Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeSep 11th 2011
   • (edited Sep 11th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI 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 ?

   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 ?

2. • CommentRowNumber2.
   • CommentAuthorjim_stasheff
   • CommentTimeSep 11th 2011
   • PermaLink
   Author: jim_stasheff
   Format: TextI 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.

   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.
3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 11th 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThe original discussion was [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=1122&Focus=9046#Comment_9046).

   The original discussion was here.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 3rd 2012
   • (edited Feb 3rd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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](http://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf#page=325), but don't quite feel the energy right now to instiki-fy this.

   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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 3rd 2012
   • (edited Feb 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to [this subsection](http://ncatlab.org/nlab/show/G-structure#ExamplesOfReductionsOfNonTangentBundles) of _[[G-structure]]_ the examples * [[Cartan geometry]]; * [[generalized complex geometry]]; * [[exceptional generalized geometry]]

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

   • Cartan geometry;

   • generalized complex geometry;

   • exceptional generalized geometry

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeFeb 9th 2012
   • (edited Feb 9th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThanks. Link in 5 should be with lowercase "structure", with uppercase it does not work.

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

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 9th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, fixed.

   Thanks, fixed.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexI had added in the notion of [(B,f)-structure](https://ncatlab.org/nlab/show/G-structure#InTermsOfBfStructures), 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](https://ncatlab.org/nlab/show/G-structure#Kochmann96) there is something slightly weaker.

   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.

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 2nd 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexPresumably 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](https://nforum.ncatlab.org/discussion/6475/higher-cartan-geometry/?Focus=63685#Comment_63685)) -- [Rational curves and prolongations of G-structures](https://arxiv.org/abs/1703.03160). Why so rare?

   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?

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 2nd 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexOne paper -- [On the complexication of the classical geometries and exceptional numbers](http://web.maths.unsw.edu.au/~norman/papers/L.pdf) -- has a section 2. G-structures on complex manifolds. I wonder what 'exceptional numbers' are.

    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.

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 2nd 2017
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexOne 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 $SL_2(\mathbb{Z})$.

    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 $SL_2(\mathbb{Z})$.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2017
    • (edited Jul 2nd 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> 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 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.

13. • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 2nd 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexI found some further references and add some cases [here](https://ncatlab.org/nlab/show/G-structure#complex_geometric_examples).

    I found some further references and add some cases here.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 9th 2018
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to today's * [[George Papadopoulos]], _Geometry and symmetries of null G-structures_ ([arXiv:1811.03500](https://arxiv.org/abs/1811.03500)) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/27">v27</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    added pointer to today’s

    • George Papadopoulos, Geometry and symmetries of null G-structures (arXiv:1811.03500)

    diff, v27, current

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded these pointers: * [[Jerome Gauntlett]], Dario Martelli, Stathis Pakis, [[Daniel Waldram]], _G-Structures and Wrapped NS5-branes_, Commun.Math.Phys. 247 (2004) 421-445 ([arxiv:hep-th/0205050](https://arxiv.org/abs/hep-th/0205050)) * Cezar Condeescu, Andrei Micu, Eran Palti, _M-theory Compactifications to Three Dimensions with M2-brane Potentials_, JHEP 04 (2014) 026 ([arxiv:1311.5901](https://arxiv.org/abs/1311.5901)) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/31">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/31">v31</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    added these pointers:

    • Jerome Gauntlett, Dario Martelli, Stathis Pakis, Daniel Waldram, G-Structures and Wrapped NS5-branes, Commun.Math.Phys. 247 (2004) 421-445 (arxiv:hep-th/0205050)

