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\begin{document}

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\section*{vielbein}

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

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

[[!include synthetic differential geometry - contents]]

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

[[!include cohomology - 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{InTermsOfCartanGeometry}{In terms of Cartan geometry}\dotfill \pageref*{InTermsOfCartanGeometry} \linebreak
\noindent\hyperlink{InTermsOfOrthogonalStructure}{In terms of orthogonal structure}\dotfill \pageref*{InTermsOfOrthogonalStructure} \linebreak
\noindent\hyperlink{ClassOfTheTangentBundle}{The class of the tangent bundle}\dotfill \pageref*{ClassOfTheTangentBundle} \linebreak
\noindent\hyperlink{ReductionOfTheStructureGroup}{Reduction of the structure group}\dotfill \pageref*{ReductionOfTheStructureGroup} \linebreak
\noindent\hyperlink{ModuliSpaceOfOrhtogonalStructures}{Moduli space of orthogonal structures: twisted cohomology}\dotfill \pageref*{ModuliSpaceOfOrhtogonalStructures} \linebreak
\noindent\hyperlink{moduli_stack_of_orthogonal_structures}{Moduli stack of orthogonal structures}\dotfill \pageref*{moduli_stack_of_orthogonal_structures} \linebreak
\noindent\hyperlink{differential_refinement_spin_connection}{Differential refinement: Spin connection}\dotfill \pageref*{differential_refinement_spin_connection} \linebreak
\noindent\hyperlink{generalized_and_exceptional_vielbein_fields}{Generalized and exceptional vielbein fields}\dotfill \pageref*{generalized_and_exceptional_vielbein_fields} \linebreak
\noindent\hyperlink{InTermsOfBasicFormsOnFrameBundle}{In terms of basic forms on the frame bundle}\dotfill \pageref*{InTermsOfBasicFormsOnFrameBundle} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{vielbein} or \emph{solder form} on a [[manifold]] $X$ is a linear identification of a [[tangent bundle]] with a [[vector bundle]] with explicit ([[pseudo-orthogonal structure|pseudo-]])[[orthogonal structure]].

Any such choice encodes a ([[pseudo-Riemannian metric|pseudo-]])[[Riemannian metric]] on $X$.

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

There are different equivalent perspectives on the notion of vielbein that are closely related:

\begin{itemize}%
\item \hyperlink{InTermsOfCartanGeometry}{In terms of Cartan geometry}


\item \hyperlink{InTermsOfOrthogonalStructure}{In terms of orthogonal structure}



\end{itemize}
\hypertarget{InTermsOfCartanGeometry}{}\subsubsection*{{In terms of Cartan geometry}}\label{InTermsOfCartanGeometry}

Let $X$ be a [[smooth manifold]] of [[dimension]] $d$. For definitness we assume here that $X$ is [[orientation|oriented]], but this is not necessary.

A [[Lie algebra valued 1-form]]

\begin{displaymath}
(E,\Omega) : T X \to \mathfrak{iso}(d)
\end{displaymath}
with values in the [[Poincaré Lie algebra]] encodes a [[pseudo-Riemannian metric]] on $X$ (if non-degenerate, at least). In this context the component

\begin{displaymath}
E : T X \to \mathbb{R}^d
\end{displaymath}
of the [[connection on a bundle|connection]] 1-form is called the \emph{vielbein} . It encodes the metric by

\begin{displaymath}
g = \langle E \otimes E\rangle  \in Sym^2_{C^\infty(X)} \Gamma(T^* X)
  \,,
\end{displaymath}
where $\langle -,-\rangle : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ is the canonical [[bilinear form]].

In other words, Given an $(SO(d) \hookrightarrow ISO(d))$-[[Cartan connection]] on $X$, the \emph{vielbein} is the isomorphism in the definition of Cartan connection.

For $d=4$ this is the \emph{vierbein} , for $d = 3$ the \emph{dreibein} , etc.

This terminology is used notably in the context of the \emph{[[first-order formulation of gravity]]}.

