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\begin{document}

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\section*{G-structure}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher 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{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{InTermsOfBfStructures}{In terms of $(B,f)$-structures}\dotfill \pageref*{InTermsOfBfStructures} \linebreak
\noindent\hyperlink{InTermsOfSubbundlesOfTheFrameBundle}{In terms of subbundles of the frame bundle}\dotfill \pageref*{InTermsOfSubbundlesOfTheFrameBundle} \linebreak
\noindent\hyperlink{in_terms_of_cartan_connections}{In terms of Cartan connections}\dotfill \pageref*{in_terms_of_cartan_connections} \linebreak
\noindent\hyperlink{in_higher_differential_geometry}{In higher differential geometry}\dotfill \pageref*{in_higher_differential_geometry} \linebreak
\noindent\hyperlink{structure_on_a_principal_bundle}{$G$-structure on a $K$-principal bundle}\dotfill \pageref*{structure_on_a_principal_bundle} \linebreak
\noindent\hyperlink{structure_on_an_etale_groupoid}{$G$-Structure on an etale $\infty$-groupoid}\dotfill \pageref*{structure_on_an_etale_groupoid} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{integrability_of_structure}{Integrability of $G$-structure}\dotfill \pageref*{integrability_of_structure} \linebreak
\noindent\hyperlink{relation_to_special_holonomy}{Relation to special holonomy}\dotfill \pageref*{relation_to_special_holonomy} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{TheStandardFlatGStructure}{The standard flat $G$-structure}\dotfill \pageref*{TheStandardFlatGStructure} \linebreak
\noindent\hyperlink{reduction_of_tangent_bundle_structure}{Reduction of tangent bundle structure}\dotfill \pageref*{reduction_of_tangent_bundle_structure} \linebreak
\noindent\hyperlink{lift_of_tangent_bundle_structure}{Lift of tangent bundle structure}\dotfill \pageref*{lift_of_tangent_bundle_structure} \linebreak
\noindent\hyperlink{ExamplesOfReductionsOfNonTangentBundles}{Reduction of more general bundle structure}\dotfill \pageref*{ExamplesOfReductionsOfNonTangentBundles} \linebreak
\noindent\hyperlink{complex_geometric_examples}{Complex geometric examples}\dotfill \pageref*{complex_geometric_examples} \linebreak
\noindent\hyperlink{special_holonomy_examples}{Special holonomy examples}\dotfill \pageref*{special_holonomy_examples} \linebreak
\noindent\hyperlink{structures_on_8manifolds}{$G$-Structures on 8-manifolds}\dotfill \pageref*{structures_on_8manifolds} \linebreak
\noindent\hyperlink{higher_geometric_examples}{Higher geometric examples}\dotfill \pageref*{higher_geometric_examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{traditional}{Traditional}\dotfill \pageref*{traditional} \linebreak
\noindent\hyperlink{in_supergeometry}{In supergeometry}\dotfill \pageref*{in_supergeometry} \linebreak
\noindent\hyperlink{in_complex_geometry}{In complex geometry}\dotfill \pageref*{in_complex_geometry} \linebreak
\noindent\hyperlink{in_higher_geometry}{In higher geometry}\dotfill \pageref*{in_higher_geometry} \linebreak


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

A \textbf{G-structure} on an $n$-[[manifold]] $M$, for a given structure group $G$, is a $G$-subbundle of the [[frame bundle]] (of the [[tangent bundle]]) of $M$.

Equivalently, this means that a $G$-structure is a choice of [[reduction of structure groups|reduction]] of the canonical structure group $GL(n)$ of the [[principal bundle]] to which the [[tangent bundle]] is [[associated bundle|associated]] along the given inclusion $G \hookrightarrow GL(n)$.

More generally, one can consider the case $G$ is not a [[subgroup]] but equipped with any group homomorphism $G \to GL(n)$. If this is instead an [[epimorphism]] one speaks of a [[lift of structure groups]].

Both cases, in turn, can naturally be understood as special cases of [[twisted differential c-structures]], which is a notion that applies more generally to [[principal infinity-bundles]].

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Given a [[smooth manifold]] $X$ of [[dimension]] $n$ and given a [[Lie group|Lie]] [[subgroup]] $G \hookrightarrow GL(n)$ of the [[general linear group]], then a \emph{$G$-structure} on $X$ is a [[reduction of structure groups|reduction of the structure group]] of the [[frame bundle]] of $X$ to $G$.

