Some remarks on generalized Grassmann manifolds
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 \(\dim M'\leq \dim M-4.\)
scientific article; zbMATH DE number 4136909