On some properties of the curvature and Ricci tensors in complex affine geometry
type: Item
instance of: scholarly article
title: On some properties of the curvature and Ricci tensors in complex affine geometry
zbMATH Open document ID: 0840.53009
DOI: 10.1007/BF01264926
author: Barbara Opozda
published in: Geometriae Dedicata
publication date: 1996-06-18T00:00:00Z
review text: Let \(f:(M^n, \nabla) \to \mathbb{C}^{n + 1}\) be a complex affine hypersurface immersion and let \(\xi\) be any transversal vector field along \(f\). Then the transversal complex plane bundle \(\mathcal N\) spanned by \(\xi\) and \(J \xi\) induces a complex affine connection \(\nabla\) on \(M^n\), in the sense that \(\nabla J = 0\), where \(J\) is the complex structure on \(M^n\). The author defines for an abstract complex manifold with complex affine connection a complex Ricci tensor ric, which, in case the connection is holomorphic, is the complex trace of the curvature tensor \(R\). Next, the author assumes the induced connection to be holomorphic, or affine Kähler, and proves some properties of ric. In the third section, it is proved that a nondegenerate complex affine hypersurface \(M^n\), equipped with a complex transversal bundle is semi-symmetric (i.e. \(R \cdot R = 0\)) if and only if ric is symmetric (if \(n = 2)\) or ric is proportional to the complex second fundamental form (if \(n = 2\)). If \(f\) is degenerate (i.e. the second fundamental form is degenerate), then the result is no longer true. In the next section, it is proved that a nondegenerate complex affine hypersurface \(M^n\) is locally symmetric if and only if \(M^n\) is a quadric or flat. If \(f\) is degenerate, then the author proves a cylindrical decomposition theorem. The final section contains some instructive examples.
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53C56
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 806225
zbMATH Keywords: complex affine hypersurfaces
zbMATH Keywords: complex Ricci tensor
zbMATH Keywords: semi-symmetric
zbMATH Keywords: locally symmetric
reviewed by: Franki Dillen
MaRDI profile type: Publication
cites work: Q3831592
cites work: Q3210472
cites work: Q5565587
cites work: On affine hypersurfaces with parallel nullity
cites work: Q4032066
cites work: On the geometry of affine Kähler immersions
cites work: Equivalence theorems for complex affine hypersurfaces
cites work: Q4296709
cites work: Hypersurface with parallel Ricci tensor in a space of constant holomorphic sectional curvature
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description: scientific article; zbMATH DE number 806225