    • Cezar Condeescu, Andrei Micu, Eran Palti, M-theory Compactifications to Three Dimensions with M2-brane Potentials, JHEP 04 (2014) 026 (arxiv:1311.5901)

    diff, v31, current

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded 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&#8211;615, (exposition in [arXiv:0108125](http://arxiv.org/abs/math/0108125)) In relation to [[BPS states]]/partial reduction of [[supersymmetry]]: * [[Jerome Gauntlett]], Dario Martelli, Stathis Pakis, [[Daniel Waldram]], _G-Structures and Wrapped NS5-branes_, Commun.Math.Phys. 247 (2004) 421-445 ([arxiv:hep-th/0205050](https://arxiv.org/abs/hep-th/0205050)) and specifically so for [[M-theory on 8-manifolds]]: * Cezar Condeescu, [[Andrei Micu]], [[Eran Palti]], _M-theory Compactifications to Three Dimensions with M2-brane Potentials_, JHEP 04 (2014) 026 ([arxiv:1311.5901](https://arxiv.org/abs/1311.5901)) * Daniël Prins, [[Dimitrios Tsimpis]], _IIA supergravity and M-theory on manifolds with $SU(4)$ structure_, Phys. Rev. D 89.064030 ([arXiv:1312.1692](https://arxiv.org/abs/1312.1692)) * [[Elena Babalic]], [[Calin Lazaroiu]], _Foliated eight-manifolds for M-theory compactification_, JHEP 01 (2015) 140 ([arXiv:1411.3148](https://arxiv.org/abs/1411.3148)) * C. S. Shahbazi, _M-theory on non-Kähler manifolds_, JHEP 09 (2015) 178 ([arXiv:1503.00733](https://arxiv.org/abs/1503.00733)) * [[Elena Babalic]], [[Calin Lazaroiu]], _The landscape of $G$-structures in eight-manifold compactifications of M-theory_, JHEP 11 (2015) 007 ([arXiv:1505.02270](https://arxiv.org/abs/1505.02270)) * [[Elena Babalic]], [[Calin Lazaroiu]], _Internal circle uplifts, transversality and stratified $G$-structures_, JHEP 11 (2015) 174 ([arXiv:1505.05238](https://arxiv.org/abs/1505.05238)) See also * {#Papadopoulos18} [[George Papadopoulos]], _Geometry and symmetries of null $G$-structures_ ([arXiv:1811.03500](https://arxiv.org/abs/1811.03500)) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/33">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/33">v33</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    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:

    • Jerome Gauntlett, Dario Martelli, Stathis Pakis, Daniel Waldram, G-Structures and Wrapped NS5-branes, Commun.Math.Phys. 247 (2004) 421-445 (arxiv:hep-th/0205050)

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

    • Cezar Condeescu, Andrei Micu, Eran Palti, M-theory Compactifications to Three Dimensions with M2-brane Potentials, JHEP 04 (2014) 026 (arxiv:1311.5901)

    • Daniël Prins, Dimitrios Tsimpis, IIA supergravity and M-theory on manifolds with $SU(4)$ structure, Phys. Rev. D 89.064030 (arXiv:1312.1692)

    • Elena Babalic, Calin Lazaroiu, Foliated eight-manifolds for M-theory compactification, JHEP 01 (2015) 140 (arXiv:1411.3148)

    • C. S. Shahbazi, M-theory on non-Kähler manifolds, JHEP 09 (2015) 178 (arXiv:1503.00733)

    • Elena Babalic, Calin Lazaroiu, The landscape of $G$-structures in eight-manifold compactifications of M-theory, JHEP 11 (2015) 007 (arXiv:1505.02270)

    • Elena Babalic, Calin Lazaroiu, Internal circle uplifts, transversality and stratified $G$-structures, JHEP 11 (2015) 174 (arXiv:1505.05238)

