We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Q^m = SO(m+2)/SO(m)SO(2), m ≥ 3. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space ℂP^k which is embedded canonically in Q^{2k} as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q^{2k+1}, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow.
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