Some remarks on generalized Grassmann manifolds
type: Item
instance of: scholarly article
title: Some remarks on generalized Grassmann manifolds
zbMATH Open document ID: 0694.53028
DOI: 10.1007/BF01950711
author: Oleg Mushkarov
published in: Acta Mathematica Hungarica
publication date: 1989-01-01T00:00:00Z
review text: Let M be an almost Hermitian manifold with an almost Hermitian structure (g,J). This structure is called a \(G_ 1\)-structure if we have \[ g(N(X,Y),Z)=g(N(Y,Z),X) \] for all X,Y,Z\(\in {\mathcal X}(M)\) where N is the Nijenhuis tensor on M. The generalized Grassmann manifold in the form of a homogeneous space \(G(n_ 1,...,n_ k)=U(n)/U(n_ 1)\times...\times U(n_ k),\) where U(n) is the unitary group of order n and \(n_ 1\leq n_ 2\leq...\leq n_ k\) are positive integers with sum n is studied. A similar family of almost Hermitian structures on \(G(n_ 1,...,n_ k)\) for \(k\geq 3\) is defined and considered. Such almost Hermitian manifolds have a lot of almost Hermitian submanifolds and are not nearly Kähler manifolds. These two properties give the possibility to construct examples which show that in certain dimensions the estimates given in the following theorems are sharp. Theorem A. Let \(M'\) be a Hermitian submanifold of a \(G_ 1\)-space M through a point \(x\in M\). Then \[ \dim M'\leq \dim M+1- \sqrt{1+4 rank_ xN} \] where N is the Nijenhuis tensor of M. Theorem B. Let \(M'\) be an almost Hermitian submanifold of a strictly \(G_ 1\)-space M. Then \(\dim M'\leq \dim M-4.\)
Mathematics Subject Classification ID: 53C10
Mathematics Subject Classification ID: 53B35
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 4136909
zbMATH Keywords: generalized Grassmann manifold
zbMATH Keywords: almost Hermitian structures
zbMATH Keywords: Nijenhuis tensor
zbMATH Keywords: almost Hermitian submanifold
reviewed by: A. Fleischer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4136909