There are many more explicit (and more abstract) equivalent ways to say this, which we discuss below. There are also many evident variants and generalizations.

Notably one may consider reductions of the frames in the $k$th order [[jet bundle]]. (e. g. \hyperlink{Alekseevskii}{Alekseevskii}) This yields order $k$ $G$-structure and the ordinary $G$-structures above are then first order.

Moreover, the definition makes sense for generalized [[manifolds]] modeled on other base spaces than just [[Cartesian spaces]]. In particular there are evident generalizations to [[supermanifolds]] and to [[complex manifolds]].

\hypertarget{InTermsOfBfStructures}{}\subsubsection*{{In terms of $(B,f)$-structures}}\label{InTermsOfBfStructures}

\begin{defn}
\label{BfStructure}\hypertarget{BfStructure}{}
A \textbf{$(B,f)$-structure} is

\begin{enumerate}%
\item for each $n\in \mathbb{N}$ a [[pointed topological space|pointed]] [[CW-complex]] $B_n \in Top_{CW}^{\ast/}$


\item equipped with a pointed [[Serre fibration]]

\begin{displaymath}
\itexarray{
    B_n
    \\
    \downarrow^{\mathrlap{f_n}}
    \\
    B O(n)
  }
\end{displaymath}
to the [[classifying space]] $B O(n)$ (\href{classifying+space#EOn}{def.});


\item for all $n_1 \leq n_2$ a pointed continuous function

$\iota_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}$

which is the identity for $n_1 = n_2$;



\end{enumerate}
such that for all $n_1 \leq n_2 \in \mathbb{N}$ these [[commuting square|squares commute]]

\begin{displaymath}
\itexarray{
    B_{n_1} &\overset{\iota_{n_1,n_2}}{\longrightarrow}& B_{n_2}
    \\
    {}^{\mathllap{f_{n_1}}}\downarrow && \downarrow^{\mathrlap{f_{n_2}}}
    \\
    B O(n_1) &\longrightarrow& B O(n_2)
  }
  \,,
\end{displaymath}
where the bottom map is the canonical one (\href{classifying+space#InclusionOfBOnIntoBOnPlusOne}{def.}).

The $(B,f)$-structure is \textbf{multiplicative} if it is moreover equipped with a system of maps $\mu_{n_1,n_2} \colon B_{n_1}\times B_{n_2} \to B_{n_1 + n_2}$ which cover the canonical multiplication maps (\href{classifying+space#WhitneySumMapOnClassifyingSpaces}{def.})

\begin{displaymath}
\itexarray{
    B_{n_1} \times B_{n_2}
      &\overset{\mu_{n_1, n_2}}{\longrightarrow}&
    B_{n_1 + n_2}
    \\
    {}^{\mathllap{f_{n_1} \times f_{n_2}}}\downarrow 
      && 
    \downarrow^{\mathrlap{f_{n_1 + n_2}}}
    \\
    B O(n_1) \times B O(n_2)
     &\longrightarrow&
    B O(n_1 + n_2)
  }
\end{displaymath}
and which satisfy the evident [[associativity]] and [[unitality]], for $B_0 = \ast$ the unit, and, finally, which commute with the maps $\iota$ in that all $n_1,n_2, n_3 \in \mathbb{N}$ these squares commute:

\begin{displaymath}
\itexarray{
    B_{n_1} \times B_{n_2}
      &\overset{id \times \iota_{n_2,n_2+n_3}}{\longrightarrow}&
    B_{n_1} \times B_{n_2 + n_3}
    \\
    {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{n_1,n_2 + n_3}}}
    \\
    B_{n_1 + n_2}
      &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}&
    B_{n_1 + n_2 + n_3}
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    B_{n_1} \times B_{n_2}
      &\overset{\iota_{n_1,n_1+n_3} \times id}{\longrightarrow}&
    B_{n_1+n_3} \times B_{n_2 }
    \\
    {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{n_1 + n_3 , n_2}}}
    \\
    B_{n_1 + n_2}
      &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}&
    B_{n_1 + n_2 + n_3}
  }
  \,.
\end{displaymath}
Similarly, an \textbf{$S^2$-$(B,f)$-structure} is a compatible system

\begin{displaymath}
f_{2n} \colon B_{2n} \longrightarrow B O(2n)
\end{displaymath}
indexed only on the even natural numbers.

