## Geometry of $$G_{2}$$ orbits and isoparametric hypersurfaces.(English)Zbl 1231.53051

Author’s abstract: “We characterize the adjoint $$G_{2}$$-orbits in the Lie algebra $$\mathfrak {g}$$ of $$G_{2}$$ as fibered spaces over $$S^{6}$$ with fibers given by the complex Cartan hypersurfaces. This combines the isoparametric hypersurfaces of case $$(g,m)=(6,2)$$ with the case $$(3,2)$$. The fibrations on two singular orbits turn out to be diffeomorphic to the twistor fibrations of $$S^{6}$$ and $$G_{2}/SO(4)$$. From the symplectic point of view, we show that there exists a 2-parameter family of Lagrangian submanifolds on every orbit.”

### MSC:

 53C40 Global submanifolds 53C30 Differential geometry of homogeneous manifolds 32L25 Twistor theory, double fibrations (complex-analytic aspects)
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### References:

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