We study conditions for which the mapping torus of a 6-manifold endowed with an $SU(3)$-structure is a locally conformal calibrated $G_2$-manifold, that is, a 7-manifold endowed with a $G_2$-structure $varphi$ such that $d varphi = - theta wedge varp
hi$ for a closed non-vanishing 1-form $theta$. Moreover, we show that if $(M, varphi)$ is a compact locally conformal calibrated $G_2$-manifold with $mathcal{L}_{theta^{#}} varphi =0$, where ${theta^{#}}$ is the dual of $theta$ with respect to the Riemannian metric $g_{varphi}$ induced by $varphi$, then $M$ is a fiber bundle over $S^1$ with a coupled $SU(3)$-manifold as fiber.
We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost Kahler manifolds. We give an explicit non-compact example of an Einstein almost cokahler manifold that is not cokahler. We prove that compact Einstein almost
cokahler manifolds with non-negative $*$-scalar curvature are cokahler (indeed, transversely Calabi-Yau); more generally, we give a lower and upper bound for the $*$-scalar curvature in the case that the structure is not cokahler. We prove similar bounds for almost Kahler Einstein manifolds that are not Kahler.
We study the existence of left invariant closed $G_2$-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these $G_2$-structures, we show long time existence and uniqueness of solution for
the Laplacian flow on the noncompact manifold. Moreover, considering the Laplacian flow on the associated Lie algebra as a bracket flow on $R^7$ in a similar way as in [23] we prove that the underlying metrics $g(t)$ of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in the nilpotent Lie group, as $t$ goes to infinity.
We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the cla
ssification of seven dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.