Generally, an \textbf{$S^k$-$(B,f)$-structure} for $k \in \mathbb{N}$, $k \geq 1$ is a compatible system

\begin{displaymath}
f_{k n} \colon B_{ kn} \longrightarrow B O(k n)
\end{displaymath}
for all $n \in \mathbb{N}$, hence for all $k n \in k \mathbb{N}$.

\end{defn}
(\hyperlink{Lashof63}{Lashof 63}, \hyperlink{Stong68}{Stong 68, beginning of chapter II}, \hyperlink{Kochmann96}{Kochmann 96, section 1.4})

See also at \emph{[[B-bordism]]}.

\begin{example}
\label{ExamplesOfBfStructures}\hypertarget{ExamplesOfBfStructures}{}
Examples of $(B,f)$-structures (def. \ref{BfStructure}) include the following:

\begin{enumerate}%
\item $B_n = B O(n)$ and $f_n = id$ is \textbf{orthogonal structure} (or ``no structure'');


\item $B_n = E O(n)$ and $f_n$ the [[universal principal bundle]]-projection is \textbf{[[framing]]-structure};


\item $B_n = B SO(n) = E O(n)/SO(n)$ the classifying space of the [[special orthogonal group]] and $f_n$ the canonical projection is \textbf{[[orientation]] structure};


\item $B_n = B Spin(n) = E O(n)/Spin(n)$ the classifying space of the [[spin group]] and $f_n$ the canonical projection is \textbf{[[spin structure]]}.



\end{enumerate}
Examples of $S^2$-$(B,f)$-structures include

\begin{enumerate}%
\item $B_{2n} = B U(n) = E O(2n)/U(n)$ the classifying space of the [[unitary group]], and $f_{2n}$ the canonical projection is \textbf{[[almost complex structure]]}.

\end{enumerate}
\end{example}
\begin{defn}
\label{}\hypertarget{}{}
Given a [[smooth manifold]] $X$ of [[dimension]] $n$, and given a $(B,f)$-structure as in def. \ref{BfStructure}, then a \textbf{$(B,f)$-structure on the manifold} is an [[equivalence class]] of the following structure:

\begin{enumerate}%
\item an [[embedding]] $i_X \; \colon \; X \hookrightarrow \mathbb{R}^k$ for some $k \in \mathbb{N}$;


\item a [[homotopy class]] of a [[lift]] $\hat g$ of the classifying map $g$ of the [[tangent bundle]]

\begin{displaymath}
\itexarray{
    && B_{n}
    \\
    &{}^{\mathllap{\hat g}}\nearrow& \downarrow^{\mathrlap{f_n}}
    \\
    X &\overset{g}{\hookrightarrow}& B O(n)
  }
  \,.
\end{displaymath}


\end{enumerate}
The equivalence relation on such structures is to be that generated by the relation $((i_{X})_1, \hat g_1) \sim ((i_{X})_,\hat g_2)$ if

\begin{enumerate}%
\item $k_2 \geq k_1$


\item the second inclusion factors through the first as

\begin{displaymath}
(i_X)_2 
    \;\colon\; 
  X 
    \overset{(i_X)_1}{\hookrightarrow}
  \mathbb{R}^{k_1} 
    \hookrightarrow
  \mathbb{R}^{k_2}
\end{displaymath}

\item the lift of the classifying map factors accordingly (as homotopy classes)

\begin{displaymath}
\hat g_2
    \;\colon\;
  X 
    \overset{\hat g_1}{\longrightarrow}
  B_{n}
    \longrightarrow
  B_{n}
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\hypertarget{InTermsOfSubbundlesOfTheFrameBundle}{}\subsubsection*{{In terms of subbundles of the frame bundle}}\label{InTermsOfSubbundlesOfTheFrameBundle}

\begin{defn}
\label{}\hypertarget{}{}
Given a [[smooth manifold]] $X$ of [[dimension]] $n$ with [[frame bundle]] $Fr(X)$, and given a [[Lie group]] [[monomorphism]]

\begin{displaymath}
G \longrightarrow GL(\mathbb{R}^n)
\end{displaymath}
into the [[general linear group]], then a \emph{$G$-structure} on $X$ is an $G$-[[principal bundle]] $P \to X$ equipped with an inclusion of [[fiber bundles]]

\begin{displaymath}
\itexarray{
    P &&\hookrightarrow&& Fr(X)
    \\
    & \searrow && \swarrow
    \\
    && X
  }
\end{displaymath}
which is $G$-equivariant.