\hypertarget{InTermsOfOrthogonalStructure}{}\subsubsection*{{In terms of orthogonal structure}}\label{InTermsOfOrthogonalStructure}

We discuss here how a choice of vielbein on a [[manifold]] is equivalently the [[reduction of structure groups|reduction of the structure group]] of the [[tangent bundle]] from the [[general linear group]] $GL(n)$ to its [[maximal compact subgroup]], the [[orthogonal group]].

The following also introduces the description of this in terms of [[smooth infinity-groupoid|smooth]] [[twited cohomology|twisted]] [[cohomology]]. While of course this is not necessary to understand vielbeins, it does give a very natural conceptual description with the advantage that it seamlessly generalizes to notions of \emph{[[generalized vielbein]]} fields and generally to [[twisted differential c-structures]].

\hypertarget{ClassOfTheTangentBundle}{}\paragraph*{{The class of the tangent bundle}}\label{ClassOfTheTangentBundle}

For completeness, we first review how the [[tangent bundle]] of a [[smooth manifold]] is naturally incarnated as a [[cocycle]] in $GL(n)$-valued [[Cech cohomology]] and how this in turn is naturally formulated in terms of [[Lie groupoids]]/[[differentiable stack|smooth]] [[moduli stacks]]. The reader familiar with these basics should skip to the \hyperlink{ReductionOfTheStructureGroup}{next section}.

Let $X$ be a [[smooth manifold]] of [[dimension]] $n$.

By definition this means that there is an [[atlas]] $\{\phi_i^{-1} : \mathbb{R}^n \simeq U_i \hookrightarrow X\}$ of [[coordinate charts]]. On each overlap $U_i \cap U_j$ of two [[coordinate charts]] the [[partial derivatives]] of the corresponding [[coordinate transformations]]

\begin{displaymath}
\phi_j\circ \phi_i^{-1}
  : 
  U_i \cap U_j \subset \mathbb{R}^n \to \mathbb{R}^n
\end{displaymath}
form the [[Jacobian matrix]] of [[smooth functions]]

\begin{displaymath}
((\lambda_{i j})^{\mu}{}_{\mu})
  \coloneqq
  \left[\frac{d}{d x^\nu} \phi_j \circ \phi_i^{-1} (x^\mu)
  \right]
  :
  U_i \cap U_j 
  \to 
  GL_n
\end{displaymath}
with values in invertible [[matrices]], hence in the [[general linear group]] $GL(n)$. By construction (by the [[chain rule]]), these functions satisfy on triple overlaps of coordinate charts the [[matrix product]] equations

\begin{displaymath}
(\lambda_{i j})^\mu{}_\lambda (\lambda_{j k})^\lambda{}_{\nu} 
  = 
  (\lambda_{i k})^\mu{}_{\nu}
  \,,
\end{displaymath}
hence the equation

\begin{displaymath}
\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}
\end{displaymath}
in the [[group]] $C^\infty(U_i \cap U_j, GL(n))$ of smooth $GL(n)$-valued functions on the chart overlap.

This is the [[cocycle]] condition for a smooth [[Cech cohomology|Cech cocycle]] in degree 1 with coefficients in $GL(n)$ (precisely: with coefficients in the [[sheaf]] of smooth functions with values in $GL(n)$ ):

\begin{displaymath}
[(\lambda_{i j})] \in H^1_{smooth}(X, GL_n)
  \,.
\end{displaymath}
It is useful to formulate this statement in the language of [[Lie groupoids]]/[[differentiable stacks]].

\begin{itemize}%
\item $X$ itself is trivially a Lie groupoid $(X \stackrel{\to}{\to} X)$;


\item from the atlas $\{U_i \to X\}$ we get the corresponding [[Cech groupoid]]

\begin{displaymath}
C(\{U_i\}) = 
  (\coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to} \coprod_i U_i)
\end{displaymath}

\item any [[Lie group]] $G$ induces its [[delooping]] [[Lie groupoid]]

\begin{displaymath}
\mathbf{B}G
  = 
  \left(
     G \stackrel{\to}{\to} * 
  \right)
  \,.
\end{displaymath}


\end{itemize}
The above situation is neatly encoded in the existence of a [[diagram]] of Lie groupoids of the form

\begin{displaymath}
\itexarray{
     C(\{U_i\}) &\stackrel{\lambda}{\to}& \mathbf{B} GL(n).
     \\
     {}^{\mathllap{\simeq}}\downarrow
     \\
     X
  }
  \,,
\end{displaymath}
where

\begin{itemize}%
\item the left morphism is [[stalk]]-wise (around small enough [[neighbourhoods]] of each point) an [[equivalence of groupoids]];


\item the horizontal functor has as components the functions $\lambda_{i j}$ and its functoriality is the cocycle condition $\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}$.



