# Killing vector fields on a compact $G_2$ manifold

I am trying to show that there can not be any nonvanishing Killing vector fields on a compact $G_2$ manifold.

For the definition of a $G_2$ manifold just see the Wikipedia page.

I know that since the manifold is Ricci-flat, any Killing vector field must be a parallel vector field, but I am unsure how to prove that these can not be nonvanishing.

I have read a lot of things in the literature about Betti numbers. For example the article "The structure of compact Ricci-flat Riemannian manifolds" by Fischer and Wolf.

But it's pretty over my head. How should I go about this problem? Should I continue to learn more about the Betti numbers of compact $G_2$ manifolds or is there a more simple (maybe even obvious) thing that I am missing?

• Could you give a reference about: If manifold is Ricci-flat, then any Killing vector field must be a parallel vector field, i.e. does not exist zero point? Does it mean that any Ricc-flat manifold admits a Killing field?
– DLIN
Apr 5, 2019 at 11:36

A parallel vector field implies a reduction of holonomy. Any form of $G_2$ does not preserve any nonzero vector when acting in its nontrivial 7-dimensional representation. So the holonomy must be the subgroup of $G_2$ preserving a nonzero vector, i.e. $SU(3)$. The reduction of holonomy group will, by deRham's splitting theorem, give the manifold a product structure, at least locally, so a local product of the line and a Calabi-Yau. The parallel vector field is dual to a parallel closed 1-form. If the manifold is compact, that implies that the first Betti number is nonzero. Moreover, applying the Cheeger-Gromoll splitting theorem, after taking a finite covering of the compact manifold, it splits into a product of a torus and a manifold of holonomy a subgroup of $G_2$. Strictly speaking, if you follow the definition in Wikipedia page (as in the question above), these products are actually $G_2$ manifolds, so the statement in the question, that there are no $G_2$ manifolds with parallel vector fields, is not correct. The correct statement is that the holonomy of a compact $G_2$ manifold is a proper subgroup of $G_2$ (up to conjugacy in $SO(7)$) if and only if the manifold has a finite Riemannian covering by a product of a torus and a compact manifold of dimension less than $7$ with holonomy a subgroup of $SU(3)$ (up to conjugacy in $G_2$).