# Intrinsic holonomy and curved cosets of Cartan geometries

Jacob W. Erickson

###### Abstract.

We provide an intrinsic notion of curved cosets for arbitrary Cartan
geometries, simplifying the existing construction of curved orbits for a given
holonomy reduction. To do this, we define an intrinsic holonomy group, which
is shown to coincide precisely with the standard definition of the holonomy
group for Cartan geometries in terms of an associated principal connection.
These curved cosets retain many characteristics of their homogeneous
counterparts, and they behave well under the action of automorphisms. We
conclude the paper by using the machinery developed to generalize the de Rham
decomposition theorem for Riemannian manifolds and give a potentially useful
characterization of inessential automorphism groups for parabolic geometries.

##  1\. Introduction

In motivating the study of Cartan geometries, we are often told that Cartan
geometries resemble Lie groups in a particular way, allowing us to extend the
spirit of Klein’s Erlangen program to geometric structures like Riemannian
manifolds that are not necessarily homogeneous. From this, an astonishingly
extensive portion of the vast menagerie of geometric structures studied in
differential geometry can be brought under a single unified perspective.

The recent renaissance in the study of parabolic geometries over the past few
decades bears testament to the enormous benefit of this frame of mind. For
example, using this resemblance of Cartan geometries to Lie groups, sequences
of natural differential operators analogous to the (dual) Bernstein-Gelfand-
Gelfand resolutions in representation theory were constructed in [11] (later
simplified by [7]) for parabolic geometries of arbitrary type.

And yet, in the long shadow cast by these towering achievements of
differential geometry by way of representation theory, several fundamental
questions still seem astoundingly unexplored. For example, if a Cartan
geometry is like a Lie group, then what do its subgroups look like? Would the
notion of subgroup even be useful for Cartan geometries, let alone make sense?
If so, then how much like actual subgroups can we expect them to behave?

As we will see shortly, the answers to these questions come, at least in part,
from holonomy reductions. These were explored for Cartan geometries in [8],
but while the constructions in that paper were fairly straightforward to
implement in most applications, the approach was extrinsic, relying on an
extension of the Cartan connection to a principal connection on a larger
bundle, and the role of the analogy between Cartan geometries and Lie groups
was, in our opinion, somewhat obscured by this. Thus, the first part of this
paper will be devoted to providing an intrinsic reframing of the machinery of
[8]. To do this, we will give an intrinsic definition of holonomy, then use
this intrinsic definition to specify “subgroups” and “cosets” of the Cartan
geometry.

Excitingly, our notion of intrinsic holonomy allows us to make the analogy
between Cartan geometries and Lie groups quite explicit, and to highlight
this, we will attempt to fully commit to the pretense of doing “Lie theory” on
Cartan geometries. Nearly all results in the main body of the paper are
directly motivated by analogous results for Lie groups, and in some cases,
proofs will deliberately mirror well-known proofs from elementary Lie theory.
We have found that this “pretend Lie theory” perspective is quite appealing to
those not already familiar with Cartan geometries, and hope that presenting
our ideas in this somewhat avant-garde way both clarifies their role in the
subject and encourages others to explore this area of geometry.

We have organized the paper as follows. After giving some preliminary
definitions in Section 2, we start in Section 3 by giving a more natural
definition of holonomy, and then show that this intrinsic holonomy is actually
equivalent to the extrinsic holonomy used in [8]. In Section 4, we introduce
the notion of curved cosets and demonstrate some ways in which they behave
like genuine cosets of Lie groups. Section 5 deals with tractor bundles and
tractor connections, and while none of the results in that section are
particularly new, we think that the perspective used there is aesthetically
pleasant and connects well with the machinery developed in Section 4. The way
that automorphisms of Cartan geometries interact with holonomy and curved
cosets is explored in Section 6. Finally, to exhibit possible applications
beyond those described in [8], we end the paper in Section 7 with a
generalization of the de Rham decomposition theorem for Riemannian manifolds
and a new situationally useful approach to essential automorphisms of
parabolic geometries.

######  Acknowledgements.

The author would like to thank Andreas Čap for his helpful commentary on
several drafts of this paper, in particular for his critique of earlier
drafts.

##  2\. Preliminaries

We begin by recalling some basic definitions from the theory of Cartan
geometries that will be used throughout the paper. The definitions of tractor
connections, automorphisms, extension functors, and completeness, while
standard aspects of the subject, have been deferred to later sections for
clarity of exposition. For a more detailed introduction, the author recommends
[9]. Alternatively, an intuitive but slightly less scrupulous introduction can
be found in [16].

