Reduction of $\beta$-integrable 2-Segre structures

We show that locally every $\beta$-integrable $(2,n)$-Segre structure can be reduced to a torsion-free $S^1\cdot \,\mathrm{GL}(n,\mathbb{R})$-structure. This is done by observing that such reductions correspond to sections with holomorphic image of a certain ‘twistor bundle.’ For the homogeneous $(2,n)$-Segre structure on the oriented $2$-plane Grassmannian, the reductions are shown to be in one-to-one correspondence with the smooth quadrics $Q \subset \mathbb{CP}^{n+1}$ without real points.

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Published 9 April 2013