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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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