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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
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References:

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