\end{defn}
(\hyperlink{Sternberg64}{Sternberg 64, section VII, def. 2.1}).

\begin{remark}
\label{}\hypertarget{}{}
From this perspective, a $G$-structure consists of the collection of all $G$-[[frame field|frames]] on a manifold. For instance for an [[orthogonal structure]] it consists of all those frames which are pointwise an [[orthonormal basis]] of the [[tangent bundle]] (with respect to the [[Riemannian metric]] which is defined by the orthonormal structure).

\end{remark}
Accordingly:

\begin{defn}
\label{GStructureGeneratedByFrameField}\hypertarget{GStructureGeneratedByFrameField}{}
Given $G \hookrightarrow GL(n)$ and given any one [[frame field]] $\sigma \colon X \to Fr(X)$ over a [[manifold]] $X$, then acting with $G$ on $\sigma$ at each point produces a $G$-subbundle. This is called the $G$-structure \emph{generated} by the frame field $\sigma$.

\end{defn}
\hypertarget{in_terms_of_cartan_connections}{}\subsubsection*{{In terms of Cartan connections}}\label{in_terms_of_cartan_connections}

A $G$-structure equipped with compatible [[connection on a bundle|connection]] data is equivalently a [[Cartan connection]] for the inclusion $(G \hookrightarrow \mathbb{R}^n \rtimes G)$.

See at \emph{\href{Cartan+connection#ExampleGStructures}{Cartan connection -- Examples -- G-structures}}

\hypertarget{in_higher_differential_geometry}{}\subsubsection*{{In higher differential geometry}}\label{in_higher_differential_geometry}

\hypertarget{structure_on_a_principal_bundle}{}\paragraph*{{$G$-structure on a $K$-principal bundle}}\label{structure_on_a_principal_bundle}

We give an equivalent definition of $G$-structures in terms of [[higher differential geometry]] (``from the [[nPOV]]''). This serves to clarify the slightly subtle but important difference between existence and choice of $G$-structure, and seamlessly embeds the notion into the more general context of [[twisted differential c-structures]].

\begin{defn}
\label{}\hypertarget{}{}
Let $G \to K$ be a [[homomorphism]] of [[Lie groups]]. Write

\begin{displaymath}
\mathbf{c} : \mathbf{B}G \to \mathbf{B}K
\end{displaymath}
for the morphism of [[delooping]] [[Lie groupoids]] ( the [[smooth infinity-groupoid|smooth]] [[moduli stacks]] of smooth $K$- and $G$-[[principal bundles]], respectively).

For $X$ a [[smooth manifold]] (or generally an [[orbifold]] or [[Lie groupoid]], etc.) Let $P \to X$ be a $K$-[[principal bundle]] and let

\begin{displaymath}
k \colon X \longrightarrow \mathbf{B}K
\end{displaymath}
be any choice of morphism [[modulating morphism|modulating]] it.

Write $\mathbf{H}(X, \mathbf{B}G)$ etc. for the [[derived hom space|hom-groupoid]] of [[smooth infinity-groupoid|smooth groupoids]] / smooth stacks . This is equivalently the groupoid of $G$-principal bundles over $X$ and smooth [[gauge transformations]] between them.

Then the \textbf{groupoid of $G$-structure on $P$} (with respect to the given morphism $G \to K$) is the [[homotopy pullback]]

\begin{displaymath}
\mathbf{c}Struc_{[P]}(X)
  :=
  \mathbf{H}(X, \mathbf{B}G)
   \times_{\mathbf{H}(X, \mathbf{B}K)}
  \{k\}
  \,.
\end{displaymath}
\begin{displaymath}
\itexarray{
    \mathbf{c}Struc_{[P]}(X)
    &\longrightarrow&
    \ast
    \\
    \downarrow & \swArrow_\simeq& \downarrow^{\mathrlap{k}}
    \\
    \mathbf{H}(X, \mathbf{B}G)
     &\stackrel{\mathbf{H}(X, \mathbf{c})}{\longrightarrow}&
    \mathbf{H}(X, \mathbf{B}K)
  }
\end{displaymath}
(the groupoid of \emph{[[twisted differential c-structure|twisted c-structures]]}).

