Reduction of beta-integrable 2-Segre structures

Details Reduction of beta-integrable 2-Segre structures Reduction of beta-integrable 2-Segre structures

2011-10-14

arxiv.org/abs/1110.3279v3

Mathematics

Differential Geometry

28 pages. Version 3 corrects a mistake in section two

Scientific paper

It is shown that over a 2n-manifold M equipped with a beta-integrable 2-Segre structure S, there exists a quasiholomorphic fibre bundle rho : X -> M with fibre CP^1\RP^1. We prove that rho-sections having holomorphic image are in one-to-one correspondence with reductions of S to torsion-free S^1\cdot GL(n,R)-structures on M. Consequently every beta-integrable 2-Segre structure can locally be reduced to a torsion-free S^1\cdot GL(n,R)-structure. In the homogeneous case of the oriented 2-plane Grassmannian, the reductions are in one-to-one correspondence with the smooth quadrics in CP^{n+1} without real points.

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