A characterization of constant isotropic immersions by circles
type: Item
instance of: scholarly article
title: A characterization of constant isotropic immersions by circles
zbMATH Open document ID: 1048.53042
author: Sadahiro Maeda
published in: Archiv der Mathematik
publication date: 2004-03-07T00:00:00Z
review text: Motivated by the well-known result of Nomizu and Yano on circles and spheres in Riemannian geometry, the author provides a characterization of constant isotropic immersions into an arbitrary Riemannian manifold by circles on the submanifolds. Let \(M^n\) be an \(n\)-dimensional connected Riemannian submanifold of an \((n + p)\)-dimensional Riemannian manifold \(\widetilde M^{n+p}\) through an isometric immersion \(f\). Then the following are equivalent: (i) \(M^n\) is a constant (\(\lambda\)-) isotropic submanifold of \(\widetilde M^{n+p}\). (ii) There exists \(\kappa > 0\) such that for each circle \(\gamma\) of curvature \(\kappa\) on the submanifold \(M^n\) the curve \(f\circ\gamma\) in \(\widetilde M^{n+p}\) has constant first curvature \(\kappa_1\) along this curve. As an immediate consequence of this result, the author characterizes Veronese imbeddings of complex projective spaces into complex projective spaces which are typical examples of Kähler immersions. Let \(f : M_n\to M_N(c)\) be a Kähler isometric full immersion of an \(n\)-dimensional Kähler manifold \(M_n\) into an \(N\)-dimensional complex space form \(M_N(c)\) of constant holomorphic sectional curvature \(c > 0\). Then the following conditions are equivalent. (i) For some positive integer \(k\), the submanifold \(M_n\) is locally congruent to \(\mathbb CP^n( c/k)\), \(N=\binom{n+k}{n}-1\) and \(f\) is locally equivalent to the \(k\)-th Veronese imbedding \(f_k\). (ii) There exists \(\kappa > 0\) such that for each circle \(\gamma\) of curvature \(\kappa\) on the submanifold \(M_n\) the curve \(f\circ\gamma\) in \(M_N(c)\) has constant first curvature \(\kappa_1\) along this curve.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 2051580
zbMATH Keywords: constant isotropic immersions
zbMATH Keywords: circles on submanifolds
zbMATH Keywords: ambient Riemannian manifold
zbMATH Keywords: extrinsic shape
zbMATH Keywords: Veronese imbeddings of complex projective spaces
zbMATH Keywords: Kähler isometric full immersion
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
full work available at URL: s00013-003-4677-1
OpenAlex ID: W2019204407
DOI: 10.1007/S00013-003-4677-1
description: scientific article; zbMATH DE number 2051580