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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
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)
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:
In relation to BPS states/partial reduction of supersymmetry:
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
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)
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)
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)
and this one:
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)
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)
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)
added pointer to
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