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\begin{document}

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\section*{integrability of G-structures}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{DefinitionTraditional}{Traditional}\dotfill \pageref*{DefinitionTraditional} \linebreak
\noindent\hyperlink{IntermOfDifferentialCohesion}{In higher differential cohesive geometry}\dotfill \pageref*{IntermOfDifferentialCohesion} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{existence_and_torsion}{Existence and torsion}\dotfill \pageref*{existence_and_torsion} \linebreak
\noindent\hyperlink{in_terms_of_adapted_coordinate_systems}{In terms of adapted coordinate systems}\dotfill \pageref*{in_terms_of_adapted_coordinate_systems} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{ExampleComplexStructure}{Complex structure}\dotfill \pageref*{ExampleComplexStructure} \linebreak
\noindent\hyperlink{ExampleSymplecticStructure}{Symplectic structure}\dotfill \pageref*{ExampleSymplecticStructure} \linebreak
\noindent\hyperlink{ExamplesOrthogonalStructure}{Orthogonal structure}\dotfill \pageref*{ExamplesOrthogonalStructure} \linebreak
\noindent\hyperlink{UnitaryStructure}{Unitary structure}\dotfill \pageref*{UnitaryStructure} \linebreak
\noindent\hyperlink{ExampleG2Structure}{$G_2$-Structure}\dotfill \pageref*{ExampleG2Structure} \linebreak
\noindent\hyperlink{further_examples}{Further examples}\dotfill \pageref*{further_examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

Given a [[G-structure]], it is \emph{integrable} or \emph{locally flat} if over [[germs]] it restricts to the canonical (trivial) $G$-structure. If it restricts to the canonical structure only over order-$k$ [[infinitesimal disks]], then it is called \emph{order $k$ integrable}. The [[obstruction]] to first-order integrability is the [[torsion of a G-structure]], and hence first-order integrable $G$-structures are also called \emph{torsion-free $G$-structures}.

Beware that some authors use the term ``integrable'' for ``torsion-free''. This originates in concentration on the case of [[almost symplectic structure]], i.e. $G$-structure for $G = Sp(2n)$ the [[symplectic group]], in which case the [[Darboux theorem]] gives that first order integrability (to [[symplectic structure]]) already implies full integrability. However, in general this is not the case. For instance for [[orthogonal structure]], i.e. $G$-structure for $G = O(n)$ the [[orthogonal group]], then the [[fundamental theorem of Riemannian geometry]] gives that the torsion obstruction to first-order integrability vanishes, exhibited by the [[Levi-Civita connection]], but full integrability here is equivalent to this being a [[flat connection]], which is a strong additional constraint. This is the case from which the terminology ``locally flat'' for ``integrable'' derives from.

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\hypertarget{DefinitionTraditional}{}\subsubsection*{{Traditional}}\label{DefinitionTraditional}

Let $V$ be a linear local model space, e.g. a [[vector space]] in plain [[differential geometry]] or [[super vector space]] in [[supergeometry]], etc.. Write $GL(V)$ for its [[general linear group]]. Consider a [[group]] [[homomorphism]] $G \longrightarrow GL(V)$.

Write $\mathbf{c}_0$ for the \emph{standard flat $G$-structure} on $V$ (see at \emph{\href{G-structure#TheStandardFlatGStructure}{G-Structure -- Examples -- Standard flat G-structure}}).

\begin{defn}
\label{}\hypertarget{}{}
A [[G-structure]] $\mathbf{c}$ on a [[manifold]] $X$ modeled on $V$ (e.g. a [[smooth manifold]] or [[supermanifold]]) is called integrable if

\begin{enumerate}%
\item there exists [[cover]] $\{U_i \hookrightarrow X\}$ by [[open subsets]] $U_i \hookrightarrow V$;


\item such that the $G$-structure $\mathbf{c}$ on $X$ restricts on each patch to the default $G$-structure $\mathbf{c}_0$ on $V$:

\begin{displaymath}
\mathbf{c}|_{U_i} \simeq \mathbf{c}_0|_{U_i}
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
This is due to (\hyperlink{Sternberg64}{Sternberg 64, section VII, def. 2.4}, \hyperlink{Guillemin65}{Guillemin 65, section 3}). For review see also (\hyperlink{Alekseevskii}{Alekseevskii}, \hyperlink{Lott90}{Lott 90, page 4 of the exposition}).

