Infinitesimal Isospectral Deformations of the Grassmannian of 3-Planes in $ Mathbb {{R}}^6$

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 $\bar G^\mathbb{{R}}_{{3,3}}$...
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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 non-trivial 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.