Characterization of reductive Klein geometries

Question

In my struggle to understand Cartan/Klein geometries, I have the intuition that reductive Klein geometries are the link to connect the "classical" differential geometry approach with this "modern" approach. So I want to understand them better.

A Klein geometry $(G,H)$ with $M=G/H$ is called reductive if the Lie algebra $\mathfrak{g}$, seen as an $\operatorname{Ad}(H)$-module, can be decomposed as $$ \mathfrak{g}=\mathfrak{h}\oplus\mathfrak{p}, $$ with $\mathfrak g$ and $\mathfrak h$ being the corresponding Lie algebras.

I interpret a reductive Klein geometry like one in which there is a "canonical sense of translation". Let's focus on a point $x\in M$, we know that a vector $v\in \mathfrak g$ gives rise to a one-parameter subgroup of $G$ and, in particular, a fundamental vector field on $M$ that we can understand like a "little displacement" of $x$ to $x'\in M$. But several different elements of $\mathfrak g$ can give rise to the same "little displacement" at $x$.

If the Klein geometry is reductive you can fix a choice of $\mathfrak p$ and get a projection $f:\mathfrak g \to \mathfrak h$. This way, the Maurer–Cartan form $A$ of $G$ can be split $$ A=(f+\mathrm{id}-f)A=A^{\mathfrak h}+A^{\mathfrak p}. $$ The 1-form $\omega=A^{\mathfrak h}$ is the 1-form of a principal connection on the principal bundle $\pi:G\to G/H$, and $\ker(A^{\mathfrak h})$ describe the horizontal subspaces. For $\pi(p)=x$, the map $d\pi_p:\ker(A^{\mathfrak h}_p)\to T_{x}M$ is an isomorphism, since it is surjective, and dimensions agree. So we have the isomorphism$$ T_{x}M \xrightarrow{d\pi_p^{-1}}\ker(A^{\mathfrak h}_p)\xrightarrow{A^{\mathfrak p}_p} \mathfrak p. $$

The meaning of this isomorphism is to assign a "canonical" one-parameter group of transformations to a little displacement from $x$ to $x'\in M$.

This is clearly visualized for the Euclidean plane: given an infinitesimal translation (a tangent vector) of a point, there are "lots" of transformations tangent to this vector: the translation itself and an infinite number of rotations of different radius. But we have a canonical euclidean transformation associated.

My question is: is this what characterize the reductive Klein geometries? That is, given a Klein geometry in which there are isomorphism from every tangent space of the base manifold to a fixed subspace of $\mathfrak g$, and assuming that these isomorphisms are consistent (in a sense that I can't make precise yet), can we conclude it is a reductive Klein geometry? Can we say, loosely speaking, that reductive Klein geometries are those in which little displacements on $M$ can be associated canonically with special one-parameter subgroups of $G$?

Added: Once I know the answer is yes, I wonder if these "special one-parameter subgroup" will eventually be the geodesics of the geometry (whatever they are in this context). Should I expect geodesics to exist only if we are in a reductive geometry?

Answer 1

Yes. Recall that the Cartan geometry on $M=G/H$ has principal right $H$-bundle $E$ defined to be $E=G$ with bundle map $\pi\colon g \in G\mapsto gH\in G/H=M$.

Recall that any submersion $\pi\colon E\to M$ has a vertical bundle $V\subset TE$, the vector bundle whose each fiber $V_x$ at $x\in E$ is the kernel of $\pi'(x)\colon T_x E\to T_m M$ where $m=\pi(x)$. Recall that a connection on a principal $H$-bundle $\pi\colon E \to M$ is precisely an $H$-invariant splitting $TE=V\oplus W$.

Suppose that we have a principal right $H$-bundle $\pi\colon E\to M$, with an action of a Lie group $G$ by bundle automorphisms, acting transitively on $E$. Any connection $W$ is completely determined by the linear subspace $W_{x_0}\subset T_{x_0}E_H$ at any one given point $x_0\in E$, by $G$-invariance. Let $m_0=\pi(x_0)$. Let $K\subseteq G$ be the stabilizer of $m_0$. Each $k\in K$ moves $x_0$ to some point of the fiber $E_{m_0}$, say to $kx_0$. The group $H$ acts transitively on the fibers of $E\to M$, so some $h\in H$ has inverse $h^{-1}$ which takes $kx_0$ back to $x_0=kx_0h^{-1}$. This maps $K\to H$, easily seen to be a Lie group injection $\phi\colon K\to H$. We thus act on $E$ by a new $K$-action, $k\cdot x=kx\phi(k)^{-1}$, fixing $x_0$. Hence a $G$-invariant connection $W$ is precisely determined by one linear subspace $W_{x_0}$, which must be invariant under the new $K$-action, since it is both left $G$ and right $H$ invariant. Reversing the steps, any $K$-invariant linear subspace $W_{x_0} \subset TE_{x_0}$ complementary to $V_{x_0}$ determines a unique $G$-invariant connection.

Taking the special case of $M=G/H$, $x_0=1\in G$, $m_0=1\cdot H\in G/H$, we find $K=H$, $\phi\colon H\to H$ is the identity map, so invariant connections are identified with $\operatorname{Ad}H$ invariant complements $\mathfrak{p}=W_{x_0}$ to $\mathfrak{h}=V_{x_0}$ in $TE_{x_0}=\mathfrak{g}$. So a reductive homogeneous space, in the sense of Sharpe's book, is precisely a homogeneous space with invariant connection on the bundle $G\to G/H$, i.e. precisely a homogeneous space with an $H$-module decomposition $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{p}$.