Dedicated to Professor Tongde Zhong on the occasion of his 80th birthday * Supported in part by NSF-0801056. the Chern-Moser-Weyl curvature tensor and the mappings to be involved are CR mappings. Unfortunately, there is no monotonicity phenomenon in general. Our crucial observation is that the monotonicity exists along directions in the null space of the Levi-form. Since the null space of the Levi-from may be regarded as the 'largest' holomoprhic subset inside T (1,0) M , our result may be

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... dered as a generalization of those results on complex manifolds. Unfortunately, in our investigation, we have to exclude the important strongly pseudoconvex case: ℓ = 0; for the null space of the Levi-form in this setting is the 0-space. Since the hyperquarics have vanishing Chern-Moser-Weyl tensor, our criterion makes it possible to construct many algebraic Levi non-degenerate hypersurfaces which can not be embedded into a hyperquadric of the same signature ℓ > 0 in a complex space of higher dimension. However, it still remains to be an open question to answer if any algebraic strongly pseudoconvex hypersurface M ℓ can be embedded into H N ℓ for some N with ℓ = 0.