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We study classifying problems for real hypersurfaces in a complex two-plane Grassmannian  G  2 (ℂ  m +2 ). In relation to the generalized Tanaka–Webster connection, we consider a new concept of parallel normal Jacobi operator for real hypersurfaces in  G  2 (ℂ  m +2 ) and prove that a real hypersurface in  G  2 (ℂ  m +2 ) with generalized Tanaka–Webster  ⊥ -parallel normal Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍ P n   in  G  2 (ℂ  m +2 ), where  m  = 2 n .

𝔇 ⊥ -parallel normal Jacobi operators for Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster connection