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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?

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  • $\begingroup$ 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? $\endgroup$
    – DLIN
    Apr 5, 2019 at 11:36

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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.
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  • $\begingroup$ Thank you so much for your insightful answer. Do you have a good reference for the splitting theorems you mentioned? Right now my only reference is Dominic Joyce's book and it's very difficult to understand. $\endgroup$
    – user117832
    Mar 4, 2018 at 12:48
  • $\begingroup$ You might look at Eschenburg and Heintze, A $\endgroup$
    – Ben McKay
    Mar 4, 2018 at 15:26

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