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We show the following theorem: Let  X  be an irreducible compact Hermitian symmetric manifold of complex dimension  n  whose bisectional curvature is ( s  − 1)-nondegenerate. Then in  X  there exists no smooth Levi-flat  CR  manifold  M  with real co-dimension  n – s  and  CR  dimension  s  ≧ 2, such that the determinant ℂ-line bundle ofis smoothly trivial.

Nonexistence of higher codimensional Levi-flat CR manifolds in symmetric spaces