Conformally parallel \(G_2\) structures on a class of solvmanifolds
A seven-dimensional Riemannian manifold \((Y,g)\) is referred to as a \(G_2\)-structure if the structure group of its tangent bundle is reducible to the exceptional group \(G_2\), and presence of this \(G_2\)-structure is equivalent to the existence of a three-form \(\zeta\) on the manifold. When \(\zeta\) is covariantly constant, the holonomy group is naturally contained in \(G_2\) and the manifold is called parallel. If the metric can be modified to a metric with holonomy a subgroup of \(G_2\) by a transformation \(g\to e^{2f}g\), where \(f\) is some function, then \(G_2\) is said to be conformally parallel. As was done by \textit{C. Will} [Differ. Geom. Appl. 19, No. 3, 307--318 (2003; Zbl 1045.53032)] it is natural to study such \(G_2\)-structures on a rank-one solvable extension of a metric 6-dimensional nilpotent Lie algebra \(n\) endowed with an SU(3)-structure \((\omega,\psi^+)\) and a non-singular self-adjoint derivation \(D\) diagonalized by a unitary basis. Such an extension is given by a metric Lie algebra \(s=n\oplus \mathbb RH\) with bracket \[ [H,U]=DU,\;[U,V]= [U,V]_{n\times n} \] where \(U,V\in n\) and \(H \perp n\), \(\|H\|=1\). Let \(N\) be the 2-step nilotent isometry group with Lie algebra \(n\), then there is a natural \(G_2\)-structure on a manifold \(Y=N\times \mathbb R\) corresponding to the 3-form \[ \zeta= \psi^++\omega \wedge H^b\in\Lambda^3T^*Y, \] where \(b\) is the isomorphism of the tangent bundle \(T\) of \(N\) onto the cotangent bundle \(T^*\) induced by the metric. Clearly the Lie algebra \(\sigma\) is isomorphic to each fibre of the principal fibration \(T^*Y\to Y\). As result the author presents as main theorem the following conclusion: \((Y,\zeta)\) is conformally parallel if and only if \(n\) is either \(\mathbb R^6\), or 2-step nilpotent but not isomorphic to the Lie algebra \({\mathfrak h}_3\oplus{\mathfrak h}_3\), where \({\mathfrak h}_3\) denotes the real 3-dimensional Heisenberg algebra. In the remaining parts of the paper, the SU(3)-structure is deformed to describe the non-homogeneous Ricci-flat structure with holonomy containded in \(G_2\), and the authors find a metric with exceptional holonomy.
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