Abstract
We prove that there do not exist CR submanifolds Mn of maximal CR dimension of a complex projective space $${\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}$$ with flat normal connection D of M, when the distinguished normal vector field is parallel with respect to D. If D is lift-flat, then there exists a totally geodesic complex projective subspace $${\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}$$ of $${\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}$$ such that M is a real hypersurface of $${\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}$$ .
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