Timeline for Does diff$(M)$ act transitively on the set of integrable $G$-structures?
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| Jul 2, 2014 at 19:26 | comment | added | Robert Bryant | Just a comment on terminology: You should be careful about using the term 'integrable $G$-structure', since this means different things to different authors. For example, while Guillemin, in his papers, takes 'integrable $G$-structure' to be synonymous with 'locally flat $G$-structure', this conflicts with an older usage by authors such as Chern, for whom an integrable $G$-structure is one whose intrinsic torsion functions are constant (not necessarily zero). Thus, for example, for Chern, a contact structure is an integrable $G$-structure while it is not for Guillemin. | |
| Jul 2, 2014 at 18:12 | comment | added | Ben McKay | Consider $G=SL(n,R)$. Then $G$-structures are volume forms. On a compact manifold, the volume of the manifold is an invariant of the $G$-structure, even after diffeomorphism. So the answer is no. Similarly for the symplectic group. Ever for a flat Riemannian metric, on a torus, there are different volumes and different sets of lengths of closed geodesics. But in general, no one knows what the diffeomorphism invariants of flat $G$-structures are. | |
| Jul 2, 2014 at 18:07 | comment | added | Qiaochu Yuan | Surely not; wouldn't that imply that e.g. the isomorphism type of a complex manifold is completely determined by its underlying smooth manifold? | |
| Jul 2, 2014 at 17:59 | history | asked | Jon Middleton | CC BY-SA 3.0 |