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We classify real hypersurfaces with isometric Reeb flow in the complexquadrics Q^m for m > 2. We show that m is even, say m = 2k, and any suchhypersurface is an open part of a tube around a k-dimensional complexprojective space CP^k which is embedded canonically in Q^{2k} as a totallygeodesic complex submanifold. As a consequence we get the non-existence of realhypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics.

Real hypersurfaces with isometric Reeb flow in complex quadrics