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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Cartan connection} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory} [[!include infinity-Chern-Weil theory - contents]] \hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology} [[!include differential cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{traditional}{Traditional}\dotfill \pageref*{traditional} \linebreak \noindent\hyperlink{InTermsOfSmoothModuliStacks}{Synthetically in terms of differential cohesion}\dotfill \pageref*{InTermsOfSmoothModuliStacks} \linebreak \noindent\hyperlink{weaker_definitions_pre_and_semicartan_geometry}{Weaker definitions (pre- and semi-Cartan geometry)}\dotfill \pageref*{weaker_definitions_pre_and_semicartan_geometry} \linebreak \noindent\hyperlink{precartan_geometry}{Pre-Cartan geometry}\dotfill \pageref*{precartan_geometry} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{TableOfExamples}{Table of example}\dotfill \pageref*{TableOfExamples} \linebreak \noindent\hyperlink{pseudoriemannian_geometry}{(pseudo-)Riemannian geometry}\dotfill \pageref*{pseudoriemannian_geometry} \linebreak \noindent\hyperlink{ExampleGStructures}{$G$-Structures}\dotfill \pageref*{ExampleGStructures} \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{Cartan connection} is a \emph{[[principal connection]]} on a [[smooth manifold]] equipped with a certain compatibility condition with the [[tangent bundle]] of the manifold. It combines the concept of [[G-structure]] with that of [[soldering form]]. This combination allows us to express various types of geometric structures on $X$ -- such as notably ([[pseudo-Riemannian metric|pseudo]]-)[[Riemannian geometry]], [[conformal geometry]] and many more (see \hyperlink{TableOfExamples}{below}) -- in terms of [[connection on a bundle|connection]] data, i.e. in terms of [[nonabelian cohomology|nonabelian]] [[differential cohomology]]-data. In particular the [[first order formulation of gravity]] in terms of Cartan connections has been one of the historical motivations (\hyperlink{Cartan23}{Cartan 23}). In a little bit more detail, a Cartan connection on a manifold $X$ for a given [[subgroup]] inclusion $H \hookrightarrow G$ is data that identifies all the [[tangent spaces]] $T_x X$ of $X$ with the tangent space $\mathfrak{g}/\mathfrak{h} = T_{e H} (G/H)$ of the [[coset space]] [[Klein geometry]] $G/H$, such that the choice of these identifications is transported along compatibly. Therefore a manifold equipped with a Cartan connection is also called a \emph{[[Cartan geometry]]} (see also there), a generalization (globalization) of the concept of [[Klein geometry]]. In yet a little bit more detail, an \emph{$(H \hookrightarrow G)$-Cartan connection} on $X$ is a $G$-[[principal connection]] on $X$ equipped with a [[reduction of structure groups|reduction of its structure group]] along $H \to G$ and such that the connection 1-form linearly identifies each [[tangent space]] $T_x X$ of $X$ with the tangent space $\mathfrak{g}/\mathfrak{h} = T_{e H} (G/H)$ of the [[coset space]]. \hypertarget{history}{}\subsection*{{History}}\label{history} The concept essentially originates around (\hyperlink{Cartan23}{Cartan 23}), but the formulation in terms of [[principal connections]] and in fact the terminology ``Cartan connection'' is due to [[Charles Ehresmann]] who formulated principal