Holomorphic differential invariants for an ellipsoidal real hypersurface
The author computes the fourth-order curvature tensor \(S\) for an ellipsoidal real hypersurface \(M=\{z\in\mathbb C^n\mid r(z,\overline z)=0\}\), where \(r(z,\zeta)=\sum_{j=1}^n{z_j\zeta_j+A_j(z_j^2+\zeta_j^2)}-1\). It is called generic when \(0<A_1<\cdots <A_n<\frac{1}{2}\). He uses the method of complexification. He shows that for a generic ellipsoidal real hypersurfaces \(M\) of \(\mathbb C^n\), with \(n\geq 3\), the tensor \(S\) is different from zero at every point of \(M\). Using the holomorphically invariant principal contact form \(\Theta\) with \(\|S\|_{\Theta}=1\), he defines the principal characteristic vector field \(V\) by \(\iota_V\Theta=1\), \(\iota_Vd\Theta=0\). He proves that on a generic ellipsoidal real hypersurfaces in \(\mathbb C^n\), for \(n\geq 3\), the flow of the principal characteristic vector is completely integrable (in a suitable sense).
scientific article; zbMATH DE number 1529066