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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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