The rigidity for real hypersurfaces in a complex projective space
type: Item
instance of: scholarly article
title: The rigidity for real hypersurfaces in a complex projective space
zbMATH Open document ID: 0931.53025
author: In Bae Kim
author: Byung Hak Kim
author: Ryoichi Takagi
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2000-02-23T00:00:00Z
review text: Let \(M\) be a \((2n-1)\)-dimensional connected Riemannian manifold and \(\iota, \hat\iota\) two isometric immersions from \(M\) into the \(n\)-dimensional complex projective space \(P_n(\mathbb{C})\), \(n\geq 4\). The authors first prove that the two structure vector fields on \(M\) induced by these immersions coincide up to sign. Combining this with a previous result by the authors they conclude that if the type number of \(\iota\) or \(\hat\iota\) is not equal to 2 at every point of \(M\), then there exists an isometry \(\phi\) of \(P_n(\mathbb{C})\) such that \(\phi\circ \iota= \hat\iota\). As an application, they obtain that, if \(M\) is a \((2n-1)\)-dimensional connected homogeneous Riemannian manifold and if it admits an isometric immersion \(\iota\) into \(P_n(\mathbb{C})\), then \(\iota(M)\) is extrinsically homogeneous. The analogous statements for complex hyperbolic space are also valid.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1275224
zbMATH Keywords: complex projective space
zbMATH Keywords: real hypersurfaces
zbMATH Keywords: rigidity
zbMATH Keywords: type number
zbMATH Keywords: homogeneous submanifolds
zbMATH Keywords: complex hyperbolic space
reviewed by: Jürgen Berndt
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178224896
OpenAlex ID: W1968991478
cites work: Rigidity theorems for real hypersurfaces in a complex projective space
cites work: The type number of real hypersurfaces in \(P_n(\mathbb{C})\)
cites work: On geodesic hyperspheres in a complex projective space
cites work: Q4954887
cites work: A rigidity for real hypersurfaces in a complex projective space
cites work: On homogeneous real hypersurfaces in a complex projective space
cites work: Real hypersurfaces in a complex projective space with constant principal curvatures. II
DOI: 10.2748/TMJ/1178224896
description: scientific article; zbMATH DE number 1275224