Each Cartan geometry is modeled on a homogeneous geometric structure in the
sense of Klein.

######  Definition 2.1.

A _model_ or _model geometry_ is a pair $(G,H)$, where $G$ is a Lie group and
$H$ is a closed subgroup of $G$ such that $G/H$ is connected.

If $(G,H)$ is a model, then $G$ is a principal $H$-bundle over $G/H$. The
geometric information of the model is encoded by the Maurer-Cartan form
$\omega_{{}_{G}}$ on $G$, which takes each $X_{g}\in T_{g}G$ to
$\operatorname{L}_{g^{-1}*}X_{g}\in\mathfrak{g}$. To extend this perspective,
we replace $G$ with a principal $H$-bundle over a smooth manifold $M$ and
$\omega_{{}_{G}}$ with a $\mathfrak{g}$-valued $1$-form called a _Cartan
connection_ that satisfies a few similar properties.

######  Definition 2.2.

Suppose $M$ is a smooth manifold and $(G,H)$ is a model. Then, a _Cartan
geometry of type $(G,H)$_ over $M$ is a pair $(\mathscr{G},\omega)$, where
$q_{{}_{H}}:\mathscr{G}\to M$ is a principal $H$-bundle over $M$ and $\omega$
is a $\mathfrak{g}$-valued $1$-form on $\mathscr{G}$ such that the following
are satisfied:

  * •

For every $\mathscr{g}\in\mathscr{G}$,
$\omega_{\mathscr{g}}:T_{\mathscr{g}}\mathscr{G}\to\mathfrak{g}$ is a linear
isomorphism;

  * •

For every $h\in H$,
$\operatorname{R}_{h}^{*}\omega=\mathrm{Ad}_{h^{-1}}\omega$ ;

  * •

For every $Y\in\mathfrak{h}$ and $\mathscr{g}\in\mathscr{G}$, the flow of the
vector field $\omega^{-1}(Y)$ starting at $\mathscr{g}$ is given by
$\exp(t\omega^{-1}(Y))\mathscr{g}=\mathscr{g}\exp(tY)$.

This description of geometric structure encompasses many of the various
flavors of modern differential geometry, putting them all under a common
framework. As a motivating example, an $m$-dimensional Riemannian manifold
$(M,\mathrm{g})$ determines a unique (torsion-free) Cartan geometry of type
$(\operatorname{\mathrm{I}}(m),\operatorname{\mathrm{O}}(m))$ over $M$, where
$\operatorname{\mathrm{I}}(m)$ is the isometry group of $m$-dimensional
Euclidean space, and a (torsion-free) Cartan geometry of type
$(\operatorname{\mathrm{I}}(m),\operatorname{\mathrm{O}}(m))$ over $M$
determines a Riemannian metric on $M$ that is unique up to scaling.

Given a model $(G,H)$, a helpful example of a Cartan geometry of type $(G,H)$
to keep in mind is the corresponding _Klein geometry_ $(G,\omega_{{}_{G}})$ of
type $(G,H)$ over $G/H$, which gives the model as a Cartan geometry. When a
Cartan geometry is not equivalent to the Klein geometry, its curvature
indicates how they differ locally.

######  Definition 2.3.

Let $(\mathscr{G},\omega)$ be a Cartan geometry of type $(G,H)$. The
_curvature_ $\Omega$ of $(\mathscr{G},\omega)$ is given by
$\Omega=\mathrm{d}\omega+\frac{1}{2}[\omega,\omega]$. In particular, for
$X,Y\in\mathfrak{g}$, the curvature applied to
$\omega^{-1}(X)\wedge\omega^{-1}(Y)$ is given by

$\displaystyle\Omega(\omega^{-1}(X)\wedge\omega^{-1}(Y))$
$\displaystyle=\mathrm{d}\omega(\omega^{-1}(X)\wedge\omega^{-1}(Y))+[X,Y]$
$\displaystyle=\omega^{-1}(X)\omega(\omega^{-1}(Y))-\omega^{-1}(Y)\omega(\omega^{-1}(X))$
$\displaystyle\hskip 50.00008pt-\omega([\omega^{-1}(X),\omega^{-1}(Y)])+[X,Y]$
$\displaystyle=[X,Y]-\omega([\omega^{-1}(X),\omega^{-1}(Y)]).$

We will frequently use the notation $\Omega^{\omega}(X\wedge
Y):=\Omega(\omega^{-1}(X)\wedge\omega^{-1}(Y))$. When $\Omega$ vanishes
identically, we say the Cartan geometry is _flat_.

