Infinitesimal isospectral deformations of the Grassmannian of 3-planes in \(\mathbb R^6\).

*(English)*Zbl 1152.53040Let \((X,g)\) be a Riemannian symmetric space of compact type and \((g_t)\) be a family of Riemannian metrics on \(g\) with \(g_0 = g\). V. Guillemin [Semin. on micro-local analysis, Inst. adv. Study, Princeton/New Jersey 1977–78, Ann. Math. Stud. 93, 79–111 (1979; Zbl 0425.58020)] proved that if \((g_t)\) is an isospectral deformation, then the corresponding infinitesimal deformation \(h = \frac{dg_t}{dt}|_{t=0}\) belongs to the kernel \({\mathcal N}_2\) of a certain Radon transform defined on the space of symmetric \(2\)-forms on \(X\) in terms of integration over the maximal flat totally geodesic tori of \(X\). The infinitesimal deformation \(h\) is trivial if and only if it is a Lie derivative of \(g\). As the set of such Lie derivatives is a subspace of \({\mathcal N}_2\), one can define \(I(X)\) to be the orthogonal complement of this subspace in \({\mathcal N}_2\). The symmetric space \((X,g)\) is said to be infinitesimally rigid in the sense of Guillemin if \(I(X)\) is trivial.

R. Michel [Bull. Soc. Math. Fr. 101, 17–69 (1973; Zbl 0265.53041)] and C. Tsukamoto [Ann. Sci. Éc. Norm. Supér. (4) 14, 339–356 (1981; Zbl 0481.53041)] proved that compact Riemannian symmetric spaces of rank one are infinitesimally rigid in the sense of Guillemin. The authors proved in [Radon transforms and the rigidity of the Grassmannians, Princeton, NJ: Princeton University Press (2004; Zbl 1051.44003)] that a product of irreducible symmetric spaces of compact type is not infinitesimally rigid in the sense of Guillemin.

The present monograph is devoted to the study of infinitesimal rigidity in the sense of Guillemin for irreducible symmetric spaces of compact type. An irreducible symmetric space \((X,g)\) of compact type is said to be reduced if it is not a non-trivial covering space of another symmetric space of compact type. The first result in this monograph asserts that if \((X,g)\) is infinitesimally rigid in the sense of Guillemin, then \((X,g)\) must necessarily be reduced. This leads to the problem to determine the space of infinitesimal isospectral deformations of an irreducible reduced symmetric space. The authors investigate this problem for particular real Grassmannians.

Let \(G_{n,n}^{\mathbb R}\) be the real Grassmannian of all \(n\)-dimensional linear subspaces in \({\mathbb R}^{2n}\). The reduced space \(\overline{G}_{n,n}^{\mathbb R}\) of \(G_{n,n}^{\mathbb R}\) is the quotient of \(G_{n,n}^{\mathbb R}\) by the isometry of \(G_{n,n}^{\mathbb R}\) which maps an \(n\)-plane to its orthogonal complement in \({\mathbb R}^{2n}\). The main result in this monograph asserts that \(\overline{G}_{3,3}^{\mathbb R}\) is not infinitesimally rigid in the sense of Guillemin. This gives the first example of an irreducible reduced symmetric space of compact type which admits non-trivial infinitesimal isospectral deformations. The rigidity problem for the other Grassmannians \(\overline{G}_{n,n}^{\mathbb R}\), \(n \geq 4\), remains open.

The other main result in this monograph asserts that every one-form \(u\) on \(\overline{G}_{n,n}^{\mathbb R}\) satisfying \(\int_Z u(\zeta)\,dZ = 0\) for every maximal flat totally geodesic torus \(Z\) in \(\overline{G}_{n,n}^{\mathbb R}\) and every parallel vector field \(\zeta\) on \(Z\) is exact. The kernel of the Radon transform for one-forms consists precisely of those one-forms satisfying this integrability condition, which is also known as the Guillemin condition.

R. Michel [Bull. Soc. Math. Fr. 101, 17–69 (1973; Zbl 0265.53041)] and C. Tsukamoto [Ann. Sci. Éc. Norm. Supér. (4) 14, 339–356 (1981; Zbl 0481.53041)] proved that compact Riemannian symmetric spaces of rank one are infinitesimally rigid in the sense of Guillemin. The authors proved in [Radon transforms and the rigidity of the Grassmannians, Princeton, NJ: Princeton University Press (2004; Zbl 1051.44003)] that a product of irreducible symmetric spaces of compact type is not infinitesimally rigid in the sense of Guillemin.

The present monograph is devoted to the study of infinitesimal rigidity in the sense of Guillemin for irreducible symmetric spaces of compact type. An irreducible symmetric space \((X,g)\) of compact type is said to be reduced if it is not a non-trivial covering space of another symmetric space of compact type. The first result in this monograph asserts that if \((X,g)\) is infinitesimally rigid in the sense of Guillemin, then \((X,g)\) must necessarily be reduced. This leads to the problem to determine the space of infinitesimal isospectral deformations of an irreducible reduced symmetric space. The authors investigate this problem for particular real Grassmannians.

Let \(G_{n,n}^{\mathbb R}\) be the real Grassmannian of all \(n\)-dimensional linear subspaces in \({\mathbb R}^{2n}\). The reduced space \(\overline{G}_{n,n}^{\mathbb R}\) of \(G_{n,n}^{\mathbb R}\) is the quotient of \(G_{n,n}^{\mathbb R}\) by the isometry of \(G_{n,n}^{\mathbb R}\) which maps an \(n\)-plane to its orthogonal complement in \({\mathbb R}^{2n}\). The main result in this monograph asserts that \(\overline{G}_{3,3}^{\mathbb R}\) is not infinitesimally rigid in the sense of Guillemin. This gives the first example of an irreducible reduced symmetric space of compact type which admits non-trivial infinitesimal isospectral deformations. The rigidity problem for the other Grassmannians \(\overline{G}_{n,n}^{\mathbb R}\), \(n \geq 4\), remains open.

The other main result in this monograph asserts that every one-form \(u\) on \(\overline{G}_{n,n}^{\mathbb R}\) satisfying \(\int_Z u(\zeta)\,dZ = 0\) for every maximal flat totally geodesic torus \(Z\) in \(\overline{G}_{n,n}^{\mathbb R}\) and every parallel vector field \(\zeta\) on \(Z\) is exact. The kernel of the Radon transform for one-forms consists precisely of those one-forms satisfying this integrability condition, which is also known as the Guillemin condition.

Reviewer: Jürgen Berndt (Cork)

##### MSC:

53C35 | Differential geometry of symmetric spaces |

58A10 | Differential forms in global analysis |

58J53 | Isospectrality |

44A12 | Radon transform |