On the flatness of a class of metric \(f\)-manifolds
Let \(M\) be a \((2n+s)\)-dimensional manifold with an \(f.pk\)-structure, i.e., \(M\) has an \(f\)-structure \(\varphi\) (\(\varphi^3+\varphi=0\)) with parallelizable kernel. In particular there exist vector fields \(\xi_1,\dots,\xi_s\), 1-forms \(\eta^1,\dots,\eta^s\) such that \(\varphi\xi_i=0\), \(\eta^i\circ\varphi=0\), \(\varphi^2=-I+\sum_{j=1}^s\eta^j\otimes\xi_j\), \(\eta^i(\xi_j)=\delta^i_j\). Let \(g\) be a compatible metric, i.e. \(g(X,Y)=g(\varphi X,\varphi Y)+\sum_{j=1}^s\eta^j(X)\eta^j(Y)\); the resulting structure then is called a metric \(f.pk\)-structure. Define a 2-form \(F\) by \(F(X,Y)=g(X,\varphi Y)\). If \(F=d\eta^1=\cdots=d\eta^s\), the metric \(f.pk\)-structure is called an almost \(\mathcal S\)-structure. If \(s=1\), this is a contact metric structure (cf. the reviewer's book [Riemannian geometry of contact and symplectic manifolds, Birkhäuser (2002; Zbl 1011.53001)]). The notion of normality is not needed here, so we simply remark that an \(\mathcal S\)-\textit{structure} is a normal almost \(\mathcal S\)-structure. In [Tôhoku Math. J. 28, 373--379 (1976; Zbl 0364.53013)] the reviewer proved in dimensions \(2n+1\geq 5\), a contact metric manifold cannot be flat. \({\mathbb R}^3\) does carry a flat contact metric structure. In the present paper the authors generalize this result to almost \(\mathcal S\)-manifolds. They prove that if \(M^{2n+s}\) is a flat almost \(\mathcal S\)-manifold, then \(n=1\). The authors close their paper by constructing a flat almost \(\mathcal S\)-manifold of dimension \(2+s\) with \(s\) arbitrary.
scientific article; zbMATH DE number 2031037