Infinitesimal isospectral deformations of the Grassmannian of 3-planes in ${\mathbb{R}}^6$

Infinitesimal isospectral deformations of the Grassmannian of 3-planes in

ℝ^{6}

[Déformations infinitésimales isospectrales de la grassmannienne des 3-plans dans

ℝ^{6}

]

Gasqui, Jacques ; Goldschmidt, Hubert

Mémoires de la Société Mathématique de France, no. 108 (2007) , 98 p.

Résumé
Abstract

Ce mémoire a pour cadre la grassmannienne $G_{n, n}^{ℝ}$ des $n$ -plans de $ℝ^{2 n}$ , avec $n \geq 3$ , et son espace réduit ${\bar{G}}_{n, n}^{ℝ},$ qui est l’espace symétrique irréductible, quotient de $G_{n, n}^{ℝ}$ 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 ${\bar{G}}_{3, 3}^{ℝ},$ 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 ${\bar{G}}_{n, n}^{ℝ},$ mettant en jeu la nullité d’une transformée de Radon.

We study the real Grassmannian $G_{n, n}^{ℝ}$ of $n$ -planes in $ℝ^{2 n}$ , with $n \geq 3$ , and its reduced space. The latter is the irreducible symmetric space ${\bar{G}}_{n, n}^{ℝ}$ , which is the quotient of the space $G_{n, n}^{ℝ}$ 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 ${\bar{G}}_{3, 3}^{ℝ}$ 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 ${\bar{G}}_{n, n}^{ℝ}$ in terms of a Radon transform.

MR Zbl | 1 citation dans Numdam

DOI : 10.24033/msmf.420

Classification : 44A12, 53C35, 58A10, 58J53
Keywords: Symmetric space, Grassmannian, Radon transform, infinitesimal isospectral deformation, symmetric form, Guillemin condition
Mot clés : Espace symétrique, grassmannienne, transformée de Radon, déformation isospectrale, forme symétrique, condition de Guillemin

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     title = {Infinitesimal isospectral deformations of the {Grassmannian} of 3-planes in ${\mathbb{R}}^6$},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {108},
     year = {2007},
     doi = {10.24033/msmf.420},
     mrnumber = {2447005},
     zbl = {1152.53040},
     language = {en},
     url = {http://archive.numdam.org/item/MSMF_2007_2_108__1_0/}
}

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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, Série 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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