When the curvature vanishes everywhere, the Cartan geometry is locally
equivalent to the model geometry. In other words, a flat Cartan geometry is
equivalent to a locally homogeneous geometric structure along the lines of
those used in Thurston’s geometrization conjecture.

One particularly important tool for Cartan geometries is the idea of
development, which allows us to compare paths within a geometry with
corresponding paths within the model.

######  Definition 2.4.

Suppose $\gamma:[0,1]\to\mathscr{G}$ is a path in a Cartan geometry
$(\mathscr{G},\omega)$ of type $(G,H)$. Then, the _development_ of $\gamma$ on
$G$ is the unique path $\gamma_{{}_{G}}:[0,1]\to G$ starting at the identity
in $G$ such that $\gamma^{*}\omega=\gamma_{{}_{G}}^{*}\omega_{{}_{G}}$.

Figure 1. A drawing of a path in the orthonormal frame bundle of the sphere
equipped with a Cartan connection of type
$(\operatorname{\mathrm{I}}(2),\operatorname{\mathrm{O}}(2))$, and its
development in $\operatorname{\mathrm{I}}(2)$ (thought of as the orthonormal
frame bundle over the Euclidean plane)

Intuitively, $\gamma$ gives us instructions for a path on $\mathscr{G}$ in
terms of our Cartan connection, and $\gamma_{{}_{G}}$ is the path we take if
we try to follow these same instructions using the Maurer-Cartan form on $G$,
starting at the identity. For example, if $\gamma$ is already a path on $G$,
then $\gamma_{{}_{G}}$ is just the left-translate
$\operatorname{L}_{\gamma(0)^{-1}}\circ\,\gamma$.

We now turn to the machinery described in [8]. Given a Cartan geometry
$(\mathscr{G},\omega)$ of type $(G,H)$ over $M$, we can construct a principal
$G$-bundle $\widehat{\mathscr{G}}=\mathscr{G}\times_{H}G$ and a principal
connection $\widehat{\omega}$ on $\widehat{\mathscr{G}}$ that is characterized
by $j^{*}\widehat{\omega}=\omega$, where
$j:\mathscr{G}\to\widehat{\mathscr{G}},\leavevmode\nobreak\
\mathscr{g}\mapsto(\mathscr{g},e)$. This allows us to define an extrinsic
notion of holonomy.

######  Definition 2.5.

The _extrinsic holonomy group_ of $(\mathscr{G},\omega)$ at
$\widehat{u}\in\widehat{\mathscr{G}}$ is the holonomy group
$\operatorname{\mathrm{Hol}}_{\widehat{u}}(\widehat{\mathscr{G}},\widehat{\omega})$
of the principal connection $\widehat{\omega}$ at $\widehat{u}$.

In [5] and [16], a more natural definition of holonomy was given, where the
holonomy group was defined to be the subgroup of $G$ whose elements were
developments along loops within the Cartan geometry. While this idea is on the
right track, it has several issues that make it less viable than the extrinsic
holonomy given above. For example, this alternative definition does not
capture the orientation-reversing part of the holonomy of the locally
Euclidean Klein bottle, since the corresponding loops on the Klein bottle do
not lift to loops in the orthonormal frame bundle. In the next section, we
will give a corrected version of this idea.

With holonomy in hand, we can define the notion of holonomy reduction used in
[8].

######  Definition 2.6.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$ over $M$
and $X$ is a homogeneous space for $G$. Then, a _holonomy reduction of
$G$-type $X$_ for $(\mathscr{G},\omega)$ is defined to be a section
$\sigma:M\to\widehat{\mathscr{G}}\times_{G}X$ that is parallel with respect to
the connection induced by $\widehat{\omega}$.

This is called a holonomy reduction because, for $X=G/J$, it is equivalent to
a reduction of structure group on $\widehat{\mathscr{G}}$ from $G$ to $J$ such
that $\widehat{\omega}$ pulls back to a principal $J$-connection under the
corresponding inclusion, and the smallest choice of $J$ is the holonomy group.
In other words, it is equivalent to a holonomy reduction of the extended
principal bundle with the extended principal connection.

