On \(2\)-almost contact Tachibana manifolds
A Tachibana manifold is a \((2n+2)\)-dimensional Riemannian manifold equipped with \(2\) structure vector fields \(\xi _{2m+1}\), \(\xi _{2m+2}\), such that the \(2\)-form \(\varphi =\eta ^{2m+1}\wedge \eta ^{2m+2}\) is harmonic, \(\eta ^r\) being the \(1\)-form dual to \(\xi _r\), \(r+2m+1,2m+2\) [\textit{S. Tachibana}, Tensor New Ser. 27, 123-130 (1973; Zbl 0249.53035)]. In the paper under review, the authors study \(2\)-almost contact Tachibana manifolds, i.e., Tachibana manifolds where \(\xi_r\) defines a \(2\)-almost contact structure and there exists a horizontal vector field \(A\) such that \(\mathcal{L} _A \eta ^r =\|A\|^2\eta ^r\). It is proved that the \(1\)-form \(\alpha\) of \(A\) satisfies \(\mathcal{L} _A \alpha =4l \alpha\) and \(\Delta \alpha =-4lm\alpha\). In particular, since \(-4ml <0\), \(M\) cannot be compact. It is also proved that \(\xi _{2m+1}\) and \(\xi _{2m+2}\) commute and define infinitesimal transformations of \(A\). Moreover, all horizontal vector fields orthogonal to \(A\) are skew-symmetric Killing vector fields. The authors also study infinitesimal automorphisms of the symplectic form \(\Omega _{\varphi}\) of \(M\), when \(M\) is equipped with an \(f\)-structure [\textit{K. Yano} and \textit{M. Kon}, Structures on Manifolds, World Scientific, Singapore (1984; Zbl 0557.53001)].
scientific article; zbMATH DE number 1846805
scientific article