Infinitesimal isospectral deformations of the Grassmannian of 3-planes in 6
Mémoires de la Société Mathématique de France, no. 108 (2007) , 98 p.

We study the real Grassmannian G n,n of n-planes in 2n , with n3, and its reduced space. The latter is the irreducible symmetric space G ¯ n,n , which is the quotient of the space G n,n under the action of its isometry which sends a n-plane into its orthogonal complement. One of the main results of this monograph asserts that the irreducible symmetric space G ¯ 3,3 possesses non-trivial infinitesimal isospectral deformations; it provides us with the first example of an irreducible reduced symmetric space which admits such deformations. We also give a criterion for the exactness of a form of degree one on G ¯ n,n in terms of a Radon transform.

Ce mémoire a pour cadre la grassmannienne G n,n des n-plans de 2n , avec n3, et son espace réduit G ¯ n,n , qui est l’espace symétrique irréductible, quotient de G n,n par l’involution envoyant un n-plan sur son orthogonal. Un de nos principaux résultats est la construction de déformations infinitésimales isospectrales non triviales sur G ¯ 3,3 , obtenant ainsi le premier exemple d’espace symétrique irréductible réduit et non infinitésimalement rigide. Nous donnons aussi un critère d’exactitude pour les formes différentielles de degré 1 sur G ¯ n,n , mettant en jeu la nullité d’une transformée de Radon.

DOI: 10.24033/msmf.420
Classification: 44A12,  53C35,  58A10,  58J53
Keywords: Symmetric space, Grassmannian, Radon transform, infinitesimal isospectral deformation, symmetric form, Guillemin condition
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Gasqui, Jacques; Goldschmidt, Hubert. Infinitesimal isospectral deformations of the Grassmannian of 3-planes in ${\mathbb{R}}^6$. Mémoires de la Société Mathématique de France, Serie 2, , no. 108 (2007), 98 p. doi : 10.24033/msmf.420. http://numdam.org/item/MSMF_2007_2_108__1_0/

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