connections as what, in turn, today are called \emph{[[Ehresmann connections]]} (\hyperlink{Ehresmann50}{Ehresmann 50}). In (\hyperlink{Ehresmann50}{Ehresmann 50}) Cartan's ideas are formalized (see \hyperlink{Marle14}{Marle 14, page 9, 10} for review) by saying that an $(G \hookrightarrow H)$-Cartan connection is a $G$-Ehresmann connection on a $G$-[[principal bundle]] $P$ equipped with an $H$-principal subbundle $Q$, such that the restriction of the connection form along this inclusion yields a form that determines an isomorphism of each tangent space of $Q$ with $\mathfrak{g}$. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{traditional}{}\subsubsection*{{Traditional}}\label{traditional} Let $G$ be a [[Lie group]] and $H \hookrightarrow G$ a sub-Lie group. (So that we may think of the [[coset space]] $G/H$ as a [[Klein geometry]].) Write $\mathfrak{h} \hookrightarrow \mathfrak{g}$ for the corresponding [[Lie algebras]]. There are various equivalent forms of the definition of Cartan connections. The following one characterizes it as a $G$-[[principal connection]] equipped with extra [[stuff, structure and property|structure and property]]. \begin{defn} \label{Traditional}\hypertarget{Traditional}{} A $(H \hookrightarrow G)$-Cartan connection over a [[smooth manifold]] $X$ is; \begin{itemize}% \item a $G$-[[principal connection]] $\nabla$ on $X$; \item such that \begin{enumerate}% \item there is a [[reduction of structure groups]] along $H \hookrightarrow G$; \item for each point $x \in X$ the canonical composite (for any local trivialization) \begin{displaymath} T_x X \stackrel{\nabla}{\to} \mathfrak{g} \to \mathfrak{g}/\mathfrak{h} \end{displaymath} is an [[isomorphism]]. \end{enumerate} \end{itemize} \end{defn} \begin{remark} \label{}\hypertarget{}{} More explicitly for various component-realizations of principal connections, this means the following: \begin{enumerate}% \item In terms of \textbf{[[Ehresmann connection]]-data} def. \ref{Traditional} says that an $(H\hookrightarrow G)$-Cartan connection is an Ehresmann connection form $A$ on a $G$-principal bundle $P$ together with an $H$-principal bundle $Q$ and an $H$-equivariant map $i \colon Q\to P$ such that $i^\ast A$ yields an isomorphism $T Q \simeq Q\times \mathfrak{g}$. In this form the definition is due to (\hyperlink{Ehresmann50}{Ehresmann 50}), recalled in (\hyperlink{Marle14}{Marle 14, def. 4}). Often this definition is stated by describing $i^\ast A$ directly without mentioning of $A$, e.g. (\hyperlink{Sharpe}{Sharpe 97, section 5.3, def. 3.1}, \hyperlink{CapSlovak09}{Cap-Slov\'a{}k 09, 1.5.1}). Beware that $A$ is a [[principal connection]] but $i^\ast A$ is not. \item In terms of \textbf{[[Cech cohomology|Cech]] [[cocycle]] data}, def. \ref{Traditional} says that an $(H\hookrightarrow G)$-Cartan connection is a cover $\{U_i \to X\}$ equipped with 1-forms $A_i \in \Omega^1(U_i, \mathfrak{g})$ and with transition functions $h_{i j} \in C^\infty(U_i \cap U_j, H)$ such that \begin{itemize}% \item $h_{i j} h_{j k} = h_{i k}$ on $U_i \cap U_j \cap U_k$; \item $A_j = h_{i j}^{-1}(A_i + \mathbf{d})h_{i j}$ on $U_i \cap U_j$; \item $A_i(-)_x \colon T_x U_i \stackrel{\simeq}{\longrightarrow} \mathfrak{g}/\mathfrak{h}$. \end{itemize} In this form, the definition appears in (\hyperlink{Sharpe}{Sharpe 97, section 5.1 def. 1.3 together with section 5.2}, \hyperlink{CapSlovak09}{Cap-Slov\'a{}k 