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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeSep 11th 2011
• (edited Sep 11th 2011)

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 ?

• CommentRowNumber2.
• CommentAuthorjim_stasheff
• CommentTimeSep 11th 2011
I think it is a historical accident that G-structure is, as usual, defined there as the principal
G-subbundle of the frame bundle
cf the term reduction of structural group' which was originally meant for subgroups
but later was realized to make sense, mathematically if not linguistically, for any
homomorphism from G
Recall how spin 1/2 representations' was used or the orthogonal group.
• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeSep 11th 2011

The original discussion was here.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 3rd 2012
• (edited Feb 3rd 2012)

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 3rd 2012
• (edited Feb 9th 2012)

I have added to this subsection of G-structure the examples

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeFeb 9th 2012
• (edited Feb 9th 2012)

Thanks. Link in 5 should be with lowercase “structure”, with uppercase it does not work.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 9th 2012

Thanks, fixed.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMay 30th 2016

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.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeJul 2nd 2017

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?

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeJul 2nd 2017

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.

• CommentRowNumber11.
• CommentAuthorDavidRoberts
• CommentTimeJul 2nd 2017

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 $SL_2(\mathbb{Z})$.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJul 2nd 2017
• (edited Jul 2nd 2017)

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.

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeJul 2nd 2017

I found some further references and add some cases here.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeNov 9th 2018