higher geometry / derived geometry
geometric little (∞,1)-toposes
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The notion of Klein geometry is essentially that of 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 History.
Klein geometries form the local models for Cartan geometries.
For the generalization of Klein geometry to higher category theory see higher Klein geometry.
A 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
is connected. $G$ acts 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 Examples – Stabilizers of shapes / Klein geometry.
In (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
by which the geometric properties of configurations in space remain entirely unchanged
is called the Hauptgruppe, principal group.
Then in (Klein 1872, end of section 5) it says:
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.
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 Stabilizer of shapes – Klein geometry.
The text goes on to argue that spaces of this form $G/Stab_G(S)$ are of fundamental importance:
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.
For $G = E(n)$, the Euclidean group in $n$-dimensions; $H = O(n)$, the orthogonal group; then, $X$ is $n$-dimensional Cartesian space.
Analogously, for $G = Iso(d,1)$ the Poincare group of $(d+1)$-dimensional Minkowski space, and $H = SO(d,1)$ the special orthogonal group of rotations and Lorentz boosts, then $X = \mathbb{R}^{d+1}$ is Minkowski space itself.
Passing to the corresponding Cartan geometry – by what physicists call gauging – yields the first order formulation of gravity.
The notion of Klein geometry goes back to
Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872)
translation by M. W. Haskell, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, retyped pdf, scan of original)
in the context of what came to be known as the Erlangen program.
A review is for instance in