    See also

    • {#Papadopoulos18} George Papadopoulos, Geometry and symmetries of null $G$-structures (arXiv:1811.03500)

    diff, v33, current

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 21st 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexfinally added also pointer to the two classics: * [[Chris Isham]], [[Christopher Pope]], _Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings_, Class. Quant. Grav. 5 (1988) 257 ([spire:251240](http://inspirehep.net/record/251240), [doi:10.1088/0264-9381/5/2/006](https://doi.org/10.1088/0264-9381/5/2/006)) (focus on [[Spin(7)-structure]]) * [[Chris Isham]], [[Christopher Pope]], [[Nicholas Warner]], _Nowhere-vanishing spinors and triality rotations in 8-manifolds_, Classical and Quantum Gravity, Volume 5, Number 10, 1988 ([doi:10.1088/0264-9381/5/10/009](https://iopscience.iop.org/article/10.1088/0264-9381/5/10/009)) (focus on [[Spin(7)-structure]]) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/34">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/34">v34</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    finally added also pointer to the two classics:

    • Chris Isham, Christopher Pope, Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings, Class. Quant. Grav. 5 (1988) 257 (spire:251240, doi:10.1088/0264-9381/5/2/006)

      (focus on Spin(7)-structure)

    • Chris Isham, Christopher Pope, Nicholas Warner, Nowhere-vanishing spinors and triality rotations in 8-manifolds, Classical and Quantum Gravity, Volume 5, Number 10, 1988 (doi:10.1088/0264-9381/5/10/009)

      (focus on Spin(7)-structure)

    diff, v34, current

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2019
    • (edited Nov 23rd 2019)
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to this nice one: * [[Jérôme Gaillard]], _On $G$-structures in gauge/string duality_, 2011 ([cronfa:42569](https://cronfa.swan.ac.uk/Record/cronfa42569) [spire:1340775](http://inspirehep.net/record/1340775), [[GaillardGStructure.pdf:file]]) (with application to [[holographic QCD]]) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/35">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/35">v35</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    added pointer to this nice one:

    • Jérôme Gaillard, On $G$-structures in gauge/string duality, 2011 (cronfa:42569 spire:1340775, GaillardGStructure.pdf:file)

      (with application to holographic QCD)

    diff, v35, current

19. • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexalso this one: * [[Ulf Danielsson]], [[Giuseppe Dibitetto]], Adolfo Guarino, _KK-monopoles and $G$-structures in M-theory/type IIA reductions_, JHEP 1502 (2015) 096 ([arXiv:1411.0575](https://arxiv.org/abs/1411.0575)) (with application to [[D6-branes]]/[[KK-monopoles]] in [[M-theory]]) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/35">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/35">v35</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    also this one:

    • Ulf Danielsson, Giuseppe Dibitetto, Adolfo Guarino, KK-monopoles and $G$-structures in M-theory/type IIA reductions, JHEP 1502 (2015) 096 (arXiv:1411.0575)

      (with application to D6-branes/KK-monopoles in M-theory)

    diff, v35, current

20. • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexand this one: * [[Paul Koerber]], _Lectures on Generalized Complex Geometry for Physicists_, Fortsch. Phys. 59: 169-242, 2011 ([arXiv:1006.1536](https://arxiv.org/abs/1006.1536)) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/35">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/35">v35</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    and this one:

    • Paul Koerber, Lectures on Generalized Complex Geometry for Physicists, Fortsch. Phys. 59: 169-242, 2011 (arXiv:1006.1536)

    diff, v35, current

21. • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to discussion of $G$-structures on orbifolds: * A. V. Bagaev, N. I. Zhukova , _The Automorphism Groups of Finite Type $G$-Structures on Orbifolds_, Siberian Mathematical Journal 44, 213–224 (2003) ([arXiv:10.1023/A:1022920417785](https://doi.org/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](https://arxiv.org/abs/1605.04190)) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/37">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/37">v37</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    added pointer to discussion of $G$-structures on orbifolds:

    • A. V. Bagaev, N. I. Zhukova , The Automorphism Groups of Finite Type $G$-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

22. • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to * [[Marius Crainic]], Chapters 3 and 4 of: _[Differential geometry course](https://webspace.science.uu.nl/~crain101/DG-2015/)_, 2015 ([pdf](https://webspace.science.uu.nl/~crain101/DG-2015/main10.pdf), [[CrainicDifferentialGeometry15.pdf:file]]) * [[Federica Pasquotto]], _Linear $G$-structures by examples_ ([pdf](http://www.few.vu.nl/~pasquott/course16.pdf), [[PasquottoGStructures.pdf:file]]) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/39">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/39">v39</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    added pointer to

    • Marius Crainic, Chapters 3 and 4 of: Differential geometry course, 2015 (pdf, CrainicDifferentialGeometry15.pdf:file)

    • Federica Pasquotto, Linear $G$-structures by examples (pdf, PasquottoGStructures.pdf:file)

    diff, v39, current

23. • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJul 17th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * P. Molino, _Theorie des G-Structures: Le Probleme d'Equivalence_, Lecture Notes in Mathematics, Springer (1977) ([ISBN:978-3-540-37360-5](https://www.springer.com/de/book/9783540082460)) * P. Molino, _Sur quelques propriétés des G-structures_, J. Differential Geom. Volume 7, Number 3-4 (1972), 489-518 ([euclid:jdg/1214431168](https://projecteuclid.org/euclid.jdg/1214431168)) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/40">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/40">v40</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    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

24. • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to * [[Shoshichi Kobayashi]], _Transformation Groups in Differential Geometry_ 1972, reprinted as: Classics in Mathemaitics Vol. 70, Springer 1995 ([doi:10.1007/978-3-642-61981-6](https://link.springer.com/book/10.1007/978-3-642-61981-6)) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/41">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/41">v41</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    added pointer to

    • Shoshichi Kobayashi, Transformation Groups in Differential Geometry 1972, reprinted as: Classics in Mathemaitics Vol. 70, Springer 1995 (doi:10.1007/978-3-642-61981-6)

    diff, v41, current

25. • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded mentioning of the example of the canonical H-structure on $G/H$ ([here](https://ncatlab.org/nlab/show/G-structure#CanonicalHStructureOnFModH)) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/44">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/44">v44</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    added mentioning of the example of the canonical H-structure on $G/H$ (here)

    diff, v44, current

26. • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeMar 28th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded one more reference using $G$-structures in analysis of supergravity solutions: * [[Ruben Minasian]], [[Daniël Prins]], [[Hagen Triendl]], _Supersymmetric branes and instantons on curved spaces_, JHEP 10 (2017) 159 ([arXiv:1707.07002](https://arxiv.org/abs/1707.07002)) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/45">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/45">v45</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    added one more reference using $G$-structures in analysis of supergravity solutions:

    • Ruben Minasian, Daniël Prins, Hagen Triendl, Supersymmetric branes and instantons on curved spaces, JHEP 10 (2017) 159 (arXiv:1707.07002)

    diff, v45, current

27. • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * [[Robin Terrisse]], [[Dimitrios Tsimpis]], _$SU(3)$ structures on $S^2$ bundles over four-manifolds_, JHEP09 (2017) 133 ([arXiv:1707.04636](https://arxiv.org/abs/1707.04636)) <a href="https://ncatlab.org/nlab/revision/diff/G-structure/46">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/46">v46</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    added pointer to:

    • Robin Terrisse, Dimitrios Tsimpis, $SU(3)$ structures on $S^2$ bundles over four-manifolds, JHEP09 (2017) 133 (arXiv:1707.04636)

    diff, v46, current

28. • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to today's * Mateo Galdeano, *The Geometry and Superconformal Algebras of String Compactifications with a $G$-structure* &lbrack;[arXiv:2212.13781](https://arxiv.org/abs/2212.13781)&rbrack;\ <a href="https://ncatlab.org/nlab/revision/diff/G-structure/51">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-structure/51">v51</a>, <a href="https://ncatlab.org/nlab/show/G-structure">current</a>

    added pointer to today’s

    • Mateo Galdeano, The Geometry and Superconformal Algebras of String Compactifications with a $G$-structure [arXiv:2212.13781]\

    diff, v51, current