Of course, a section $\sigma$ of $\widehat{\mathscr{G}}\times_{G}X$ is
equivalent to a $G$-equivariant map $\sigma:\widehat{\mathscr{G}}\to X$, and
both are equivalent to an $H$-equivariant map $\sigma:\mathscr{G}\to X$; we
will follow the practice of not distinguishing between these maps.

We can now define curved orbits as collections of points whose fibers in
$\mathscr{G}$ have the same image under a holonomy reduction $\sigma$.

######  Definition 2.7.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$ over $M$
together with a holonomy reduction $\sigma:\mathscr{G}\to X$. The _$H$ -type_
of a point $p\in M$ is the image $\sigma(q_{{}_{H}}^{-1}(p))\subseteq X$ of
the fiber of $q_{{}_{H}}:\mathscr{G}\to M$ over $p$, and the collection of all
$p\in M$ with the same $H$-type is called a _curved orbit_.

From this definition, it is not immediately apparent whether a curved orbit is
anything more than a subset of $M$. Nevertheless, it was proved in [8] that
each curved orbit is an initial submanifold of $M$, and that each of them
carries a canonical geometric structure. Later, we will give an alternative
perspective that makes this inherent structure obvious.

##  3\. Intrinsic holonomy

We begin with a corrected definition of holonomy.

######  Definition 3.1.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$ over $M$.
The _(intrinsic) holonomy_ with respect to $\mathscr{g}\in\mathscr{G}$ of a
loop $\gamma:[0,1]\to M$ such that
$\gamma(0)=\gamma(1)=q_{{}_{H}}(\mathscr{g})$ is the element
$\widehat{\gamma}_{{}_{G}}(1)h_{\widehat{\gamma}}\in G$, where
$\widehat{\gamma}$ is a lift of $\gamma$ to $\mathscr{G}$ starting at
$\widehat{\gamma}(0)=\mathscr{g}$ and $h_{\widehat{\gamma}}\in H$ is such that
$\widehat{\gamma}(1)h_{\widehat{\gamma}}=\mathscr{g}=\widehat{\gamma}(0)$. The
_(intrinsic) holonomy group_
$\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)$ at
$\mathscr{g}$ is the subgroup of $G$ consisting of all such holonomies. In
other words,

$\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega):=\left\\{\gamma_{{}_{G}}(1)h\in
G:\begin{matrix}h\in H\text{ and }\gamma:[0,1]\to\mathscr{G}\\\ \text{ such
that }\gamma(0)=\mathscr{g}=\gamma(1)h\end{matrix}\right\\}.$

Note that the holonomy of a loop $\gamma$ in the base manifold only depends on
that loop and the chosen basepoint $\mathscr{g}\in\mathscr{G}$, since if
$\widehat{\gamma}$ is our chosen lift starting at $\mathscr{g}$, then every
other lift starting at $\mathscr{g}$ will be of the form
$\operatorname{R}_{\beta}(\widehat{\gamma})$ for some path $\beta$ in $H$
starting at the identity, and
$\operatorname{R}_{\beta}(\widehat{\gamma})_{{}_{G}}=\operatorname{R}_{\beta}(\widehat{\gamma}_{{}_{G}})$.
Furthermore, this definition remains natural while still fixing the issues in
the definitions given by [5] and [16]. Moreover, it is actually equivalent to
the extrinsic holonomy.

######  Proposition 3.2.

For every Cartan geometry $(\mathscr{G},\omega)$ of type $(G,H)$,

$\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)=\operatorname{\mathrm{Hol}}_{(\mathscr{g},e)}(\widehat{\mathscr{G}},\widehat{\omega}).$

###### Proof.

The horizontal curves of $(\widehat{\mathscr{G}},\widehat{\omega})$ are
precisely those curves that take the form
$t\mapsto(\gamma(t),(\gamma_{{}_{G}}(t))^{-1}g^{\prime})$. Thus, if
$\gamma:[0,1]\to\mathscr{G}$ is a lift of a loop in $M$ and $h\in H$ is such
that $\gamma(0)=\gamma(1)h$, then the horizontal path
$t\mapsto(\gamma(t),(\gamma_{{}_{G}}(t))^{-1})$ starts at $(\gamma(0),e)$ and
ends at

$(\gamma(1),(\gamma_{{}_{G}}(1))^{-1})=(\gamma(1)h,h^{-1}(\gamma_{{}_{G}}(1))^{-1})=(\gamma(0),(\gamma_{{}_{G}}(1)h)^{-1}).$

