Isotropic totally real submanifolds
type: Item
instance of: scholarly article
title: Isotropic totally real submanifolds
zbMATH Open document ID: 0677.53064
DOI: 10.1007/BF01160209
author: Francisco Urbano
author: Sebastián Montiel
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173950
review text: The authors study n-dimensional totally real isotropic submanifolds of a complex manifold. A submanifold of a Riemannian manifold is called isotropic [\textit{B. O'Neill}, Can. J. Math. 17, 907-915 (1965; Zbl 0171.205)] if \(\| h(v,v)\|^ 2=\lambda (p),\) where h denotes the second fundamental form, is independent of the unit tangent vector v at the point p. If \(\lambda\) is also independent of the point p, then the submanifold is called a constant isotropic submanifold. The main results are the following. (i) Let \(M^ n\) (n\(\geq 3)\) be a minimal, totally real submanifold isometrically immersed in a Kaehler manifold \(\tilde M^ n\). If \(M^ n\) is isotropic, then either \(M^ n\) is totally geodesic or \(n=5,8,14\) or 26. (ii) Every n-dimensional, constant isotropic, totally real submanifold of a complex space form, with constant holomorphic sectional curvature c, \(\tilde M^ n(c)\), where \(c\leq 0\), is totally geodesic. (iii) The authors give a complete classification of complete, constant isotropic, totally real submanifolds of \({\mathbb{C}}P^ n(c)\).
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 4109568
zbMATH Keywords: minimal submanifold
zbMATH Keywords: totally geodesic submanifold
zbMATH Keywords: totally real isotropic submanifolds
zbMATH Keywords: second fundamental form
MaRDI profile type: Publication
cites work: Isotropic submanifolds with parallel second fundamental form in \(P^ m(\)c)
cites work: Isotropic and Kähler Immersions
cites work: Planar geodesic immersions
Wikidata QID: Q125360418
OpenAlex ID: W1991642501
description: scientific article; zbMATH DE number 4109568