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\begin{document}

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\section*{Klein geometry}

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\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{History}{History}\dotfill \pageref*{History} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

The notion of \emph{Klein geometry} is essentially that of \emph{[[homogeneous space]]} ([[coset space]]) $G/H$ in the context of [[differential geometry]]. This is named ``Klein geometry'' due to its central role in [[Felix Klein]]`s [[Erlangen program]], see below at \emph{\hyperlink{History}{History}}.

Klein geometries form the \emph{local models} for [[Cartan geometries]].

For the generalization of Klein geometry to [[higher category theory]] see [[higher Klein geometry]].

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

A \textbf{Klein geometry} is a pair $(G, H)$ where $G$ is a [[Lie group]] and $H$ is a closed Lie subgroup of $G$ such that the (left) [[coset space]]

\begin{displaymath}
X \coloneqq G/H
\end{displaymath}
is [[connected space|connected]]. $G$ [[action|acts]] [[transitive action|transitively]] on the homogeneous space $X$. We may think of $H\hookrightarrow G$ as the [[stabilizer subgroup]] of a point in $X$.

See there at \emph{\href{stabilizer+group#KleinGeometry}{Examples -- Stabilizers of shapes / Klein geometry}}.

\hypertarget{History}{}\subsection*{{History}}\label{History}

In (\hyperlink{Klein1872}{Klein 1872}) (the ``[[Erlangen program]]'') is first of all, in section 1, considered the general idea of (what in modern language one would call) the [[action]] of a [[Lie group]] ``of transformations'' on a [[smooth manifold]]. The group of all such transformations

\begin{quote}%
by which the geometric properties of configurations in space remain entirely unchanged


\end{quote}
is called the \emph{Hauptgruppe}, \emph{principal group}.

Then in (\hyperlink{Klein1872}{Klein 1872, end of section 5}) it says:

\begin{quote}%
Suppose in space some group or other, the principal group for instance, be given. Let us then select a single configuration, say a point, or a straight line, or even an ellipsoid, etc., and apply to it all the transformations of the principal group. We thus obtain an infinite manifoldness with a number of dimensions in general equal to the number of arbitrary parameters contained in the group, but reducing in special cases, namely, when the configuration originally selected has the property of being transformed into itself by an infinite number of the transformations of the group. Every manifoldness generated in this way may be called, with reference to the generating group, a body.


\end{quote}
This means in modern language, that if $G$ is the given group acting on a given space $X$, and if $S \hookrightarrow X$ is a given subspace (a configuration), then the ``body'' generated by this is the [[coset]] $G/Stab_G(S)$ of $G$ by the [[stabilizer subgroup]] $Stab_G(X)$ of that configuration. See also there at \emph{\href{stabilizer%20group#KleinGeometry}{Stabilizer of shapes -- Klein geometry}}.

The text goes on to argue that spaces of this form $G/Stab_G(S)$ are of fundamental importance:

\begin{quote}%
If now we desire to base our investigations upon the group, selecting at the same time certain definite configurations as space-elements, and if we wish to represent uniformly things which are of like characteristics, we must evidently choose our space-elements in such a way that their manifoldness either is itself a body or can be decomposed into bodies.


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

\begin{itemize}%
\item For $G = E(n)$, the [[Euclidean group]] in $n$-dimensions; $H = O(n)$, the [[orthogonal group]]; then, $X$ is $n$-dimensional [[Cartesian space]].


\item Analogously, for $G = Iso(d,1)$ the [[Poincare group]] of $(d+1)$-dimensional [[Minkowski space]], and $H = O(d,1)$ the [[Lorentz group]], then $X = \mathbb{R}^{d+1}$ is [[Minkowski space]] itself.

Passing to the corresponding [[Cartan geometry]] -- by what physicists call \emph{gauging} -- yields the [[first order formulation of gravity]].



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

[[!include local and global geometry - table]]

\begin{itemize}%
\item [[Clifford-Klein space form]]


\item [[spherical space form]]



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

The notion of \emph{Klein geometry} goes back to

\begin{itemize}%
\item [[Felix Klein]], \emph{Vergleichende Betrachtungen \"u{}ber neuere geometrische Forschungen} (1872)

translation by M. W. Haskell, \emph{A comparative review of recent researches in geometry} , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (\href{http://math.ucr.edu/home/baez/erlangen/erlangen_tex.pdf}{retyped pdf}, [[KleinRetyped.pdf:file]], \href{http://math.ucr.edu/home/baez/erlangen/erlangen.pdf}{scan of original})



\end{itemize}
in the context of what came to be known as the [[Erlangen program]].

A review is for instance in

\begin{itemize}%
\item Vladimir Kisil, \emph{Erlangen Programme at Large: An Overview} (\href{http://arxiv.org/abs/1106.1686}{arXiv:1106.1686})

\end{itemize}
[[!redirects Klein geometries]]



\end{document}