Since
$\operatorname{\mathrm{Hol}}_{(\mathscr{g},e)}(\widehat{\mathscr{G}},\widehat{\omega})$
is a subgroup,
$(\operatorname{\mathrm{Hol}}_{(\mathscr{g},e)}(\widehat{\mathscr{G}},\widehat{\omega}))^{-1}=\operatorname{\mathrm{Hol}}_{(\mathscr{g},e)}(\widehat{\mathscr{G}},\widehat{\omega})$,
so the extrinsic holonomy group is equal to the intrinsic holonomy group.∎

The straightforwardness of this proof makes it particularly difficult for the
author to believe that our small adjustment to the definition of holonomy used
in [5] and [16] was not present in any of the previous literature that he
could find. Nonetheless, while there are results in the literature111See, for
example, Proposition 2.1.1 of [5], which notes that the subgroups
corresponding to holonomy from contractible loops coincide. that relate the
extrinsic holonomy group to the holonomy group consisting of developments of
closed loops in $\mathscr{G}$ that lift from loops in $M$, our definition does
not seem to have been considered before in the form presented above.

Figure 2. Even if a loop $\gamma$ in $M$ does not lift to a loop in
$\mathscr{G}$, we can still lift to a path $\widehat{\gamma}$ and then “close
the loop” by right translating by an element of $H$. Developing along
$\widehat{\gamma}$ and then right translating by that element of $H$ gives the
intrinsic holonomy of $\gamma$ with respect to $\widehat{\gamma}(0)$.

In a similar way, we can also simplify what we mean by a holonomy reduction.

######  Proposition 3.3.

Suppose $M$ is connected. Then, a choice of basepoint
$\mathscr{e}\in\mathscr{G}$ together with a closed subgroup $J\leq G$
containing the holonomy group
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$ uniquely
determines a holonomy reduction $\sigma:\mathscr{G}\to G/J$. Conversely, a
holonomy reduction $\sigma:\mathscr{G}\to G/J$ determines a set of possible
basepoints in $\mathscr{G}$.

###### Proof.

Parallel sections of $\widehat{\mathscr{G}}\times_{G}(G/J)$ are precisely
those which are constant on the horizontal curves. In other words, all
holonomy reductions are of the form

$\sigma:(\gamma(1)h,g^{\prime})\mapsto(g^{\prime})^{-1}(\gamma_{{}_{G}}(1)h)^{-1}\sigma(\gamma(0),e).$

Thus, if we pick a point $\mathscr{e}\in\mathscr{G}$ and a closed subgroup
$J\geq\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$, and we
set $\sigma(\mathscr{e})=eJ$, then $\sigma$ is simply the map
$\sigma:\gamma(1)h\mapsto(\gamma_{{}_{G}}(1)h)^{-1}J$, where
$\gamma(0)=\mathscr{e}$. Note that this is a well-defined map, since every
element of $\mathscr{G}$ can be written as $\gamma(1)h$ for a path
$\gamma:[0,1]\to\mathscr{G}$ starting at $\mathscr{e}$, and if
$\gamma_{1}(1)h_{1}=\gamma_{2}(1)h_{2}$, then
$\gamma_{1{}_{G}}(1)h_{1}(\gamma_{2{}_{G}}(1)h_{2})^{-1}\in\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)\leq
J$, so

$\sigma(\gamma_{1}(1)h_{1})=(\gamma_{1{}_{G}}(1)h_{1})^{-1}J=(\gamma_{2{}_{G}}(1)h_{2})^{-1}J=\sigma(\gamma_{2}(1)h_{2}).$

Conversely, if $\sigma:\mathscr{G}\to G/J$ is a holonomy reduction, then
$\sigma^{-1}(eJ)$ gives the set of all basepoints which characterize $\sigma$
in the way described above.∎

Note that, if $\sigma^{-1}(eJ)$ is empty, then we can use a $g\in G$ such that
$\sigma^{-1}(g^{-1}J)$ is not empty; the corresponding closed subgroup will
change from $J$ to $g^{-1}Jg$.

