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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 21st 2014
• (edited Dec 21st 2014)

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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeDec 21st 2014
• (edited Dec 21st 2014)

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

$\array{ \tau_{\mathbb{A}^n_i} && \longrightarrow && \tau_X \\ & {}_{\mathllap{\mathbf{c}_0}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{\mathbf{c}}} \\ && G \mathbf{Struc} }$