On vertical skew symmetric almost contact 3-structures (Q2573757)

The authors consider a \((2m+3)\)-dimensional Riemannian manifold \(M(g, \xi _r,\eta ^r)\) endowed with three Reeb vector fields \(\xi _r, r= 2m+1, 2m+2, 2m+3.\) Then the tangent space \(T_pM\) for all \(p\in M\), decomposes as \(T_pM = H_p\oplus + H_p^{\perp} \) where \(H_p^{\perp} \) is the distribution spanned by (\(\xi _r\)), called the vertical distribution. For \(\xi = \sum \lambda _r \xi_r \in H_p^{\perp}\), they assume that the vertical connection forms \(\theta ^r_s\) have the form \(\theta ^r_s = \lambda _s\eta ^r - \lambda _r\eta ^s\) and say that \(M(g, \xi _r,\eta ^r)\) is endowed with a vertical skew symmetric almost contact 3-structure. Under these hypotheses they prove that \(M(g, \xi _r,\eta ^r)\) is foliated by 3-dimensional submanifolds \(M^{\perp}\) of constant curvature, tangent to the vertical distribution and \(\| \xi \| ^2 \) is an isoparametric function. Next, the authors assume that \(M(g, \xi _r,\eta ^r)\) is equipped with a framed f-structure \(\varphi\) and prove that the 2-form \(\Omega \) associated to it is closed and \(M(\varphi \Omega, g, \xi _r,\eta ^r)\) is a CR-manifold that may be viewed as the local Riemannian product \(M = M^T \otimes M^{\perp} \) such that i) \(M^T\) is a Kählerian submanifold tangent to the horizontal distribution \(H\) of \(M\) and is totally geodesic immersed in \(M\) and ii) \(M^{\perp}\) is a 3-dimensional manifold of constant curvature tangent to the vertical distribution \(H^{\perp}\) and totally geodesic immersed in \(M\). When \(M(g, \xi _r,\eta ^r)\) is not compact, a set of local Hamiltonian vector fields for the presymplectic 2-form \(\Omega\) is found.