09, 1.5.4}). \end{enumerate} \end{remark} See also \href{http://en.wikipedia.org/wiki/Cartan_connection#Cartan_connections_as_principal_connections}{Wikipedia -- Cartan connection -- As principal connections}. \begin{defn} \label{Torsion}\hypertarget{Torsion}{} Given a Cartan connection $\nabla$, def. \ref{Traditional}, its [[torsion of a Cartan connection]] is the image of its [[curvature]] under the projection $\mathfrak{g} \to \mathfrak{g}/\mathfrak{h}$. \end{defn} (\hyperlink{Sharpe}{Sharpe, section 5.3, below def. 3.1}, \hyperlink{CapSlovak09}{Cap-Slov\'a{}k 09, section 1.5.7, p. 85}, \hyperlink{Lott}{Lott 01, section 3}). \begin{remark} \label{}\hypertarget{}{} In the case of vanishing torsion, the resulting [[flat connection|flat]] [[parallel transport]] with values in $G/H$ identifies an [[open neighbourhood]] of each point of $X$ with an open neighbourhood in $G/H$. \end{remark} \begin{remark} \label{}\hypertarget{}{} The last clause in def. \ref{Traditional} says that the [[tangent space]] of $X$ at any point $x$ is being identified with the tangent space of the [[homogeneous space]] $G/H$ at the base point $e H$. This may be visualized by imagining that $X$ ``tangentially touches'' $G/H$ at $x\in X$ and $e H \in G/H$. But by homogeneity, all the tangent spaces of $G/H$ are [[isomorphism|isomorphic]], and canonically so by [[invariant differential form|left translation]]. Hence by the path-lifting property of [[principal connections]], one may (at least for vanishing [[torsion]], def. \ref{Torsion}) visualize the Cartan connection as describing how the $G/H$ touching $X$ at $x$ ``rolls'' along paths (infinitesimal paths, vectors) through $x$. This picture of ``rolling'' is particularly vivid for the case that $(H \hookrightarrow G) = (O(n)\hookrightarrow O(n+1))$ is the inclusion of [[orthogonal groups]], which gives that $G/H = S^n$ is the [[n-sphere]] (for more on this see at \emph{[[conformal connection]]}). This picture of model spaces rolling along was influential in the historical development of the concept of Cartan geometry in the spirit of [[Klein geometry]]. \end{remark} \hypertarget{InTermsOfSmoothModuliStacks}{}\subsubsection*{{Synthetically in terms of differential cohesion}}\label{InTermsOfSmoothModuliStacks} We discuss a [[synthetic mathematics|synthetic]] formulation of Cartan connections in terms of [[differential cohesion]]. \begin{quote}% under construction \end{quote} Write \begin{itemize}% \item $\Omega^1(-,\mathfrak{g})$ for the [[sheaf]] (on the [[site]] of [[formal manifold|formal]] [[smooth manifolds]]) of [[Lie algebra valued differential forms]], regarded as the [[smooth set|smooth]] [[moduli space]] of $\mathfrak{g}$-differential forms (as explained at \emph{[[geometry of physics]]} in the chapter \emph{[[geometry of physics -- differential forms|on differential forms]]}) \item $\mathbf{B} G_{conn} \simeq \Omega^1(-,\mathfrak{g})//G$ for the universal [[moduli stack of connections]], which is equivalently the [[homotopy quotient]] of $\Omega^1(-,\mathfrak{g})$ by the [[action]] of $G$ (regarded as a [[smooth group]]) by [[gauge transformations]]; \item $\Omega^1(-,\mathfrak{g})//H$ for the [[homotopy quotient]] by just the [[subgroup]] $H \hookrightarrow G$; \item $\mathbf{J} \;\colon\;\Omega^1(-,\mathfrak{g})//H \longrightarrow \Omega^1(-,\mathfrak{g})//G\simeq \mathbf{B}G_{conn}$ for the canonical