A local characterization of affine holomorphic immersions with an anti-complex and $\nabla $-parallel shape operator

Maria Robaszewska¹

¹ Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Annales Polonici Mathematici, Tome 78 (2002) no. 1, pp. 59-84.

Résumé

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 ${\mathbb C}^{n+1}$.

Détail

DOI : 10.4064/ap78-1-7

Keywords: study complex hypersurfaces mathbb which together their transversal bundles have property around point there exists local section transversal bundle inducing nabla parallel anti complex shape operator class examples hypersurfaces arbitrary rank every hypersurface positive type number locally kind modulo affine isomorphism mathbb