English translation of paper: Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)

Question

I am interested in the history of $G_2$ manifolds and want to read this paper in english:

Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)

Does anyone know where I can find a translation?

Answer 1

My library has provided a copy of the article of Bonan. Here is a summary of its contents:

Section 1: The author introduces the inner product algebra of octonions (algèbra des octaves de Cayley), which I will denote by $\mathbb{O}$ and write $\mathbb{O} = \mathbb{R}\mathbf{1}\oplus\mathrm{Im}\mathbb{O}$, and he defines $G_2$ to be the subgroup of $\mathrm{SO}\bigl(\mathrm{Im}\mathbb{O}\bigr)\simeq\mathrm{SO}(7)$ that preserves the imaginary part of multiplication. He then defines a $G_2$-invariant $3$-form $\alpha$ on $\mathrm{Im}\mathbb{O}$ by the rule $\alpha(x,y,z) = \langle x, yz\rangle$, where $\langle,\rangle$ is the inner product on $\mathbb{O}$, and its Hodge dual $\beta = \ast\alpha$, a $G_2$-invariant $4$-form on $\mathrm{Im}\mathbb{O}$. He also discusses a $G_2$-invariant decomposition of the exterior differential forms on $\mathrm{Im}\mathbb{O}$ (but does not give the full decomposition).

Section 2: The author sort of defines $\mathrm{Spin}(7)\subset\mathrm{SO}(\mathbb{O})\simeq\mathrm{SO}(8)$ as the subgroup that double covers $\mathrm{SO}(7)$, that preserves the orthogonal splitting $\mathbb{O} = \mathbb{R}\mathbf{1}\oplus\mathrm{Im}\mathbb{O}$, and that acts as $G_2$ on $\mathrm{Im}\mathbb{O}$. (Comment: These properties do not uniquely define $\mathrm{Spin}(7)$, but he then gives a more explicit description of the Lie algebra of $\mathrm{Spin}(7)$ that does uniquely specify it.) Then he states that $\mathrm{Spin}(7)$ preserves the $4$-form $\gamma = \mathbf{1}^*\wedge\alpha + \beta$. (This is true, but it would have been better to define $\mathrm{Spin}(7)$ directly as the subgroup of $\mathrm{SO}(\mathbb{O})$ that preserves $\gamma$.) He also gives some properties that provide an incomplete decomposition of the exterior forms on $\mathbb{O}$ under the action of $\mathrm{Spin}(7)$.

Section 3: The author makes some general remarks about holonomy and parallel differential forms and states that any Riemannian $7$-manifold with holonomy in $G_2$ supports a (nonzero) parallel $3$-form and a (nonzero) parallel $4$-form and that any Riemannian $8$-manifold with holonomy in $\mathrm{Spin}(7)$ supports a (nonzero) parallel $4$-form. He also uses his partial decompositions of the exterior algebra to prove some inequalities on the Betti numbers of such manifolds that are compact (analogous to the famous inequalities for Kähler manifolds). Finally, he remarks that the relations defining the Lie algebra of $\mathrm{Spin}(7)$ imply, by the Bianchi identities, relations on the Riemann curvature tensor of a Riemannian $8$-manifold with holonomy in $\mathrm{Spin}(7)$ that force the vanishing of its Ricci tensor. (Of course, this also implies the vanishing of the Ricci tensor of a Riemannian $7$-manifold with holonomy in $G_2$.)

Footnotes: There are two interesting footnotes. First, footnote (3) claims that the $3$-form $\alpha$ was constructed by Chevalley in his 1954 book "Algebraic Theory of Spinors". However, this is false; Chevalley's book contains no such construction. In any case, the fact that $G_2$ leaves invariant a $3$-form in its $7$-dimensional representation had been known for a long time before that; it appears to be due to Engel in 1900, at least in the complex setting. It remains a mystery as to where Bonan really learned about $\alpha$. Perhaps he discovered the existence of $\alpha$ on his own and then Chevalley told him that it was already known (this is speculation on my part, though). Second, footnote (4) remarks that Kostant had already proved in 1956 that both $\mathrm{Spin}(7)$ $8$-manifolds and $G_2$ $7$-manifolds had to have a nonzero parallel $4$-form (On invariant skew tensors, Proc. Nat. Acad. Sci. 42 (1956), 148–151). (Of course, by duality, this implies that any $G_2$ $7$-manifold has to also have a nonzero parallel $3$-form.) If I ever knew this before now, I had forgotten it. This is particularly interesting because, as late as 1955, Berger claimed in his thesis that manifolds with these two holonomies did not support any nontrivial parallel differential forms.