A local characterization of affine holomorphic immersions with an anti-complex and $\nabla $-parallel shape operator
Volume 78 / 2002
Annales Polonici Mathematici 78 (2002), 59-84
MSC: 53A15, 53B05, 53C56.
DOI: 10.4064/ap78-1-7
Abstract
We study the complex hypersurfaces $f:M^{(n)}\to {\mathbb C}^{n+1}$ which together with their transversal bundles have the property that around any point of $M$ there exists a local section of the transversal bundle inducing a $\nabla $-parallel anti-complex shape operator $S$. We give a class of examples of such hypersurfaces with an arbitrary rank of $S$ from 1 to $[{n/2}]$ and show that every such hypersurface with positive type number and $S\not =0$ is locally of this kind, modulo an affine isomorphism of ${\mathbb C}^{n+1}$.