\begin{remark}
\label{}\hypertarget{}{}
More concretely, if $G$-structure is modeled by $G$-subbundles $P$ of the [[frame bundle]] (as discussed at \emph{\href{G-structure#InTermsOfSubbundlesOfTheFrameBundle}{G-structure -- In terms of subbundles of the frame bundle}} ), then it is integrable if each $P \hookrightarrow Fr(X)$ restricts on each patch to $P_0 \hookrightarrow Fr(V)$

\begin{displaymath}
\itexarray{
     P_0|_{U_i} &\hookrightarrow& Fr(V)|_{U_i}
     \\
     {}^{\mathllap{\simeq}}\downarrow && \downarrow
     \\
     P|_{U_i} &\hookrightarrow& Fr(X)|_{U_i}
  }
  \,.
\end{displaymath}
\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
For $k \in \mathbb{N}$, a [[G-structure]] $\mathbf{c}$ on a [[manifold]] $X$ modeled on $V$ (e.g. a [[smooth manifold]] or [[supermanifold]]) is called \emph{order-$k$ infinitesimally integrable} if at each point $x \in X$ its restriction to the order-$k$ [[infinitesimal neighbourhood]] $\mathbb{D}^V_0 \simeq \mathbb{D}_x^X \hookrightarrow X$ is equal to the default $G$-structure $\mathbf{c}_0$.

\end{defn}
(\hyperlink{Guillemin65}{Guillemin 65, section 4})

\hypertarget{IntermOfDifferentialCohesion}{}\subsubsection*{{In higher differential cohesive geometry}}\label{IntermOfDifferentialCohesion}

One may formalize the concept of integrable $G$-structure in the generality of [[higher differential geometry]], formalized in [[differential cohesion]]. See also there at \emph{\href{differential+cohesive+%28infinity%2C1%29-topos#structures}{differential cohesion -- G-Structure}}.

Let $V$ be framed, def. \ref{Framing}, let $G$ be an [[∞-group]] and $G \to GL(V)$ a homomorphism, hence

\begin{displaymath}
G\mathbf{Struc}\colon \mathbf{B}G \longrightarrow \mathbf{B}GL(V)
\end{displaymath}
a morphism between the [[deloopings]].

\begin{defn}
\label{GStructure}\hypertarget{GStructure}{}
For $X$ a $V$-manifold, def. \ref{Manifold}, a \textbf{[[G-structure]]} on $X$ is a lift of the structure group of its [[frame bundle]], def. \ref{FrameBundleMap}, to $G$, hence a diagram

\begin{displaymath}
\itexarray{
     X && \longrightarrow&&  \mathbf{B}G
     \\
     & {}_{\mathllap{\tau_X}}\searrow && \swarrow_{\mathrlap{G\mathbf{Struc}}}
     \\
     && \mathbf{B}GL(V)
  }
\end{displaymath}
hence a morphism

\begin{displaymath}
\mathbf{c} \colon \tau_X \longrightarrow G\mathbf{Struc}
\end{displaymath}
is the [[slice (∞,1)-topos]].

\end{defn}
In fact $G\mathbf{Struc}\in \mathbf{H}_{/\mathbf{B}GL(n)}$ is the [[moduli ∞-stack]] of such $G$-structures.

The double [[slice (∞,1)-topos|slice]] $(\mathbf{H}_{/\mathbf{B}GL(n)})_{/G\mathbf{Struc}}$ is the [[(∞,1)-category]] of such $G$-structures.

\begin{example}
\label{TrivialGStructure}\hypertarget{TrivialGStructure}{}
If $V$ is framed, def. \ref{Framing}, then it carries the trivial $G$-structure, which we denote by

\begin{displaymath}
\mathbf{c}_0 \colon \tau_{V} \longrightarrow G\mathbf{Struc}
  \,.
\end{displaymath}
\end{example}
\begin{defn}
\label{IntegrableGStructure}\hypertarget{IntegrableGStructure}{}
For $V$ framed, def. \ref{Framing}, and $X$ a $V$-manifold, def. \ref{Manifold}, then $G$-structure $\mathbf{c}$ on $X$ is \emph{[[integrability of G-structure|integrable]]} (or \emph{locally flat}) if there exists a $V$-cover

\begin{displaymath}
\itexarray{
     && U
     \\
     & \swarrow && \searrow
     \\
     V && && X
  }
\end{displaymath}
such that the [[correspondence]] of [[frame bundles]] induced via remark \ref{FrameBundlesFunctorial}

