Abstract: Let (X,g) be an irreducible symmetric space of compact type. According to a result of Guillemin, the infinitesimal deformation corresponding to an isospectral deformation of the metric g belongs to the kernel of a certain Radon transform acting on the symmetric 2-forms on X. This is the motivation for defining the space I(X) of infinitesimal isospectral deformations of X as a subspace of the kernel of this Radon transform. If I(X) vanishes, an isospectral deformation of the metric g is trivial to first-order.
We shall give an overview of our results concerning the space I(X):
1) A necessary condition for the vanishing of I(X) is that it be reduced, i.e.,~it is not the cover of another symmetric space.
2) It was known that the space I(X) vanishes when X is a projective space which is not equal a sphere (using work of Duistermaat--Guillemin, this leads to spectral rigidity results for these projective spaces). We generalize this result by showing that the space I(X) vanishes when X is a Grassmannian which is reduced.
3) If X is the reduced space of the symmetric space SU(n)/SO(n) or of the unitary group SU(n), with n\ge3, the space I(X) does not vanish and we give explicit constructions of non-trivial infinitesimal deformations.
This talk will be accessible to graduate students.