On the structure and symmetry properties of almost \(\mathcal S\)-manifolds
Riemannian manifolds with structure group \(U(k) \times O(s)\) (and, in particular, \(I_s\)) have been studied by D. Blair and gave rise to the consideration of almost \(S\)-manifolds and \(S\)-manifolds. Their geometries generalize that of contact metric and Sasakian manifolds, respectively, which are obtained by putting \(s=1\). The (almost) \(S\)-manifolds have been studied by using the theory of \(f\)-structures and it turned out that their geometry may have features which are quite different from these for the case \(s=1\). These classes of manifolds may also be studied by focussing on the underlying CR-structure and this is the method used by the authors to give some more examples of differences that may appear when \(s \neq 1\). First, they prove that in contrast to the case \(s=1\), a simply connected \(S\)-manifold with \(s > 1\) is non-compact. (\(S^{2k+1}(1)\) is a counterexample for the case \(s=1\) and \(S^{2k+1}(1) \times T^{(s)}\) shows that simply connectedness is needed.) This result follows at once from the following classification theorem which the authors also prove here: A complete, simply connected \(S\)-manifold is a CR-product \(N \times \mathbb R^{s-1}\) with \(N\) a Sasakian manifold and with suitable product metric. Finally, they also extend the notions of locally and globally (strongly) \(\varphi\)-symmetric spaces for Sasakian and contact metric manifolds to the case \(s > 1\) using CR-symmetries and by introducing a Tanaka-Webster connection \(\widetilde \nabla\) for \(s > 1\). They prove that the \(S\)-manifold is locally CR-symmetric if and only if \(\widetilde \nabla \widetilde R = 0\). By providing a counterexample, they show that this characterization cannot be extended to the class of almost \(S\)-manifolds. Their example is a contact metric \((k,\mu)\)-space which is locally \(\varphi\)-symmetric but \(\widetilde \nabla \widetilde R \neq 0\).
scientific article; zbMATH DE number 2165500
scientific article