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If here $k$ is trivial in that it factors through the point, $k \colon X \to \ast \to \mathbf{B}K$ then this homotopy fiber product is $\mathbf{H}(X,K/G)$, where $K/G$ is the [[coset space]] ([[Klein geometry]]) which itself sits in the [[homotopy fiber sequence]]

\begin{displaymath}
K/G \to \mathbf{B}G \to \mathbf{B}K
  \,.
\end{displaymath}
\end{remark}
\begin{example}
\label{}\hypertarget{}{}
Specifically, when $X$ is a [[smooth manifold]] of [[dimension]] $n$, the [[frame bundle]] $Fr(X)$ is [[modulating moprhism|modulated]] by a morphism $\tau_X \colon X \to \mathbf{B} GL(n)$ into the [[moduli stack]] for the [[general linear group]] $K := GL(n)$. Then for any group homomorphism $G \to GL(n)$, a \textbf{$G$-structure} on $X$ is a $G$-structure on $Fr(X)$, as above.

\end{example}
\hypertarget{structure_on_an_etale_groupoid}{}\paragraph*{{$G$-Structure on an etale $\infty$-groupoid}}\label{structure_on_an_etale_groupoid}

We discuss the concept in the generality of [[higher differential geometry]], formalized in [[differential cohesion]].

See at \emph{\href{differential+cohesive+%28infinity%2C1%29-topos#structures}{differential cohesion -- G-Structure}}

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

\hypertarget{integrability_of_structure}{}\subsubsection*{{Integrability of $G$-structure}}\label{integrability_of_structure}

\begin{defn}
\label{Integrability}\hypertarget{Integrability}{}
A $G$-structure on a [[manifold]] $X$ is called \emph{locally flat} (\hyperlink{Sternberg64}{Sternberg 64, section VII, def. 24}) or \emph{integrable} (e.g. \hyperlink{Alekseevskii}{Alekseevskii}) if it is locally equivalent to the \emph{standard flat $G$-structure}, def. \ref{StandardFlatGStructure}.

This means that there is an [[open cover]] $\{U_i \to X\}$ by [[open subsets]] of the [[Cartesian space]] $\mathbb{R}^n$ such that the restriction of the $G$-structure to each of these is equivalent to the standard flat $G$-structure.

\end{defn}
See at \emph{[[integrability of G-structures]]} for more on this

The [[obstruction]] to integrability of $G$-structure is the \emph{[[torsion of a G-structure]]}. See there for more.

\hypertarget{relation_to_special_holonomy}{}\subsubsection*{{Relation to special holonomy}}\label{relation_to_special_holonomy}

The existence of $G$-structures on tangent bundles of [[Riemannian manifolds]] is closely related to these having [[special holonomy]].

\begin{theorem}
\label{}\hypertarget{}{}
Let $(X,g)$ be a [[connected topological space|connected]] [[Riemannian manifold]] of [[dimension]] $n$ with [[holonomy group]] $Hol(g) \subset O(n)$.

For $G \subset O(n)$ some other [[subgroup]], $(X,g)$ admits a torsion-free G-structure precisely if $Hol(g)$ is [[adjoint action|conjugate]] to a [[subgroup]] of $G$.

Moreover, the space of such $G$-structures is the [[coset]] $G/L$, where $L$ is the group of elements suchthat conjugating $Hol(g)$ with them lands in $G$.

\end{theorem}
This appears as (\hyperlink{Joyce}{Joyce prop. 3.1.8})

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

\hypertarget{TheStandardFlatGStructure}{}\subsubsection*{{The standard flat $G$-structure}}\label{TheStandardFlatGStructure}

\begin{defn}
\label{StandardFlatGStructure}\hypertarget{StandardFlatGStructure}{}
For $G \hookrightarrow GL(n)$ a subgroup, then the \emph{standard flat $G$-structure} on the [[Cartesian space]] $\mathbb{R}^n$ is the $G$-structure which is generated, via def. \ref{GStructureGeneratedByFrameField}, from the canonical [[frame field]] on $\mathbb{R}^n$ (the one which is the identity at each point, under the defining identifications).

