Where is the geometry in a Klein geometry?

Suppose we have a Klein geometry $M = G / H$ , so we have a $H$ -principal bundle $π : G \to M$ . Suppose it is a reductive Klein geometry, so we have an $A d (H)$ -invariant decomposition of the Lie algebra $g$ . Then, the Maurer-Cartan form $A$ of $G$ can be accordingly decomposed

g = h \oplus m,

where $m$ is a complementary subspace to $h$ in $g$ that is invariant under the adjoint action of $H$ , i.e., $A d (h) m \subseteq m$ for all $h \in H$ .

Consider the Maurer-Cartan form $A \in Ω^{1} (G, g)$ , an $G$ -equivariant map that associates to each $g \in G$ and $v \in T_{g} G$ an element $A_{g} (v) \in g$ . In this context, one can decompose $A$ in accordance with the Lie algebra decomposition:

A = A_{h} + A_{m},

where $A_{h} \in Ω^{1} (G, h)$ and $A_{m} \in Ω^{1} (G, m)$ are the respective components of the Maurer-Cartan form with respect to the decomposition.

If we fix a particular $m$ and a particular linear isomorphism $i : R^{n} \to m$ we have that given $p \in G$ with $π (p) = x$ the following map is a linear isomorphism

φ_{p} : R^{n} \overset{i}{\to} m \overset{(A_{m})_{p}^{- 1}}{\to} \ker ((A_{h})_{p}) \overset{d π_{p}}{\to} T_{x} M,

since $(A_{m})_{p}$ is a linear isomorphism.

On the other hand, for $M$ we have the frame bundle $π_{F M} : F M \to M$ , which is a principal bundle with the group $G L (n)$ . Recall that $F_{x} M = {b : R^{n} \to T_{x} M, b linear isomorphism}$ .

We can define a map $h : H \to G L (n)$ such that the map:

Φ : G \to F M, Φ (p) = φ_{p}

is a bundle morphism over $M$ that respects the group actions, i.e.:

$π_{F M} \circ Φ = π$ .
$Φ (p g) = Φ (p) \cdot h (g)$ , whenever $p \in G$ and $g$ is in $H$ .

Specifically, set

h (g) = i^{- 1} \circ A d (g) \circ i,

where $A d (g) : m \to m$ is the adjoint action of $g$ . Then, for $v \in R^{n}$

Φ (p g) (v) = φ_{p g} (v) = d π_{p g} \circ (A_{m})_{p g}^{- 1} \circ i (v) .

Observe that $((R_{g^{- 1}})^{*} A_{m})_{p g} = (A_{m})_{p} \circ (R_{g^{- 1}})_{*}$ by definition of pullback, and by the properties of the Maurer Cartan form $((R_{g^{- 1}})^{*} A_{m})_{p g} = A d (g) \circ (A_{m})_{p g}$ , so we have

(A_{m})_{p g}^{- 1} = (R_{g})_{*} \circ (A_{m})_{p}^{- 1} \circ A d (g) .

On the other hand, since $G$ is an $H$ -principal bundle, $d π_{p g} = d π_{p} \circ (R_{g^{- 1}})_{*}$ (see this). So finally we obtain

Φ (p g) (v) = d π_{p} \circ (A_{m})_{p}^{- 1} \circ A d (g) \circ i (v) = φ_{p} (h (g) (v)) = Φ (p) \cdot h (g) (v) .

What we have, then, is nothing but a group reduction of $F M$ , or equivalently, an H-structure on $M$ . Since they are in 1-to-1 relation with sections of $F M / H$ , what we have introduced in $M$ is a choice of a preferred basis in every point $x$ in $M$ . And the set of preferred basis is, in a sense, a geometry in $M$

Examples:

Rn with a connection (differential geometry)
Rn with a volume form
Rn with an orientation