\end{itemize}
We want to think of such a diagram as being directly a morphism of [[smooth infinity-groupoid|smooth groupoids]]

\begin{displaymath}
T X : X \to \mathbf{B} GL(n) \;\; \in \mathbf{H}
  \,.
\end{displaymath}
This is true in the [[(2,1)-category]] $\mathbf{H}$ in which stalkwise equivalences $W \subset Mor(PSh(SmthMfd, Grpd))$ have been [[simplicial localization|formally inverted]] to become [[homotopy equivalences]].

Since all real [[vector bundles]] on $X$ are encoded by such morphisms, as are their [[gauge transformations]], we say that $\mathbf{B} GL(n)$ is the [[moduli stack]] for real vector bundles.

Of course there is a ``smaller'' Lie groupoid that also classifies real vector bundles. Passing to this ``smaller'' Lie groupoid is what the choice of vielbein accomplishes, to which we now turn.

\hypertarget{ReductionOfTheStructureGroup}{}\paragraph*{{Reduction of the structure group}}\label{ReductionOfTheStructureGroup}

Consider the defining inclusion of the [[orthogonal group]] into the [[general linear group]]

\begin{displaymath}
O(n) \hookrightarrow GL(n)
  \,.
\end{displaymath}
We may understand this inclusion geometrically in terms of the canonical [[metric]] on $\mathbb{R}^n$, but we may also understand it purely Lie theoretically as the the inclusion of the [[maximal compact subgroup]] of $GL(n)$. This makes manifest that the inclusion is trivial at the level of [[homotopy theory]] (it is a [[homotopy equivalence]]) and hence \emph{only} encodes geometric information.

The inclusion induces a corresponding morphism of moduli stacks

\begin{displaymath}
\mathbf{c} : \mathbf{B} O(n) \to \mathbf{B} GL(n)
  \,.
\end{displaymath}
A choice of [[orthogonal structure]] on $T X$ a [[G-structure]] for $G = O(n)$, hence is a factorization of the above $GL(n)$-valued cocycle through $\mathbf{c}$, up to a smooth [[homotopy]].

\begin{displaymath}
\itexarray{
     X &&\stackrel{h}{\to}&& \mathbf{B} O(n)
     \\
     & {}_{\mathllap{\lambda}}\searrow &\swArrow_{E^{-1}}& \swarrow_{\mathrlap{\mathbf{c}}}
     \\
     && \mathbf{B}GL(n)
  }
  \,.
\end{displaymath}
This consists of two pieces of data

\begin{itemize}%
\item the morphism $h$ is a $O(n)$-valued 1-cocycle -- a collection of \emph{orthogonal transition functions} -- hence on each overlap of coordinate patches a smooth function

\begin{displaymath}
((h_{i j}){}^a{}_b) : U_i \cap U_j \to O(n)
\end{displaymath}
such that

\begin{displaymath}
h_{i j} \cdot h_{j k} = h_{i k}
  \,;
\end{displaymath}

\item the homotopy $E$ is on each chart a function



\end{itemize}
\begin{displaymath}
E_i  = ((E_i)^a{}_\mu) : U_i \to GL(n)
\end{displaymath}
\begin{itemize}%
\item such that on each double overlap it intertwines the transition functions $\lambda$ of the tangent bundle with the new orthogonal transition functions, meaning that the equation

\begin{displaymath}
(E_i)^a{}_{\mu} (\lambda_{i j})^{\mu}{}_\nu
  =
  (h_{i j})^a{}_b (E_j]^b{}_\nu
\end{displaymath}
holds. This exhibits the [[natural transformation|naturality]] [[diagram]] of $E$:

\begin{displaymath}
\itexarray{
     * &\stackrel{E_i}{\to}& * 
     \\
     {}^{\mathllap{h_{i j}}}\downarrow && \downarrow^{\mathrlap{\lambda_{i j}}}
     \\
     * &\stackrel{E_j}{\to}& * 
  }
\end{displaymath}