\end{defn}
\hypertarget{reduction_of_tangent_bundle_structure}{}\subsubsection*{{Reduction of tangent bundle structure}}\label{reduction_of_tangent_bundle_structure}

\begin{itemize}%
\item For the [[subgroup]] of $GL(n, \mathbb{R})$ of matrices of positive determinant, a $GL(n, \mathbb{R})^+$-structure defines an [[orientation]].


\item For the [[orthogonal group]], an $O(n)$-structure defines a [[Riemannian metric]]. (See the discussion at [[vielbein]] and at


\item For the [[special linear group]], an $SL(n,R)$-structure defines a [[volume form]].


\item For the trivial group, an $\{e\}$-structure consists of an [[absolute parallelism]] of the manifold.


\item For $n = 2 m$ even, a $GL(m, \mathbb{C})$-structure defines an [[almost complex structure]] on the manifold. It must satisfy an integrability condition to be a [[complex structure]].



\end{itemize}
\hypertarget{lift_of_tangent_bundle_structure}{}\subsubsection*{{Lift of tangent bundle structure}}\label{lift_of_tangent_bundle_structure}

An example for a lift of structure groups is

\begin{itemize}%
\item for the [[spin group]] $spin(n)$, a $G$-structure is a [[spin structure]].

\end{itemize}
This continues with lifts to the

\begin{itemize}%
\item [[string group]] giving [[string structure]];


\item [[fivebrane group]] giving [[fivebrane structure]].



\end{itemize}
\hypertarget{ExamplesOfReductionsOfNonTangentBundles}{}\subsubsection*{{Reduction of more general bundle structure}}\label{ExamplesOfReductionsOfNonTangentBundles}

\begin{itemize}%
\item For general $G \to K$, the corresponding notion of [[Cartan geometry]] involves $G$-structure on $K$-principal bundles (not necessarily underlying a tangent bundle).


\item A $U(n,n) \hookrightarrow O(2n,2n)$-structure is a [[generalized complex structure]];


\item For $H_n \to E_{n(n)}$ the inclusion of the [[maximal compact subgroup]] into the [[split real form]] of an [[exceptional Lie group]], the corresponding structure is an [[exceptional generalized geometry]].



\end{itemize}
\hypertarget{complex_geometric_examples}{}\subsubsection*{{Complex geometric examples}}\label{complex_geometric_examples}

\begin{itemize}%
\item The choice of $SO(n, \mathbb{C})$ as subgroup of $GL(n, \mathbb{C})$, determines a complex Riemannian structure;


\item $CO(n, \mathbb{C}) \hookrightarrow GL(n, \mathbb{C})$, a complex conformal structure;


\item $Sp(2n, \mathbb{C})\hookrightarrow GL(2n, \mathbb{C})$, an almost symplectic structure;


\item $GL(2, \mathbb{C}) GL(n, \mathbb{C}) \hookrightarrow GL(2n, \mathbb{C}), n \geq 3$, determines an almost quaternionic structure;


\item more generally a $GL(m, \mathbb{C}) GL(n, \mathbb{C})$-structure on a $m n$-dimensional manifold is locally identical to a Grassmannian spinor structure.



\end{itemize}
\hypertarget{special_holonomy_examples}{}\subsubsection*{{Special holonomy examples}}\label{special_holonomy_examples}

[[!include normed division algebra Riemannian geometry -- table]]

\hypertarget{structures_on_8manifolds}{}\subsubsection*{{$G$-Structures on 8-manifolds}}\label{structures_on_8manifolds}

For discussion of [[G-structures]] on [[closed manifold|closed]] [[8-manifolds]] see \href{8-manifold#GStructuresOn8Manifolds}{there}.

\hypertarget{higher_geometric_examples}{}\subsubsection*{{Higher geometric examples}}\label{higher_geometric_examples}

See the list at [[twisted differential c-structure]].

