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Radon transforms and spectral rigidity on the complex quadrics and the real Grassmannians of rank two. (English) Zbl 0861.53054
We prove that the real Grassmannians of (unoriented) 2-planes in \(\mathbb{R}^{n+2}\), with \(n\geq 3\), are rigid in the sense of Guillemin. This result implies that these spaces are infinitesimally spectrally rigid and provides us with the first examples of symmetric spaces of compact type and \(\text{rank}>1\) having this property. It is proved by studying the maximal flat Radon transform for symmetric 2-forms on these symmetric spaces and on the complex quadrics.

MSC:
53C35 Differential geometry of symmetric spaces
44A12 Radon transform
53C65 Integral geometry
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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