\((1,1)\)-geodesic maps into Grassmann manifolds
type: Item
instance of: scholarly article
title: \((1,1)\)-geodesic maps into Grassmann manifolds
zbMATH Open document ID: 0863.53046
DOI: 10.1007/BF02572619
author: J.-H. Eschenburg
author: R. Tribuzy
published in: Mathematische Zeitschrift
publication date: 1997-05-28T00:00:00Z
full work available at URL: https://eudml.org/doc/174814
review text: Let \(M\) be a Kähler manifold, \(D\) be the Levi-Civita connection extended complex-linearly to the complexification of the tangent bundle, \(N\) be a Riemannian manifold, and \(\varphi:M\to N\) be a smooth map. The Hessian \(Dd\varphi\) may be decomposed to \((2,0)\), \((1,1)\), \((0,2)\) parts. The map \(\varphi\) is called pluriharmonic if its \((1,1)\) part is zero (the definition does not depend on the choice of the metric). A pluriharmonic isometric immersion is called \((1,1)\)-geodesic. If \(N\) is also Kähler then any holomorphic or antiholomorphic immersion is \((1,1)\)-geodesic. By [\textit{S. Udagawa}, Proc. Lond. Math. Soc., III. Ser. 57, No. 3, 577-598 (1988; Zbl 0667.53046)], if the complex dimension of \(M\) exceeds some constant depending only on \(N\) then any pluriharmonic immersion is either holomorphic or antiholomorphic. The authors construct examples of \((1,1)\)-geodesic immersions to complex Grassmann manifolds which are either holomorphic or antiholomorphic, in particular in the limit dimension for \(Gr(2,2+p)\). They are of the type \((\xi+\eta)^\perp\), where \(\xi\) is a holomorphic immersion and \(\eta\) is an antiholomorphic map to suitable Grassmannians. For real Grassmannians, it is proved that the normal Gauss map of a minimally immersed Kähler manifold or an extrinsic Hermitian symmetric space in an Euclidean space is \((1,1)\)-geodesic.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53B21
zbMATH DE Number: 856569
zbMATH Keywords: totally geodesic
zbMATH Keywords: pluriharmonic immersions
zbMATH Keywords: Grassmann manifolds
reviewed by: Victor Gichev
MaRDI profile type: Publication
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OpenAlex ID: W2009671605
description: scientific article; zbMATH DE number 856569