## Cohomogeneity-one $$G_{2}$$-structures.(English)Zbl 1025.53024

Let $$M$$ be a $$7$$-dimensional manifold equipped with a $$G_2$$-structure $$\phi$$, which is a certain $$3$$-form on $$M$$ that induces an identification of each tangent space of $$M$$ with the imaginary octonions. The $$3$$-form determines a Riemannian metric $$g$$ on $$M$$ and hence a Hodge-star operator $$\star$$. If $$\phi$$ and $$\star\phi$$ are closed, then the holonomy of $$(M,g)$$ is contained in $$G_2$$, and if $$d\phi = \lambda\star\phi$$ for some nonzero constant $$\lambda$$ then $$(M,g)$$ is said to have weak holonomy $$G_2$$.
A compact Lie group $$G$$ is said to act with cohomogeneity one on a $$G_2$$-manifold $$(M,\phi)$$ if it preserves $$\phi$$ and if there exists an orbit of codimension one. In this paper the authors determine all manifolds with holonomy or weak holonomy $$G_2$$ that have a cohomogeneity one action of a connected compact simple Lie group $$G$$. In this case $$G$$ is either $$G_2$$, $$Sp(2)$$ or $$SU(3)$$. Surprisingly, in each of the $$3$$ cases there exists a unique solution with weak holonomy $$G_2$$. For $$SU(3)$$ they show that there is a one-parameter family of solutions with $$G_2$$-holonomy, whereas in the other two cases the solutions are isolated. As a special case they obtain the $$G_2$$ manifolds found by Bryant and Salamon.
The approach by the authors is to consider a geodesic that is perpendicular to the principal orbits of the action, then to determine the solutions of the symplectic resp. cosymplectic $$G_2$$ equation $$d\phi = 0$$ resp. $$d\star\phi = 0$$ along the geodesic, and finally to investigate the relevant topology and boundary conditions. The topological types of manifolds admitting a symplectic or cosymplectic $$G_2$$-structure are determined. They show that a symplectic $$G_2$$-structure exists only if $$M$$ admits a holonomy $$G_2$$-metric, and construct new examples of compact cosymplectic $$G_2$$-structures.

### MSC:

 53C29 Issues of holonomy in differential geometry 57M50 General geometric structures on low-dimensional manifolds 57S15 Compact Lie groups of differentiable transformations 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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### References:

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