\end{itemize}
Such a lift $(h,E)$ is an [[orthogonal structure]] on $T X$. The component $E$ is called the corresponding \textbf{vielbein}. It exhibits an [[isomorphism]]

\begin{displaymath}
E : T X \stackrel{\simeq}{\to} V
\end{displaymath}
between a [[vector bundle]] $V \to X$ with [[structure group]] explicitly being the [[orthogonal group]] and the [[tangent bundle]], hence it exhibits the [[reduction of structure groups|reduction of the structure group]] of $T X$ from $GL(n)$ to $O(n)$.

\hypertarget{ModuliSpaceOfOrhtogonalStructures}{}\paragraph*{{Moduli space of orthogonal structures: twisted cohomology}}\label{ModuliSpaceOfOrhtogonalStructures}

In order to understand the space of choices of vielbein fields on a given tangent bundle, hence the \emph{[[moduli space]]} or \emph{[[moduli stack]]} of [[orthogonal structures]]/[[Riemannian metrics]] on $X$, it is useful to first consider the [[homotopy fiber]] of the morphism $\mathbf{c} : \mathbf{B}O(n) \to \mathbf{B}GL(n)$. One finds that this is the [[coset]] $O(n) \backslash GL(n)$. We may think of the [[fiber sequence]]

\begin{displaymath}
\itexarray{  
    GL(n)/O(n) &\to& \mathbf{B} O(n)
    \\
    && \downarrow
    \\
    && \mathbf{B} GL(n)
  }
\end{displaymath}
as being a bundle in $\mathbf{H}$ over the [[moduli stack]] $\mathbf{B}GL(n)$ with typical fiber $GL(n)/O(n)$. It is the smooth [[associated infinity-bundle|associated bundle]] to the smooth [[universal principal bundle|universal GL(n)-bundle]] induced by the canonical [[action]] of $GL(n)$ on $O(n)\backslash GL(n)$.

This means that if the tangent bundle $T X$ is trivializable, then the coset space $O(n)\backslash GL(n)$ is the moduli space for vielbein fields on $T X$, in that the space of these is

\begin{displaymath}
\mathbf{H}(X, O(n)\backslash GL(n)) = C^\infty(X, O(n)\backslashGL(n))
  \,.
\end{displaymath}
However, if $T X$ is not trivial, then this is true only locally: there is then an [[atlas]] $\{U_i \to X\}$ such that restricted to each $U_i$ the moduli space of vielbein fields is $C^\infty(U_i, GL(n)/ O(n))$, but globally these now glue together in a non-trivial way as encoded by the tangent bundle: we may say that

the tangent bundle \emph{twists} the functions $X \to GL(n)/O(n)$. If -- as we may -- we think of an ordinary such function as a cocycle in degree-0 cohomology, then this means that a vielbein is a cocycle in $T X$-\_[[twisted cohomology]]\_ relative to the \emph{twisting coefficient bundle} $\mathbf{c}$.

We can make this more manifest by writing equivalently

\begin{displaymath}
\itexarray{
    O(n)\backslash GL(n) &\to& (O(n)\backslash GL(n)) // GL(n)
    \\
    && \downarrow   
    \\
    && \mathbf{B}GL(n)
  }
  \,,
\end{displaymath}
where now on the right we have inserted the [[fibration]] resolution of the morphism $\mathbf{c}$ as provided by the [[factorization lemma]]: this is the morphism out of the [[action groupoid]] of the action of $GL(n)$ on $O(n)\backslash GL(n)$.

