Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves
type: Item
instance of: scholarly article
title: Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves
zbMATH Open document ID: 1090.53030
author: Sadahiro Maeda
author: Hiromasa Tanabe
published in: Mathematische Zeitschrift
publication date: 2006-03-21T00:00:00Z
review text: A smooth curve \(\gamma= \gamma(s)\) in a complex \(n\)-dimensional Kähler manifold \((M,J)\) parametrized by its arc length \(s\) is called a Kähler Frenet curve, if it satisfies \(\nabla_{\dot\gamma}\dot\gamma= \kappa(s) J\dot\gamma\) or \(\nabla_{\dot\gamma}\dot\gamma= -\kappa(s)J\dot\gamma\). Let \(f\) be a Kähler isometric immersion of a Kähler manifold \(M_n\) into an arbitrary Kähler manifold \(\widetilde M_m\) (resp. into a real space form \(\widetilde M^{2n+p}(\widetilde c)\)). Assume that there exists a positive \(C^\infty\)-function \(\kappa= \kappa(s)\) such that \(f\) maps every Kähler Frenet curve \(\gamma= \gamma(s)\) of curvature \(\kappa\) on \(M_n\) to a Frenet curve of order 2 in \(\widetilde M_m\) (resp. by non-constant \(\kappa\) to a plane curve). It is proved that then \(f\) is a totally geodesic immersion. Finally, every parallel isometric immersion of a complex space form \(M_n(c)\) into a real space form \(\widetilde M^{2n+p}\) is characterized in terms of Kähler Frenet curves.
Mathematics Subject Classification ID: 53B35
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 5014031
zbMATH Keywords: Kähler manifolds
zbMATH Keywords: Kähler Frenet curves
zbMATH Keywords: totally geodesic immersions
zbMATH Keywords: parallel isometric immersions
zbMATH Keywords: real space forms
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
full work available at URL: s00209-005-0881-y
OpenAlex ID: W1969185917
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DOI: 10.1007/S00209-005-0881-Y
description: scientific article; zbMATH DE number 5014031