A \(G_2\)-structure on a smooth \(7\)-manifold \(M\) is defined by a \(3\)-form \(\phi\) such that the symmetric bilinear form \[ (u,v) \mapsto B_\phi (u,v) = (u \lrcorner \phi ) \wedge (v \lrcorner\phi ) \wedge \phi \in \wedge^7T_p^*M, \] \(u, v\in T_pM\), is definite for all \(p\in M\). The \(3\)-form \(\phi\) induces the volume form \(\nu = (\det B_\phi )^\frac{1}{9}\) and the Riemannian metric \(g= \frac{1}{\nu}B_\phi\). The curvature tensor \(R\) of \(g\) can be decomposed into 5 components, which correspond to the five irreducible \(G_2\)-submodules of the \(196\)-dimensional vector space of algebraic curvature tensors of type \(\mathfrak{so}(7)\): scalar curvature \(s\), trace-free Ricci curvature \(Ric_0\) and three components \(W_{77}\), \(W_{64}\) and \(W_{27}\) of the Weyl curvature \(W=W_{77} +W_{64} + W_{27}\). A \(G_2\)-structure \(\phi\) is parallel if and only if the holonomy group of \(g\) is a subgroup of \(G_2\). In that case we have \(R=W_{77}\).
The authors provide explicit formulas for these curvature components in terms of certain generalized Ricci contractions. In particular, \(W_{27}\) is related to a certain symmetric trace-free tensor \(Ric^{\mathcal W}\). It is proven that various curvature assumptions (such as \(W_{27}=0\)) imply that \(\phi\) is parallel. For closed \(G_2\)-structures \(\phi\), the cup product of the first Pontrjagin class of \(M\) with the de Rham cohomology class of \(\phi\) is a well-defined top cohomology class. Integrating it over a closed manifold \(M\) yields a numerical invariant \(n(M,\phi )\in \mathbb{R}\) of the \(G_2\)-structure. The authors show (among other extimates) that \(n(M,\phi )\) is bounded from below by the integral of \(\frac{3}{16}s^2 -\| W_{77}\|^2\).
It is shown that the closed \(G_2\)-structures realising equality in that inequality have parallel intrinsic torsion and that \((M,\phi )\) is locally isomorphic to a homogeneous \(G_2\)-manifold constructed by R. L. Bryant [“Some remarks on \(G_2\)-structures”, in Akbulut, Selman (ed.) et al., Proceedings of the 11th and 12th Gökova geometry-topology conferences, Cambridge, MA: International Press, 75–109 (2006; Zbl 1115.53018)]. Further examples of \(G_2\)-structures enjoying various interesting properties are constructed in the last chapter.