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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 GG-subbundle of the frame bundle which is a GL(n)GL(n)-principal bundle. I guess this makes sense for equivariant injections along any Lie group homomorphism GGL(n)G\to GL(n). The entry says something about spin structure, warning that the group Spin(n)Spin(n) is not a subgroup of GL(n)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)Spin(n)-bundle by pulling back along a fixed noninjective map Spin(n)GL(n)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)
    • 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=*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 GG-structures for any smooth cohesive setting, such as the complex analytic case. I know one has almost complex structure by reduction along GL(d,)GL(2d,)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,)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 ( *) mGL n()(\mathbb{C}^\ast)^m \to GL_n(\mathbb{C}), or to some discrete group like SL 2()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,)GL(2d,)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 VV, then the structure group of their frame bundle is Aut(𝔻 e)Aut(\mathbb{D}_e), where 𝔻\mathbb{D} is the (first order, say) infintiesimal neighbourhood of the neutral element in VV.

    If we are in the topos over the site of real smooth manifolds and take V= nV = \mathbb{R}^n with its additive group structure, then Aut(𝔻 e)=GL(n,)Aut(\mathbb{D}_e) = GL(n,\mathbb{R}).

    If we are in the topos over the site of complex analytic manifolds and take V= nV = \mathbb{C}^n with its additive group structure, then Aut(𝔻 e)=GL(n,)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 2n\mathbb{R}^{2n}-manifolds in the former, equip them with a reduction of their structure group along GL(n,)GL(2n,)GL(n,\mathbb{C}) \to GL(2n,\mathbb{R}) and regard the result as being a n\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

    added pointer to today’s

    diff, v27, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2019

    added these pointers:

    diff, v31, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2019

    added yet more pointers for G-structures used in supergravity. Now the respective References-section reads as follows:


    Discussion of G-structures in supergravity and superstring theory:

    In relation to torsion constraints in supergravity:

    • {#Lott90} John Lott, The Geometry of Supergravity Torsion Constraints, Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125)

    In relation to BPS states/partial reduction of supersymmetry:

    and specifically so for M-theory on 8-manifolds:

    See also

    diff, v33, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 21st 2019

    finally added also pointer to the two classics:

    diff, v34, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2019
    • (edited Nov 23rd 2019)

    added pointer to this nice one:

    diff, v35, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2019

    also this one:

    diff, v35, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2019

    and this one:

    diff, v35, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2020

    added pointer to discussion of GG-structures on orbifolds:

    • A. V. Bagaev, N. I. Zhukova , The Automorphism Groups of Finite Type GG-Structures on Orbifolds, Siberian Mathematical Journal 44, 213–224 (2003) (arXiv:10.1023/A:1022920417785)

    • Robert Wolak, Orbifolds, geometric structures and foliations. Applications to harmonic maps, Rendiconti del seminario matematico - Universita politecnico di Torinom vol. 73/1 , 3-4 (2016), 173-187 (arXiv:1605.04190)

    diff, v37, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2020

    added pointer to

    diff, v39, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJul 17th 2020

    added pointer to:

    • P. Molino, Theorie des G-Structures: Le Probleme d’Equivalence, Lecture Notes in Mathematics, Springer (1977) (ISBN:978-3-540-37360-5)

    • P. Molino, Sur quelques propriétés des G-structures, J. Differential Geom. Volume 7, Number 3-4 (1972), 489-518 (euclid:jdg/1214431168)

    diff, v40, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2020

    added pointer to

    diff, v41, current

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