morphism. \end{itemize} \begin{prop} \label{MCFormAsFiberOfDifferentialModuli}\hypertarget{MCFormAsFiberOfDifferentialModuli}{} There is a [[homotopy fiber sequence]] of [[smooth groupoids]] \begin{displaymath} \itexarray{ G/H &\stackrel{\theta/H}{\longrightarrow}& \Omega^1(-,\mathfrak{g})//H \\ && \downarrow^{\mathrlap{\mathbf{J}}} \\ && \mathbf{B}G_{conn} } \,, \end{displaymath} where $\theta/H$ is the $G$-[[Maurer-Cartan form]] modulo $H$. \end{prop} \begin{proof} A detailed proof for the statement as given is spelled out at \emph{\href{orbit+method#ThetaAsHomotopyFiberOfJ}{this proposition}}. But this statement holds generally in [[cohesive (∞,1)-toposes]] and an argument at this generality proceeds as follows: via the discussion at \emph{[[∞-action]]} the action of $G$ on $\Omega^1(-,\mathfrak{g})$ is exhibited by the forgetful map $\mathbf{B}G_{conn}\to \mathbf{B}G$ and since the action of $H$ on $\Omega^1(-,\mathfrak{g})$ is the restricted action, the square on the right of \begin{displaymath} \itexarray{ G/H &\stackrel{\theta/H}{\longrightarrow}& \Omega^1(-,\mathfrak{g})//H &\longrightarrow& \mathbf{B}H \\ \downarrow && \downarrow^{\mathrlap{\mathbf{J}}} && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G_{conn} &\longrightarrow& \mathbf{B}G } \end{displaymath} is a [[homotopy pullback]]. From this the [[pasting law]] implies that in the top left corner we have indeed $G/H$, this being the [[homotopy fiber]] of $\mathbf{B}H \to \mathbf{B}G$. Similar considerations show that the top left map is the abstractly defined [[Maurer-Cartan form]]. \end{proof} \begin{example} \label{}\hypertarget{}{} For $G$ a [[semisimple Lie group|semisimple]] [[compact Lie group|compact]] [[Lie group]] and $H = T\hookrightarrow G$ a [[maximal torus]], then prop. \ref{MCFormAsFiberOfDifferentialModuli} plays a central role in the stacky formulation of the [[orbit method]]. See there at \emph{\href{orbit+method#ThetaAsHomotopyFiberOfJ}{this proposition}}. \end{example} We need this and one more ingredient for synthetically formalizing Cartan connections: \begin{remark} \label{FactorizationOfConnection}\hypertarget{FactorizationOfConnection}{} Let $\mathbb{D}^d_x \hookrightarrow X$ be the first-oder [[infinitesimal neighbourhood]] of a point in a manifold $X$. This being first order means that every [[differential p-form]] for $p \geq 2$ vanishes on $\mathbb{D}^d$. In particular therefore every [[principal connection]] restricted to $\mathbb{D}^d$ becomes a [[flat connection]] and hence is indeed gauge equivalent to the trivial connection. In particular every map \begin{displaymath} \mathbb{D}^d_x \longrightarrow \Omega^1(-,\mathfrak{g})//H \to \mathbf{B}G_{conn} \end{displaymath} has a null-homotopy, hence fits into a square of the form \begin{displaymath} \itexarray{ \mathbb{D}^d_x &\stackrel{\nabla_H}{\longrightarrow}& \Omega^1(-,\mathfrak{g})//H \\ \downarrow &\swArrow_{\mathrlap{\simeq}}& \downarrow^{\mathrlap{\mathbf{J}}} \\ \ast &\longrightarrow& \mathbf{B}G_{conn} } \,. \end{displaymath} It follows by prop. \ref{MCFormAsFiberOfDifferentialModuli} that $\nabla_H$ here factors through the [[Maurer-Cartan form]] \begin{displaymath} \nabla_H|_{\mathbb{D}^d_x} \;\colon\; \mathbb{D}^d_x \stackrel{}{\longrightarrow} G/H \stackrel{\theta/H}{\longrightarrow} \Omega^1(-,\mathfrak{g})//H \,. \end{displaymath} \end{remark} The following is a synthetic