The pullback

\begin{displaymath}
\itexarray{
    T X \times_{GL(n)} (O(n)\backslash GL(n)) 
      &\to&
    O(n)\backslash GL(n) // GL(n)
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{T X}{\to}& \mathbf{B}GL(n)
  }
\end{displaymath}
give the non-linear $T X$-associated bundle whose space of sections is the ``twisted $O(n)\backslash GL(n)$-0-cohomology'', hence the space of inequivalent vielbein fields.

\hypertarget{moduli_stack_of_orthogonal_structures}{}\paragraph*{{Moduli stack of orthogonal structures}}\label{moduli_stack_of_orthogonal_structures}

The above says that the space of vielbein fields is the [[cohomology]] of $T X$ in the [[over-(infinity,1)-topos|slice (2,1)-topos]] $\mathbf{H}_{/\mathbf{B}GL(n)}$ with coefficients in $\mathbf{c} : \mathbf{B}O \to \mathbf{B}GL(n)$

\begin{displaymath}
O Struc_{TX}(X) \simeq \mathbf{H}_{/\mathbf{B}GL(n)}(T X, \mathbf{c})
  \,.
\end{displaymath}
But also this space of choices of vielbein fields has a smooth structure, it is itself a smooth [[moduli stack]]. This is obtained by forming the [[internal hom]] in the slice over $\mathbf{B}GL(n)$ of the [[locally cartesian closed (infinity,1)-category|locally cartesian closed (2,1)-category]] $\mathbf{H}$.

\begin{displaymath}
O \mathbf{Struc}_{T X}(X)
  = 
  [X, \mathbf{B}O]_{\mathbf{B}GL(n)}
\end{displaymath}
\hypertarget{differential_refinement_spin_connection}{}\paragraph*{{Differential refinement: Spin connection}}\label{differential_refinement_spin_connection}

We may further lift this discussion to [[differential cohomology]] to get genuine \emph{differential} $T X$-twisted $\mathbf{c}$-structures.

Write $\mathbf{B}G_{conn}$ for [[groupoid of Lie-algebra valued forms]]. As an object of $\mathbf{H} =$ [[smooth infinity-groupoid|SmoothGrpd]] this the [[moduli stack]] of $G$-[[connection on a bundle|connections]].

The morphism $\mathbf{c}$ has an evident differential refinement

\begin{displaymath}
\mathbf{c}_{conn} : \mathbf{B}O(n)_{conn} \to \mathbf{B}GL(n)_{conn}
  \,.
\end{displaymath}
The [[homotopy fiber]] of this differential refinement is still the same as before

\begin{displaymath}
\itexarray{
    GL(n)/ O(n) &\to& \mathbf{B} O(n)_{conn}
    \\
    && \downarrow
    \\
    && \mathbf{B} GL(n)_{conn}
  }
  \,,
\end{displaymath}
so that the moduli space of ``differential vielbein fields'' is the same as that of plain vielbein fields.

Consider an [[affine connection]]

\begin{displaymath}
\nabla_{T X} : X \to \mathbf{B}GL(n)
\end{displaymath}
hence a $GL(n)$-[[principal connection]] which locally on out atas is given by the [[Christoffel symbols]]

\begin{displaymath}
\Gamma_i = ((\Gamma_i)_\mu{}{}^{\alpha}{}_\beta)
  \in 
  \Omega^1(U_i, \mathfrak{gl}(n))
  \,.
\end{displaymath}
A lift $(\nabla_V, E)$ in

\begin{displaymath}
\itexarray{
    X &&\stackrel{\nabla_{V}}{\to}&& \mathbf{B}O_{conn}
    \\
    & {}_{\mathllap{\nabla_{T X}}}\searrow 
     &\swArrow_{E^{-1}}& 
    \swarrow_{\mathbf{c}_{conn}}
    \\
    && \mathbf{B}GL(n)_{conn}
  }
\end{displaymath}
is in components a ``[[spin connection]]''

\begin{displaymath}
\omega_\mu = E d E^{-1} + E \Gamma_\mu E^{-1}
\end{displaymath}
\begin{displaymath}
\omega_\mu{}^a{}_b 
   = 
   E^a{}_\nu \partial_\mu E^\nu{}_b
   +
   E^a{}_\nu \Gamma_\mu{}^\nu{}_\lambda E^\lambda{}_b
   \,.
\end{displaymath}
This is the standard formula for the relation between the [[Christoffel symbols]] and the [[spin connection]] in terms of the vielbein.

