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## Infinitesimal isospectral deformations of symmetric spaces

### Hubert Goldschmidt

Columbia University

###
Thursday, November 3, 2005

L01 Carson Hall, 4 pm

Tea 3:30 pm, Math Lounge

**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.