On \(K\)-left invariant almost contact 3-structures (Q1895224)

Let \(M_{2m + 3}\) be a Riemannian manifold carrying 3 vector fields \(\xi_r\) (\(r \in \{2m + 1,2m + 2, 2m + 3\})\) whose covariant derivatives \(\nabla \xi_r\) satisfy \(\nabla \xi_r = \xi_s \wedge \xi_t\), where \(\wedge\) denotes the wedge product and \((r,s,t)\) is a cyclic order of \((2m + 1, 2m + 2, 2m + 3)\). Such vector fields define a perfect group of left invariant skew symmetric Killing vector fields. If \(\eta^r(X) = g(X,\xi_r)\), then \(M_{2m + 3}\) with the structure \((\eta^r, \xi_r, g)\) is called a \(K\)-left invariant almost 3- contact manifold. If \(M_{2m + 3}\) is endowed with an \(f\)-structure \(\phi\), that is \(\phi\) is a (1,1)-tensor satisfying \(\phi^3 + \phi = 0\), then \(M_{2m + 3}\) is called a framed \(f\)-manifold. Among other interesting things, the author proves that any framed \(f\)-manifold \(M_{2m + 3}\) endowed with a \(K\)-left invariant almost contact 3- structure is a framed \(f\)-CR-manifold and may be viewed as the local product \(M_{2m + 3} = M \times M^\perp\) of two totally geodesic submanifolds such that \(M\) is a \(2n\)-dimensional Kähler manifold tangent to the horizontal distribution of \(M_{2m + 3}\) and \(M^\perp\) is a 3-dimensional manifold of constant curvature 1, tangent to the vertical distribution of \(M_{2m + 3}\), and any invariant submanifold \(\overline {M}\) of \(M_{2m +3}\) is minimal.