienne non compacte : le rapport asymptotique de courbure. On étudie son influence sur la topologie de la variété sous-jacente en présence d’autres contraintes géométrico-topologiques portant sur le volume asymptotique, la positivité de la courbure (de Ricci) et/ou la finitude du groupe fondamental (à l’infini).

Classification: 53, 58
Mot clés : géométrie riemannienne, courbure positive, cône asymptotique, effondrement à l’infini, topologie des variétés riemanniennes non compactes.
Keywords: Riemannian geometry, nonnegative curvature, asymptotic cone, collapsing at infinity, topology of noncompact Riemannian manifolds.

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