Section 2. Geometry and topology of manifolds. Generalized almost Hermitian geometry on manifolds with \(f\)-structures (Q2799767)

A generalized almost Hermitian geometry (GAH-structure) of rank \(r\) on a manifold \(M\) is a system \(\{g,J_1,\ldots,J_r,T\}\) of tensors satisfying certain conditions. Here, \(g\) is a pseudo-Riemannian metric, \(\{J_1,\dots,J_r\}\) are linear independent \((1,1)\)-tensor fields, mutually commuting, and \(T\) is a tensor field of type \((2,1)\), the `composition tensor'. GAH-structures were studied by \textit{V. F. Kirichenko} [J. Sov. Math. 42, No. 5, 1885--1919 (1988; Zbl 0715.53033); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 18, 25--71 (1986)]. Various subclasses of GAH-structures are obtained by imposing additional conditions, so for instance generalized Hermitian (GH-structures, \(T(X,Y)=0\)) for all vector fields on \(M\), also generalized \(G_1\) resp. \(G_2\)-manifolds, generalized quasi-Kähler or generalized nearly-Kähler) structures.NEWLINENEWLINENEWLINEAn \(f\)-structure on \(M\) is a \((1,1)\)-tensor field \(f\) such that \(f^3+f=0\). The author gives a survey on developments and results related to GAH-structures and extends some of his own results. He describes a revised version of a method given in [``A natural construction of a generalized Hermitian structure'', Isv. Akad. Nauk SSSR (2), 114--119 (2001)] for the construction of GAH-structures on a manifold \((M,g)\) from \(r\) metric \(f\)-structures \(f_1,\dots,f_r\). If \(f_i f_j=0\) for \(i\), \(j=1,\dots,r\) then an appropriate \((2,1)\)-tensor \(T\) is given in terms of the Nijenhuis tensors of the \(f_i\).NEWLINENEWLINENEWLINEA \(\Phi\)-space is a homogeneous space \(G/H\), of a Lie group \(G\) endowed with an automorphism \(\Phi\) such that \(G^\Phi_0\subset H\subset G^\Phi\), where \(G^\Phi,G^\Phi_0\) are the subgroup of fixed points of \(\Phi\) and its connected component of the identity elements, respectively. Based on results of \textit{V. V. Balashchenko} and \textit{A. S. Samsonov} [Dokl. Math. 81, No. 3, 386--389 (2010; Zbl 1209.53021); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 432, No. 3, 295--298 (2010)] and \textit{V. V. Balashchenko} and \textit{N. A. Stepanov} [Sb. Math. 186, No. 11, 1551--1580 (1995; Zbl 0872.53025); translation from Mat. Sb. 186, No. 11, 3--34 (1995)], conditions are given such that an invariant GAH-structure of rank \(r\) on a naturally reductive homogeneous \(\Phi\)-space of order \(k\) is a GH-structure.NEWLINENEWLINEFor the entire collection see [Zbl 1298.53003].