Mémoires de la Société Mathématique de France 2007; 92 pp; softcover Number: 108 ISBN10: 2856292321 ISBN13: 9782856292327 List Price: US$38 Individual Members: US$34.20 Order Code: SMFMEM/108
 The authors study the real Grassmannian \(G^\mathbb{R}_{n,n}\) of \(n\)planes in \(\mathbb{R}^{2n}\), with \(n\ge 3\), and its reduced space. The latter is the irreducible symmetric space \(\bar G^\mathbb{R}_{n,n}\), which is the quotient of the space \(G^\mathbb{R}_{n,n}\) 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 \(\bar G^\mathbb{R}_{3,3}\) possesses nontrivial infinitesimal isospectral deformations; it provides the first example of an irreducible reduced symmetric space which admits such deformations. The authors also give a criterion for the exactness of a form of degree one on \(\bar G^\mathbb{R}_{n,n}\) in terms of a Radon transform. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in analysis. Table of Contents  Introduction
 Symmetric spaces of compact type and the Guillemin condition
 Invariant symmetric forms on symmetric spaces
 The real Grassmannians
 The Stiefel manifolds and the real Grassmannians
 Functions and forms of degree one on the real Grassmannians
 Isospectral deformations of the real Grassmannian of 3planes in \(\mathbb{R}^6\)
 Forms of degree one
 A family of polynomials
 Exactness of the forms of degree one
 Branching laws
 The special Lagrangian Grassmannian \(SU(4)/SO(4)\)
 The complex quadric of dimension three
 Bibliography