$\backslash$linebreak

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[special holonomy]]


\item [[integrability of G-structures]]


\item [[homothety]]



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

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

The concept of topological $G$-structure (lifts of homotopy classes of classifying maps) originates with [[cobordism theory]]. Early expositions in terms of \hyperlink{InTermsOfBfStructures}{(B,f)-structures} include

\begin{itemize}%
\item [[Richard Lashof]], \emph{Poincar\'e{} duality and cobordism}, Trans. AMS 109 (1963), 257-277


\item [[Robert Stong]], beginning of chapter II of \emph{Notes on Cobordism theory}, 1968 (\href{http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf}{toc pdf}, \href{http://press.princeton.edu/titles/6465.html}{publisher page})


\item [[Stanley Kochmann]], section 1.4 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996



\end{itemize}
The concept in differential geometry originates around the work of [[Eli Cartan]] ([[Cartan geometry]]) and

\begin{itemize}%
\item [[Shiing-Shen Chern]], \emph{The geometry of G-structures}, Bull. Amer. Math. Soc. 72(2): 167--219. 1966 (\href{http://www.ams.org/journals/bull/1966-72-02/S0002-9904-1966-11473-8/S0002-9904-1966-11473-8.pdf}{pdf}, \href{http://projecteuclid.org/euclid.bams/1183527777}{Euclid})

\end{itemize}
Textbook accounts include

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


\item [[Shoshichi Kobayashi]], [[Katsumi Nomizu]], \emph{Foundations of differential geometry} , Volume 1 (1963), Volume 2 (1969), Interscience Publishers, reprinted 1996 by Wiley Classics Library



\end{itemize}
Surveys include

\begin{itemize}%
\item [[Dmitry Vladimirovich Alekseevsky]], \emph{$G$-structure on a manifold} in M. Hazewinkel (ed.) \emph{Encyclopedia of Mathematics, Volume 4}


\item \href{http://en.wikipedia.org/wiki/G-structure}{Wikipedia}



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

\begin{itemize}%
\item [[Dominic Joyce]], section 2.6 of \emph{Compact manifolds with special holonomy} , Oxford Mathematical Monogrophs (200)

\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}
\hypertarget{in_supergeometry}{}\subsubsection*{{In supergeometry}}\label{in_supergeometry}

Discussion of $G$-structures in [[supergeometry]] includes

\begin{itemize}%
\item [[Dmitri Alekseevsky]], [[Vicente Cortés]], [[Chandrashekar Devchand]], [[Uwe Semmelmann]], \emph{Killing spinors are Killing vector fields in Riemannian Supergeometry} (\href{http://arxiv.org/abs/dg-ga/9704002}{arXiv:dg-ga/9704002})

\end{itemize}
and specifically in [[supergravity]]:

\begin{itemize}%
\item [[George Papadopoulos]], \emph{Geometry and symmetries of null G-structures} (\href{https://arxiv.org/abs/1811.03500}{arXiv:1811.03500})

\end{itemize}
See also at \emph{[[torsion constraints in supergravity]]}.

\hypertarget{in_complex_geometry}{}\subsubsection*{{In complex geometry}}\label{in_complex_geometry}

\begin{itemize}%
\item Sergey Merkulov, \emph{On group theoretic aspects of the non-linear twistor transform}, (\href{http://people.maths.ox.ac.uk/lmason/Tn/40/TN40-07.pdf}{pdf})

\end{itemize}
and his chapter A in

\begin{itemize}%
\item [[Yuri Manin]], \emph{Gauge Field Theory and Complex Geometry}, Springer.


\item Norman Wildberger, \emph{On the complexication of the classical geometries and exceptional numbers}, (\href{http://web.maths.unsw.edu.au/~norman/papers/L.pdf}{pdf})


\item Jun-Muk Hwang, \emph{Rational curves and prolongations of G-structures},\href{https://arxiv.org/abs/1703.03160}{arXiv:1703.03160}



\end{itemize}
\hypertarget{in_higher_geometry}{}\subsubsection*{{In higher geometry}}\label{in_higher_geometry}

Some discussion in [[higher differential geometry]] is in section 4.4.2 of

\begin{itemize}%
\item \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
Formalization in [[modal type theory|modal]] [[homotopy type theory]] is in

\begin{itemize}%
\item [[Felix Wellen]], \emph{[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]}, 2017


\item [[Felix Wellen]], \emph{[[schreiber:Cartan Geometry in Modal Homotopy Type Theory]]} (\href{https://arxiv.org/abs/1806.05966}{arXiv:1806.05966})



\end{itemize}
[[!redirects G-structures]]

[[!redirects (B,f)-structure]] [[!redirects (B,f)-structures]]



\end{document}