Equivalence theorems for complex affine hypersurfaces
type: Item
instance of: scholarly article
title: Equivalence theorems for complex affine hypersurfaces
zbMATH Open document ID: 0840.53010
DOI: 10.1007/BF03322835
author: Barbara Opozda
published in: Results in Mathematics
publication date: 1996-06-18T00:00:00Z
review text: The author studies under which conditions two complex affine hypersurface immersions \(f_1\), \(f_2 : (M^n, \nabla) \to \mathbb{C}^{n + 1}\) are affine equivalent, i.e. when does there exist a complex affine transformation \(A\) of \(\mathbb{C}^{n + 1}\) such that \(Af_1 = f_2\). These conditions are in terms of the affine connection and the complex-valued, complex-linear second fundamental form induced by the choice of any transversal vector field \(\xi_1\) resp. \(\xi_2\) fields being holomorphic or anti-holomorphic. Next she applies her results to the case when the induced affine shape operators are anti-complex. The key result here is that, if the induced affine shape operators are anti-complex, if the induced affine connections coincide, and if the curvature tensor of the induced affine connection is non zero at some point, then the two immersions are equivalent. As an application, she recovers Calabi's theorem stating that two Kähler hypersurfaces of \(\mathbb{C}^{n + 1}\) are unitarily equivalent provided their induced metrics coincide.
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 785219
zbMATH Keywords: complex affine hypersurfaces
zbMATH Keywords: affine equivalence
reviewed by: Franki Dillen
MaRDI profile type: Publication
cites work: Q3210472
cites work: Q3831592
cites work: On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry
cites work: Q5731354
cites work: On the geometry of affine Kähler immersions
cites work: Some equivalence theorems in affine hypersurface theory
description: scientific article; zbMATH DE number 785219