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I do not understand the entry G-structure. G-structure is, as usual, defined there as the principal $G$-subbundle of the frame bundle which is a $GL(n)$-principal bundle. I guess this makes sense for equivariant injections along any Lie group homomorphism $G\to GL(n)$. The entry says something about spin structure, warning that the group $Spin(n)$ is not a subgroup of $GL(n)$. So what is meant ? The total space of a subbundle is a subspace at least. Does this mean that I consider the frame bundle first as a (non-principal) $Spin(n)$-bundle by pulling back along a fixed noninjective map $Spin(n)\to GL(n)$ and then I restrict to a chosen subspace on which the induced action of Spin group is principal ?
The original discussion was here.
I have added to G-structure an “nPOV“-discussion, which takes care of the subtleties discussed by Zoran and Jim above (also of the subtleties that the Wikipedia entry is fighting with in its first paragraphs).
I have also created reduction of structure groups with some remarks, but then I wasn’t sure anymore if this deserves an entry of its own and left it somewhat orphaned.
I have more extensive discussion of this in section 4.4.2, but don’t quite feel the energy right now to instiki-fy this.
I have added to this subsection of G-structure the examples
Thanks. Link in 5 should be with lowercase “structure”, with uppercase it does not work.
Thanks, fixed.
I had added in the notion of (B,f)-structure, too. (Hm, didn’t I already announce this edit somewhere?)
It seems to me that it is useful to add, in the definition of multiplicative (B,f)-structure, the clause that $B_0 = \ast$ actually be the point, and act as the unit. In Kochmann 96, p. 14 there is something slightly weaker.
Presumably there are $G$-structures for any smooth cohesive setting, such as the complex analytic case. I know one has almost complex structure by reduction along $GL(d, \mathbb{C}) \to GL(2d, \mathbb{R})$, but that’s taking place in the context of real manifolds. What would it be to work directly within the complex analytic setting. Or is it that when employing as canonical structure group $GL(d, \mathbb{C})$, you can just take this to be a reduction of a real group?
Googling “g-structure on a complex manifold” finds very few entries, but includes a recent paper by Hwang (mentioned elsewhere) – Rational curves and prolongations of G-structures. Why so rare?
One paper – On the complexication of the classical geometries and exceptional numbers – has a section 2. G-structures on complex manifolds. I wonder what ’exceptional numbers’ are.
One might imagine reducing structure group of a complex manifold along $(\mathbb{C}^\ast)^m \to GL_n(\mathbb{C})$, or to some discrete group like $SL_2(\mathbb{Z})$.
I know one has almost complex structure by reduction along $GL(d, \mathbb{C}) \to GL(2d, \mathbb{R})$, but that’s taking place in the context of real manifolds. What would it be to work directly within the complex analytic setting.
In general if we are talking about manifolds locally modeled on a group object $V$, then the structure group of their frame bundle is $Aut(\mathbb{D}_e)$, where $\mathbb{D}$ is the (first order, say) infintiesimal neighbourhood of the neutral element in $V$.
If we are in the topos over the site of real smooth manifolds and take $V = \mathbb{R}^n$ with its additive group structure, then $Aut(\mathbb{D}_e) = GL(n,\mathbb{R})$.
If we are in the topos over the site of complex analytic manifolds and take $V = \mathbb{C}^n$ with its additive group structure, then $Aut(\mathbb{D}_e) = GL(n,\mathbb{C})$.
Now there is some relation between these two toposes (once we had some discussion here about how to organize that best). In particular we may take $\mathbb{R}^{2n}$-manifolds in the former, equip them with a reduction of their structure group along $GL(n,\mathbb{C}) \to GL(2n,\mathbb{R})$ and regard the result as being a $\mathbb{C}^n$-manifold in the second topos.
Googling “g-structure on a complex manifold” finds very few entries
Even in real differential geometry, the perspective of G-structures and Cartan geometries is not as popular as the component-based alternatives. This is not the fault of the concept, but of the community.
I found some further references and add some cases here.
added pointer to today’s
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