## On the Chern-Moser-Weyl tensor of real hypersurfaces.(English)Zbl 1462.32051

The authors give an explicit concise formula for the Chern-Moser-Weyl tensor of nondegenerate real hypersurfaces in terms of the defining function. As an application, they also show that the CR-invariant one-form $$X_\alpha$$ is non-trivial on real ellipsoids of revolution $$E(a)$$ in $$\mathbb C^3$$ defined by $$\rho (z_1,z_2,w)= |z_1|^2+ |z_2|^2+ |w|^2 + \operatorname{Re} (aw^2)-1=0, \ a\in \mathbb R$$ unless $$a=0.$$ In this way they resolve affirmatively a question posed by Case and Gover and provide a counterexample to a recent conjecture by Hirachi. In the last section of the paper they provide a family of locally equivalent nonspherical CR manifolds with parallel Chern-Moser-Weyl tensor and show that the hypersurfaces in this family are pairwise inequivalent globally.

### MSC:

 32V40 Real submanifolds in complex manifolds 53B25 Local submanifolds

### Keywords:

Chern-Moser-Weyl tensor; pseudohermitian invariant
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### References:

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