Submanifolds of Kaehlerian manifolds and metric compound structures
\textit{D. E. Blair}, \textit{G. D. Ludden} and \textit{K. Yano} [Kodai Math. Semin. Rep. 22, 188-198 (1970; Zbl 0202.209)] have induced an (f,g,u,v,\(\lambda)\)-structure on certain submanifolds of codimension 2 in almost Hermitian manifolds or on certain hypersurfaces in almost contact metric manifolds. Here f is a (1.1) tensor, g is the metric tensor, u,v are vector fields and \(\lambda\) is a function on the submanifolds. \textit{Y. Tashiro} and the present author [Kodai Math. J. 5, 13-29 (1982; Zbl 0482.53025)] introduced the notion of metric compound structure \((f,g,v,f^{\perp})\) on a manifold M where v is a matrix of a constant rank \(\gamma\), of which the column vectors are vector fields on M. The present author considers some scalar fields which are associated with a metric compound structure of rank \(\gamma\) and these scalar fields are used to classify invariant, anti-invariant, CR and other submanifolds in almost Hermitian manifolds. For \(\gamma =2\) there is a distribution \(D^ 2\) spanned by two independent vector fields and the metric compound structure leads to a structure called an \((f,g,D^ 2,\lambda)\)- structure, which is equivalent to an (f,g,u,v,\(\lambda)\) structure if \(\lambda \neq \pm 1\). Many geometric structures of Riemannian manifolds and submanifolds of Kaehlerian manifolds with \((f,g,D^ 2,\lambda)\)- structure satisfying some additional properties are investigated. For instance some sufficient conditions are given for M being a space of constant curvature.
scientific article; zbMATH DE number 3861952
scientific article