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On the geometry of Goursat structures

William Pasillas-Lépine, Witold Respondek (2001)

ESAIM: Control, Optimisation and Calculus of Variations

A Goursat structure on a manifold of dimension n is a rank two distribution 𝒟 such that dim 𝒟 ( i ) = i + 2 , for 0 i n - 2 , where 𝒟 ( i ) denote the elements of the derived flag of 𝒟 , defined by 𝒟 ( 0 ) = 𝒟 and 𝒟 ( i + 1 ) = 𝒟 ( i ) + [ 𝒟 ( i ) , 𝒟 ( i ) ] . Goursat structures appeared first in the work of von Weber and Cartan, who have shown that on an open and dense subset they can be converted into the so-called Goursat normal form. Later, Goursat structures have been studied by Kumpera and Ruiz. In the paper, we introduce a new local invariant for Goursat structures, called...

On the Geometry of Goursat Structures

William Pasillas-Lépine, Witold Respondek (2010)

ESAIM: Control, Optimisation and Calculus of Variations

A Goursat structure on a manifold of dimension n is a rank two distribution Ɗ such that dim Ɗ(i) = i + 2, for 0 ≤ i ≤ n-2, where Ɗ(i) denote the elements of the derived flag of Ɗ, defined by Ɗ(0) = Ɗ and Ɗ(i+1) = Ɗ(i) + [Ɗ(i),Ɗ(i)] . Goursat structures appeared first in the work of von Weber and Cartan, who have shown that on an open and dense subset they can be converted into the so-called Goursat normal form. Later, Goursat structures have been studied by Kumpera and Ruiz. In the paper, we introduce...

On the group of real analytic diffeomorphisms

Takashi Tsuboi (2009)

Annales scientifiques de l'École Normale Supérieure

The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the n -dimensional torus, its identity component is a simple group. For U ( 1 ) fibered manifolds, for manifolds admitting special semi-free U ( 1 ) actions and for 2- or 3-dimensional manifolds with nontrivial U ( 1 ) actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.

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