formulation of Cartan connections, def. \ref{Traditional}. \begin{defn} \label{CartanConnectionSynthetically}\hypertarget{CartanConnectionSynthetically}{} Let $X$ be a [[smooth set]]. Then an \emph{$(H \hookrightarrow G)$-Cartan connection} on $X$ is \begin{enumerate}% \item a $G$-[[principal connection]] \begin{displaymath} \nabla \colon X \longrightarrow \mathbf{B}G_{conn} \end{displaymath} \item equipped with a [[reduction of structure groups]] given by a lift through $\mathbf{J}$ in prop. \ref{MCFormAsFiberOfDifferentialModuli} \begin{displaymath} \itexarray{ && \Omega^1(-,\mathfrak{g})//H \\ & {}^{\mathllap{\nabla^H}}\nearrow & \downarrow^{\mathrlap{\mathbf{J}}} \\ X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}G_{conn} } \end{displaymath} \item such that over each first-order [[infinitesimal neighbourhood]] $\mathbb{D}^d_x \hookrightarrow X$ any induced factorization, via remark \ref{FactorizationOfConnection}, \begin{displaymath} \mathbb{D}^d_x \stackrel{}{\longrightarrow} G/H \end{displaymath} is [[formally étale morphism|formally étale]]. \end{enumerate} \end{defn} \hypertarget{weaker_definitions_pre_and_semicartan_geometry}{}\subsection*{{Weaker definitions (pre- and semi-Cartan geometry)}}\label{weaker_definitions_pre_and_semicartan_geometry} We discuss here some weakining of the above definition of Cartan connection that have their uses. \hypertarget{precartan_geometry}{}\subsubsection*{{Pre-Cartan geometry}}\label{precartan_geometry} \ldots{}(\hyperlink{Kuranishi95}{Kuranishi 95})\ldots{} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{TableOfExamples}{}\subsubsection*{{Table of example}}\label{TableOfExamples} [[!include local and global geometry - table]] \hypertarget{pseudoriemannian_geometry}{}\subsubsection*{{(pseudo-)Riemannian geometry}}\label{pseudoriemannian_geometry} Let $G = Iso(d,1)$ be the [[Poincare group]] and $H \subset G$ the [[orthogonal group]] $O(d,1)$. Then the quotient \begin{displaymath} \mathfrak{iso}(d,1)/\mathfrak{so}(d,1) \simeq \mathbb{R}^{d+1} \end{displaymath} is [[Lorentzian spacetime]]. Therefore an $(O(d,1)\hookrightarrow Iso(d,1))$-Cartan connection is equivalently an $O(d,1)$-connection on a manifold whose [[tangent space]]s look like [[Minkowski spacetime]]: this is equivalently a [[pseudo-Riemannian manifold]] from the perspective discussed at [[first-order formulation of gravity]]: the $\mathbb{R}^{d+1}$-valued part of the connection is the [[vielbein]]. \hypertarget{ExampleGStructures}{}\subsubsection*{{$G$-Structures}}\label{ExampleGStructures} More generally, [[G-structures]] equipped with compatible principal connections are given by Cartan connections. (We will speak of ``$H$-structure'' here, since the reudced structure will correspond to the group denoted $H$ above, while what is denoted $G$ above will be the semidirect product of $H$ with the [[translation group]]). Let $H \to GL(\mathbb{R}^n)$ be a [[Lie group]] [[homomorphism]], so that [[reduction of the structure group]] of the [[frame bundle]] of a manifold of [[dimension]] $n$ along this map is an [[G-structure|H-structure]] on the manifold. Then write \begin{displaymath} G \coloneqq \mathbb{R}^n \rtimes H \end{displaymath} for the [[semidirect product]] of $H$ with the [[translation group]] $\mathbb{R}^n$, given via the induced [[action]] of $H$ on $\mathbb{R}^n$ via the canonical action of the [[general linear group]] $GL(\mathbb{R}^n)$. With this an $(H \hookrightarrow G)= (H \hookrightarrow \mathbb{R}^n \rtimes H)$-Cartan connection is equivalently an [[G-structure|H-structure]] equipped with a [[vielbein field]] and with an $H$-[[principal connection]]. (\hyperlink{CapSlovak09}{CapSlovak 09, section 1.3.6 and 1.6.1}) With this identification the [[torsion of a Cartan connection]] maps into the [[torsion of a G-structure]]. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[connection on a bundle]] \begin{itemize}% \item [[parallel transport]], [[holonomy]] \end{itemize} \item [[principal connection]] \begin{itemize}% \item [[affine connection]], [[Levi-Civita connection]], \textbf{Cartan connection} \end{itemize} \item [[connection on a 2-bundle]] \item [[connection on an ∞-bundle]] \begin{itemize}% \item [[higher Cartan geometry|∞-Cartan connection]] \item [[higher parallel transport]] \end{itemize} \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} The idea originates in [[Élie Cartan]]`s ``method of moving frames'' (cf. [[Cartan geometry]]). \begin{itemize}% \item [[Élie Cartan]] \emph{Sur les vari\'e{}t\'e{}s \`a{} connexion affine et la th\'e{}orie de la relativit\'e{} g\'e{}n\'e{}ralis\'e{}e (premi\`e{}re partie)}. Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 3, 40 (1923), p. 325-412 (\href{http://www.numdam.org/item?id=ASENS_1923_3_40__325_0}{NUMDAM}) \end{itemize} The formalization in terms of [[principal connections]] (in their incarnation as [[Ehresmann connections]]) is due to \begin{itemize}% \item [[Charles Ehresmann]], \emph{Les connexions infinitesimales dans un espace fibre diff'erentiable}, Colloque de topologie de Bruxelles, 1950, p. 29--55. \end{itemize} reviewed in \begin{itemize}% \item [[Charles-Michel Marle]], \emph{The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles} (\href{http://arxiv.org/abs/1401.8272}{arxiv:1401.8272}) \end{itemize} Textbook accounts include \begin{itemize}% \item R. Sharpe, \emph{Differential Geometry -- Cartan's Generalization of Klein's Erlagen program} Springer (1997) \item [[Andreas ?ap]], [[Jan Slovák]], chapter 1 of \emph{Parabolic Geometries I -- Background and General Theory}, AMS 2009 \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} See also \begin{itemize}% \item [[Masatake Kuranishi]], \emph{CR geometry and Cartan geometry}, Forum mathematicum (1995) Volume: 7, Issue: 2, page 147-206 (\href{https://eudml.org/doc/186418}{EuDML page}, \href{http://gdz.sub.uni-goettingen.de/dms/load/pdf/?PPN=PPN481110151_0007&DMDID=dmdlog11}{page with link to pdf}) \item [[Dmiti Alekseevesky]], [[Peter Michor]], \emph{Differential geometry of Cartan connections} Publ. Math. Debrecen 47/3-4 (1995), 349-375 (\href{http://www.mat.univie.ac.at/~michor/cartan.pdf}{pdf}) \end{itemize} Further discussion of Cartan connections as models for the [[first order formulation of gravity]] is in \begin{itemize}% \item [[Derek Wise]], \emph{MacDowell-Mansouri gravity and Cartan geometry}, Class.Quant.Grav.27:155010,2010 (\href{http://arxiv.org/abs/gr-qc/0611154/}{arXiv:gr-qc/0611154}) \item [[Gabriel Catren]], \emph{Geometrical Foundations of Cartan Gauge Gravity} (\href{http://arxiv.org/abs/1407.7814}{arXiv:1407.7814}) \end{itemize} See also \begin{itemize}% \item wikipedia \href{http://en.wikipedia.org/wiki/Cartan_connection}{Cartan connection} \end{itemize} [[!redirects Cartan connections]] \end{document}