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](https://en.wikipedia.org/wiki/G2_manifold).


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](https://pdfs.semanticscholar.org/29f6/3182818ec9ef655d389100c06c3d4dd92f40.pdf).


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? 



Killing vector fields on a compact $G_2$ manifold
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 $G\_2$).


For more information on the Cheeger--Gromoll splitting theorem, you might look at 


* the expository paper: Cheeger, *Structure theory and convergence in Riemannian geometry*, **Milan J. Math.** 78 (2010), no. 1, 221–264
* or Eschenburg and Heintze, *An elementary proof of the Cheeger Gromoll splitting theorem*, **Annals of Global Analysis and Geometry**, June 1984, Volume 2, Issue 2, pp 141–151.
* or Jeff Cheeger and Detlef Gromoll, *The splitting theorem for manifolds of nonnegative Ricci curvature*, **J. Differential Geom.**, Volume 6, Number 1 (1971), 119-128.