\begin{displaymath}
\itexarray{
     && \tau_U
     \\
     & \swarrow && \searrow
     \\
     \tau_V && && \tau_X
  }
\end{displaymath}
(a diagram in $\mathbf{H}_{/\mathbf{B}GL(V)}$) extends to a sliced correspondence between $\mathbf{c}$ and the trivial $G$-structure $\mathbf{c}_0$ on $V$, example \ref{TrivialGStructure}, hence to a diagram in $\mathbf{H}_{/\mathbf{B}GL(V)}$ of the form

\begin{displaymath}
\itexarray{
     && \tau_U
     \\
     & \swarrow && \searrow
     \\
     \tau_V && \swArrow_{\mathrlap{\simeq}}  && \tau_X
     \\
     & {}_{\mathllap{\mathbf{c}_0}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}}
     \\
     && G\mathbf{Struct}
  }
\end{displaymath}
On the other hand, $\mathbf{c}$ is called \emph{infinitesimally integrable} (or \emph{torsion-free}) if such an extension exists (only) after restriction to all [[infinitesimal disks]] in $X$ and $U$, hence after composition with the [[counit of a comonad|counit]]

\begin{displaymath}
\flat^{rel} U \longrightarrow U
\end{displaymath}
of the [[relative flat modality]], def. \ref{InducedRelativeShapeAndFlat}:

\begin{displaymath}
\itexarray{
     && \tau_{\flat^{rel} U}
     \\
     & \swarrow && \searrow
     \\
     \tau_V && \swArrow_{\mathrlap{\simeq}}  && \tau_X
     \\
     & {}_{\mathllap{\mathbf{c}_0}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}}
     \\
     && G\mathbf{Struct}
  }
 \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
As before, if the given [[reduction modality]] encodes order-$k$ infinitesimals, then the infinitesimal integrability in def. \ref{IntegrableGStructure} is order-$k$ integrability. For $k = 1$ this is [[torsion of a G-structure|torsion-freeness]].

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{existence_and_torsion}{}\subsubsection*{{Existence and torsion}}\label{existence_and_torsion}

The [[obstruction]] for a $G$-structure to be integrable to first order is its [[torsion of a G-structure]].

\hypertarget{in_terms_of_adapted_coordinate_systems}{}\subsubsection*{{In terms of adapted coordinate systems}}\label{in_terms_of_adapted_coordinate_systems}

A $G$-structure on $X$ is integrable previsely if there exists an [[atlas]] of $X$ by [[coordinate charts]] with the property that their canonical [[frame fields]] are $G$-frames.

(\hyperlink{Sternberg64}{Sternberg 64, section VII, exercise 2.1})

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{itemize}%
\item \hyperlink{ExampleComplexStructure}{Complex structure}


\item \hyperlink{ExampleSymplecticStructure}{Symplectic structire}


\item \hyperlink{ExamplesOrthogonalStructure}{Orthogonal structure}


\item \hyperlink{ExampleG2Structure}{G2-structure}



\end{itemize}
\hypertarget{ExampleComplexStructure}{}\subsubsection*{{Complex structure}}\label{ExampleComplexStructure}

A $GL(n,\mathbb{C}) \to GL(2n,\mathbb{R})$-[[G-structure|structure]] is an \emph{[[almost complex structure]]}. Its [[torsion of a G-structure]] vanishes precisely if its [[Nijenhuis tensor]] vanishes, hence, by the [[Newlander-Nirenberg theorem]], precisely if it is a [[complex structure]]. Since a [[complex manifold]] admits holomorphic [[coordinate charts]], this first-order integrability already implies full integrability.

\hypertarget{ExampleSymplecticStructure}{}\subsubsection*{{Symplectic structure}}\label{ExampleSymplecticStructure}

An $Sp(n) \hookrightarrow GL(2n)$-[[G-structure|structure]] is an \emph{[[almost symplectic structure]]}. Its [[torsion of a G-structure]] is the [[de Rham differential]] $\mathbf{d}\omega$ of the corresponding 2-form $\omega$ (recalled e.g. in \href{symplectic+manifold#AlbuquerquePicken11}{Albuquerque-Picken 11}). Hence first-order integrability here amounts precisely to [[symplectic structure]]. The [[Darboux theorem]] asserts that this is already a fully integrable structure.