\hypertarget{generalized_and_exceptional_vielbein_fields}{}\paragraph*{{Generalized and exceptional vielbein fields}}\label{generalized_and_exceptional_vielbein_fields}

The above discussion seamlessly generalizes to many other related cases. For instance

\begin{enumerate}%
\item For the coefficient bundle

\begin{displaymath}
\itexarray{
     O(n)\backslash O(n,n) /O(n) &\to& \mathbf{B} (O(n) \times O(n))
   \\
   && \downarrow
   \\
   && \mathbf{B}O(n,n)
  }
\end{displaymath}
one gets the [[generalized vielbein]] of [[type II geometry]];


\item for the coefficient bundle

\begin{displaymath}
\itexarray{
     H_n\backslash E_n &\to& \mathbf{B} H_n
   \\
   && \downarrow
   \\
   && \mathbf{B} E_n
  }
\end{displaymath}
coming from the inclusion of the [[maximal compact subgroup]] of an [[exceptional Lie group]] one gets generalized vielbein fields for [[exceptional generalized geometry]];


\item for the coefficient bundle

\begin{displaymath}
\itexarray{
    && \mathbf{B} E_8
   \\
   && \downarrow^{\mathbf{a}}
   \\
   && \mathbf{B}^3 U(1)
  }
\end{displaymath}
coming from the second smooth universal [[Chern class]] of [[E8]] one gets part of the geometry of the [[supergravity C-field]]



\end{enumerate}
and so on. More examples are discussed for instance at \emph{[[twisted smooth cohomology in string theory]]}.

\hypertarget{InTermsOfBasicFormsOnFrameBundle}{}\subsubsection*{{In terms of basic forms on the frame bundle}}\label{InTermsOfBasicFormsOnFrameBundle}

A [[G-structure]] on $X$ for $G = O(n)$ the [[orthogonal group]] is equivalently an $O(n)$-[[principal bundle|principal]] subbundle of the [[frame bundle]] $\pi \colon Fr(X)\to X$.

This frame bundle carries a universal ``basic'' $\mathbb{R}^n$-valued differential form

\begin{displaymath}
\tau_{b} \in \Omega^1(Fr(X), \mathbb{R}^n)
\end{displaymath}
defined on a [[tangent vector]] $v\in \Gamma_{f \in Fr(X)}$ by

\begin{displaymath}
\tau_b(v) \coloneqq f^{-1}(d \pi(v))
  \,,
\end{displaymath}
where $d\pi \colon T Fr(X)\to T X$ is the [[differential]] of the bundle projection $\pi$ and $f$ is the given frame regarded as a linear [[isomorphism]] $f\colon \mathbb{R}^n \stackrel{\simeq}{\longrightarrow} T_x X$.

Then given an orthogonal structure in the form of an $O(n)$-subbundle $i \colon Fr_O(X) \hookrightarrow Fr(X)$ and given finally a local section $\sigma$ of $Fr_O(X)$, then the vielbein field with respect to that local trivialization is the [[pullback of differential forms|pullback]] form

\begin{displaymath}
\tau = \sigma^\ast i^\ast \tau_b
  \,.
\end{displaymath}
(exposition of this in the wider context of [[integrability of G-structures]] includes \hyperlink{Lott90}{Lott 90, p. 4}).

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

\begin{itemize}%
\item [[reduction of structure groups]], [[orthogonal structure]]


\item \href{integrability+of+G-structures#ExamplesOrthogonalStructure}{integrability of G-structures -- Examples -- Orthogonal structure}


\item [[super-vielbein]]


\item [[generalized vielbein]], [[exceptional generalized geometry]]



\end{itemize}
See also at [[field (physics)]] the section on \emph{\href{field%20%28physics%29#OrdinaryGravity}{Ordinary gravity}}.

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

Discussion in the general context of [[G-structures]] includes

\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}
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