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Let $M$ be a compact manifold and diff$(M)$ its diffeomorphism group. Let $G$ be a Lie subgroup of $GL(n,\mathbb{R})$.

In general, the topology of a manifold may prevent it from having an integrable $G$-structure. An easy example is $M=S^2$ and $G=O(2)$. An integrable $O(2)$-structure is a flat Riemannian metric. Such a metric cannot exist due to the Gauss-Bonnet theorem.

So suppose that $M$ has at least one integrable $G$-structure. Call it $k$. Is the set of all integrable $G$-structures on $M$ the orbit of $k$ under the action of diff$(M)$?

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  • $\begingroup$ Surely not; wouldn't that imply that e.g. the isomorphism type of a complex manifold is completely determined by its underlying smooth manifold? $\endgroup$
    – Qiaochu Yuan
    Commented Jul 2, 2014 at 18:07
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    $\begingroup$ 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. $\endgroup$
    – Ben McKay
    Commented Jul 2, 2014 at 18:12
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    $\begingroup$ 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. $\endgroup$
    – Robert Bryant
    Commented Jul 2, 2014 at 19:26

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