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Feb 15, 2015 at 18:17 comment added Oldřich Spáčil The following paper might be helpful: Cadek, Crabb, Vanzura: Obstruction theory on 8-manifolds, doi: 10.1007/s00229-008-0203-x, arXiv:0710.0734. They give a necessary and sufficient condition for existence of a $U(4)$-structure (Proposition 4.1) as well as for some other $G$-structures.
Feb 14, 2015 at 22:53 answer added Qiaochu Yuan timeline score: 6
Feb 14, 2015 at 18:22 history asked Bilateral CC BY-SA 3.0