Real hypersurfaces of a complex projective space in terms of the Jacobi operators
type: Item
instance of: scholarly article
title: Real hypersurfaces of a complex projective space in terms of the Jacobi operators
zbMATH Open document ID: 0910.53038
DOI: 10.1023/A:1006585128386
author: Jong Taek Cho
author: U-Hang Ki
published in: Acta Mathematica Hungarica
publication date: 1999-03-14T00:00:00Z
review text: Let \(M\) be a connected oriented real hypersurface of the complex projective space \((\mathbb{C} P^m,J,\widetilde g)\) with Fubini-Study metric \(\widetilde g\) of constant holomorphic sectional curvature 4, and let \((\phi,\zeta,\xi,g)\) be an almost contact metric structure on \(M\) induced by \((J,\widetilde g)\). Let \(A\) denote the shape operator of \(M\) in \(\mathbb{C} P^n\) and \(\mathcal R_\xi\) the Jacobi operator with respect to the structure vector field \(\xi\). It is proved that if \(\mathcal R_\xi\phi A=A\phi\mathcal R_\xi\), where \(\phi\) is the structure tensor, then \(M\) is locally congruent to a geodesic hypersurface or a tube of radius \(r\) over a totally geodesic \(\mathbb{C} P^k\), \(1\leq k \leq n-2\), \(0<r<\frac\pi 2\). A similar result is derived from the conditions \(\mathcal R_\xi\phi=\phi\mathcal R_\xi\) and \(\mathcal R_\xi A=A\mathcal R_\xi\).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 1261212
zbMATH Keywords: real hypersurface
zbMATH Keywords: complex projective space
zbMATH Keywords: Jacobi operator
reviewed by: Lajos Tamássy
MaRDI profile type: Publication
full work available at URL: a:1006585128386
OpenAlex ID: W1710088861
description: scientific article; zbMATH DE number 1261212