\hypertarget{ExamplesOrthogonalStructure}{}\subsubsection*{{Orthogonal structure}}\label{ExamplesOrthogonalStructure}

An $O(n)\to GL(n)$-[[G-structure|structure]] is an [[orthogonal structure]], hence a [[vielbein]], hence a [[Riemannian metric]]. The [[fundamental theorem of Riemannian geometry]] says that in this case the [[torsion of a G-structure]] vanishes, exhibited by the existence of the [[Levi-Civita connection]]. The corresponding first-order integrability is the existence of [[Riemann normal coordinates]] (since these identify the given [[vielbein]] at any point to first order with the trivial (identity) vielbein). The higher order obstructions to integrability turn out to all be proportional to combinations of the [[Riemann curvature]]. Full integrability is equivalent to the vanishing of Riemann tensor, hence to the LC-connection being a [[flat connection]].

\hypertarget{UnitaryStructure}{}\subsubsection*{{Unitary structure}}\label{UnitaryStructure}

The case of unitary structure is precisely the combination of the above three cases.

By the fact (see at \emph{\href{unitary+group#RelationToOrthogonalSymplecticAndGeneralLinearGroup}{unitary group -- relation to orthogonal, symplectic and general linear group}}) that the [[unitary group]] is the intersection

\begin{displaymath}
U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})
\end{displaymath}
a $U(n) \hookrightarrow GL(2n,\mathbb{R})$-structure -- called an \emph{[[almost Hermitian structure]]} -- is precisely a joint [[orthogonal structure]], [[almost symplectic structure]] and [[almost complex structure]]. Hence if first order integrable -- called a [[Kähler manifold]] structure -- this is precisely a joint [[orthogonal structure]]/[[Riemannian manifold]] structure, [[symplectic manifold]] structure, [[complex manifold]] structure.

\hypertarget{ExampleG2Structure}{}\subsubsection*{{$G_2$-Structure}}\label{ExampleG2Structure}

A An $G_2 \to GL(7)$-[[G-structure|structure]] is a [[G2-structure]]. Its [[torsion of a G-structure]] vanishes if the corresponding definite 3-form $\omega$ is [[covariant derivative|covariantly constant]] with respect to the induced [[Riemannian metric]], in which case the structure is a [[G2-manifold]]. Beware that some authors refer to first-order integrable $G_2$-structure (or even weaker conditions) as ``integrable $G_2$-structure'' (see \hyperlink{G2+manifold#Bryant05}{Bryant 05, remark 2} for critical discussion of the terminology). The [[torsion of a G-structure|higher-order torsion invariants]] of $G_2$-structures do not in general vanish (e.g \hyperlink{Bryant05}{Bryant 05, (4.7)}) and so, contrary to the above cases of symplectic and complex structure, $G_2$-manifold structure does not imply integrable $G_2$-structure.

\hypertarget{further_examples}{}\subsubsection*{{Further examples}}\label{further_examples}

\begin{itemize}%
\item [[CR-manifold]]


\item [[torsion constraints of supergravity]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Shlomo Sternberg]], chapter VII of \emph{Lectures on differential geometry}, Prentice-Hall (1964)


\item [[Victor Guillemin]], \emph{The integrability problem for $G$-structures}, Trans. Amer. Math. Soc. 116 (1965), 544--560. (\href{http://www.jstor.org/stable/1994134}{JSTOR})


\item D. V: Alekseevskii, \emph{$G$-structure on a manifold} in M. Hazewinkel (ed.) \emph{Encyclopedia of Mathematics, Volume 4}



\end{itemize}
Lecture notes include

\begin{itemize}%
\item Federica Pasquotto, \emph{Linear $G$-structures by example} (\href{http://www.few.vu.nl/~pasquott/course16.pdf}{pdf})

\end{itemize}
Discussion with an eye towards [[torsion constraints in supergravity]] is in

\begin{itemize}%
\item [[John Lott]], \emph{The Geometry of Supergravity Torsion Constraints}, Comm. Math. Phys. 133 (1990), 563--615, (exposition in \href{http://arxiv.org/abs/math/0108125}{arXiv:0108125})

\end{itemize}
Discussion with an eye towards [[special holonomy]] is in

\begin{itemize}%
\item [[Dominic Joyce]], \emph{Compact manifolds with special holonomy}, Oxford University Press 2000

\end{itemize}
See also the references at \emph{[[torsion of a Cartan connection]]} and at \emph{[[torsion constraints in supergravity]]}.

[[!redirects integrability of G-structure]] [[!redirects integrability of G-structures]]

[[!redirects integrable G-structure]] [[!redirects integrable G-structures]]

[[!redirects integrable G-stucture]] [[!redirects integrable G-stuctures]]

[[!redirects first-order integrable G-structure]] [[!redirects first-order integrable G-structures]]



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