I might be totally confused, but in this paper of Hubert Goldschmidt (Infinitesimal isospectral deformations of symmetric spaces), he seems to construct the very deformations of the title. It is not obvious that these can be integrated, but if they can be, it gives you a family of manifolds isospectral to the symmetric space spaces and not symmetric, which would provide a negative answer to the OP (note that if you read Goldschmidt's paper, you will see that various space spaces are not NOT to be so deformable).
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I might be totally confused, but in this paper of Hubert Goldschmidt (Infinitesimal isospectral deformations of symmetric spaces), he seems to construct the very deformations of the title. It is not obvious that these can be integrated, but if they can be, it gives you a family of manifolds isospectral to the symmetric space and not symmetric, which would provide a negative answer to the OP (note that if you read Goldschmidt's paper, you will see that various space are not NOT to be so deformable). |
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