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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 21st 2014
   • (edited Dec 21st 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have been added a first approximation to an Idea-section to _[[torsion of a G-structure]]_ - Have also added a pointer to [Lott 90](http://ncatlab.org/nlab/show/torsion+of+a+G-structure#Lott90) and started a stub _[[torsion constraints in supergravity]]_, for the moment only to record some references. Have also further touched related entries such as _[[torsion of a Cartan connection]]_.

   I have been added a first approximation to an Idea-section to torsion of a G-structure -

   Have also added a pointer to Lott 90 and started a stub torsion constraints in supergravity, for the moment only to record some references.

   Have also further touched related entries such as torsion of a Cartan connection.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 21st 2014
   • (edited Dec 21st 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexLet me try to restate synthetically the characterization of integrable/torsion-free $G$-structure the way Lott says it on p. 4 of [arXiv:0108125](http://arxiv.org/abs/math/0108125): Fix a model space $\mathbb{A}^n$ with first order infinitesimal disk around the origin denoted $\mathbb{D}^n \hookrightarrow \mathbb{A}^n$. Assume that the canonical $\mathbb{D}^n$-bundle of $\mathbb{A}^n$ is trivial. Write $GL(n) := \mathbf{Aut}(\mathbb{D}^n)$. Then (as we discussed in another thread recently) if any $X$ has a formally étale cover by $\mathbb{A}^n$-s then it carries a canonical frame bundle, modulated by some map $\tau_X : X \longrightarrow \mathbf{B} GL(n)$. Now fix any group $G$ and a map $G\mathbf{Struc} : \mathbf{B}G \longrightarrow \mathbf{B}GL(n)$. Then given an $\mathbb{A}^n$-manifold $X$ as above, a $G$-structure on $X$ is equivalently a morphism $\mathbf{c} : \tau_X \longrightarrow G\mathbf{Struc}(X)$ in the slice over $\mathbf{B}GL(n)$. So far this is clear. Now regarding how to say synthetically that this $G$-structure is integrable/torsion free. To that end, fix a $G$-structure on the model space $\mathbf{c}_0 : \tau_{\mathbb{A}^n} \longrightarrow G\mathbf{Struc}$. Now I suppose we should say: the $G$-structure $\mathbf{c}$ on $X$ is integrable/torsion-free if there exists a formally étale cover $\coprod_i \mathbb{A}^n \longrightarrow X$ such that this extends to a morphism of $G$-structures, i.e. a morphism $ \coprod_i \mathbf{c}_0 \longrightarrow \mathbf{c}$ in the slice over $G\mathbf{Struc}$ (which itself is in the slice over $\mathbf{B}GL(n)$). So this just expresses that along each patch inclusion $\mathbb{A}^n_i \longrightarrow X$ the $G$-structure on $X$ restricts to the fixed one on the model space, up to equivalence $$ \array{ \tau_{\mathbb{A}^n_i} && \longrightarrow && \tau_X \\ & {}_{\mathllap{\mathbf{c}_0}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{\mathbf{c}}} \\ && G \mathbf{Struc} } $$

   Let me try to restate synthetically the characterization of integrable/torsion-free $G$-structure the way Lott says it on p. 4 of arXiv:0108125:

   Fix a model space $\mathbb{A}^n$ with first order infinitesimal disk around the origin denoted $\mathbb{D}^n \hookrightarrow \mathbb{A}^n$. Assume that the canonical $\mathbb{D}^n$-bundle of $\mathbb{A}^n$ is trivial.

   Write $GL(n) := \mathbf{Aut}(\mathbb{D}^n)$.

   Then (as we discussed in another thread recently) if any $X$ has a formally étale cover by $\mathbb{A}^n$-s then it carries a canonical frame bundle, modulated by some map $\tau_X : X \longrightarrow \mathbf{B} GL(n)$.

   Now fix any group $G$ and a map $G\mathbf{Struc} : \mathbf{B}G \longrightarrow \mathbf{B}GL(n)$.

   Then given an $\mathbb{A}^n$-manifold $X$ as above, a $G$-structure on $X$ is equivalently a morphism

   $\mathbf{c} : \tau_X \longrightarrow G\mathbf{Struc}(X)$

   in the slice over $\mathbf{B}GL(n)$.

   So far this is clear. Now regarding how to say synthetically that this $G$-structure is integrable/torsion free.

   To that end, fix a $G$-structure on the model space

   $\mathbf{c}_0 : \tau_{\mathbb{A}^n} \longrightarrow G\mathbf{Struc}$.

   Now I suppose we should say: the $G$-structure $\mathbf{c}$ on $X$ is integrable/torsion-free if there exists a formally étale cover $\coprod_i \mathbb{A}^n \longrightarrow X$ such that this extends to a morphism of $G$-structures, i.e. a morphism

   $\coprod_i \mathbf{c}_0 \longrightarrow \mathbf{c}$

   in the slice over $G\mathbf{Struc}$ (which itself is in the slice over $\mathbf{B}GL(n)$).

   So this just expresses that along each patch inclusion $\mathbb{A}^n_i \longrightarrow X$ the $G$-structure on $X$ restricts to the fixed one on the model space, up to equivalence

   $$\array{ \tau_{\mathbb{A}^n_i} && \longrightarrow && \tau_X \\ & {}_{\mathllap{\mathbf{c}_0}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{\mathbf{c}}} \\ && G \mathbf{Struc} }$$
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * [[José Figueroa-O'Farrill]], _On the intrinsic torsion of spacetime structures_ ([arXiv:2009.01948](https://arxiv.org/abs/2009.01948)) <a href="https://ncatlab.org/nlab/revision/diff/torsion+of+a+G-structure/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/torsion+of+a+G-structure/14">v14</a>, <a href="https://ncatlab.org/nlab/show/torsion+of+a+G-structure">current</a>

   added pointer to today’s

   • José Figueroa-O’Farrill, On the intrinsic torsion of spacetime structures (arXiv:2009.01948)

   diff, v14, current