It is well-known in the principal bundle context that a choice of closed
subgroup $J$ containing the holonomy group determines a holonomy reduction, so
the existence of a result like Proposition 3.3 is not particularly surprising.
However, viewed through the heuristic analogy where we pretend
$(\mathscr{G},\omega)$ is a Lie group, the idea that a holonomy reduction just
amounts to specifying a basepoint and a particular closed subgroup takes on
additional meaning. Indeed, if we pretend that $\mathscr{e}$ is the “identity
element” of $(\mathscr{G},\omega)$, then this allows us to specify a
“subgroup” $\sigma^{-1}(eJ)$ of $(\mathscr{G},\omega)$, corresponding to
$J\leq G$. In the next section, we will see that the “subgroup”
$\sigma^{-1}(eJ)$ is an example of what we call a _curved coset_.

##  4\. Curved cosets

In this section, we will define curved cosets, describe some of their
properties, and show how they are related to the curved orbits of [8]. To
start, however, we need a way to compare elements of a Cartan geometry with
elements in its corresponding model.

######  Definition 4.1.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$ and
$\mathscr{e},\mathscr{g}\in\mathscr{G}$. We say that $g\in G$ is a
_development of $\mathscr{g}$ from $\mathscr{e}$_ whenever there exists a path
$\gamma:[0,1]\to\mathscr{G}$ and $h\in H$ such that $\gamma(0)=\mathscr{e}$,
$\gamma(1)h=\mathscr{g}$, and $\gamma_{{}_{G}}(1)h=g$. When our choice of
$\mathscr{e}$ is understood, we will just say that $g$ is a development of
$\mathscr{g}$.

Unlike developments of paths, developments of points of $\mathscr{G}$ are not
necessarily unique. Indeed, suppose $\alpha:[0,1]\to\mathscr{G}$ is a lift of
a loop in $M$, so that $\alpha(0)=\mathscr{e}$ and $\alpha(1)=\mathscr{e}k$
for some $k\in H$. Then, if $\gamma:[0,1]\to\mathscr{G}$ is a path with
$\gamma(0)=\mathscr{e}$ and $\gamma(1)h=\mathscr{g}$ for some $h\in H$, then
concatenating $\alpha$ with the right-translate $\operatorname{R}_{k}(\gamma)$
gives a path from $\mathscr{e}$ to $\mathscr{g}h^{-1}k$, so both
$\gamma_{{}_{G}}(1)h$ and
$\alpha_{{}_{G}}(1)(k^{-1}\gamma_{{}_{G}}(1)k)(h^{-1}k)^{-1}=(\alpha_{{}_{G}}(1)k^{-1})(\gamma_{{}_{G}}(1)h)$
are developments of $\mathscr{g}$ from $\mathscr{e}$. More succinctly, if $g$
is a development of $\mathscr{g}$ from $\mathscr{e}$, then so is every element
of $\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)g$.

Thus, in the heuristic analogy between $(\mathscr{G},\omega)$ and $G$, if we
think of $\mathscr{e}$ as the “identity element”, then elements of
$\mathscr{G}$ correspond to elements of $G$, the elements of $G$ to which they
correspond are given by their developments from $\mathscr{e}$, and the
holonomy group $\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$
tells us exactly if and how this correspondence fails to assign a unique group
element.

In the Klein geometry $(G,\omega_{{}_{G}})$ of type $(G,H)$, where the
holonomy group is always trivial, the unique development of $g_{1}\in G$ from
$g_{0}\in G$ is $g_{0}^{-1}g_{1}$, so the (left) coset $g_{0}J$ is precisely
the set of all $g_{1}$ with development from $g_{0}$ contained in $J$.
Analogously, we define _curved (left) cosets_ of a Cartan geometry to be sets
of points with developments contained in a given subgroup of $G$.

######  Definition 4.2.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$. Let
$\mathscr{g}\in\mathscr{G}$ and let $J\leq G$ be a subgroup containing
$\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)$. Then, the
_curved (left) coset_ $\mathscr{g}\bullet J$ through $\mathscr{g}$ is given by
the set of all points in $\mathscr{G}$ with development from $\mathscr{g}$
contained in $J$. In other words,

$\mathscr{g}\bullet J:=\left\\{\gamma(1)h:\leavevmode\nobreak\
\begin{matrix}h\in H\text{ and }\gamma:[0,1]\to\mathscr{G}\text{ such}\\\
\text{that }\gamma(0)=\mathscr{g}\text{ and }\gamma_{{}_{G}}(1)h\in
J\end{matrix}\right\\}.$

While the above definition works without the assumption that $J$ contains
$\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)$,
$\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)$ is the
smallest sensible candidate for a choice of $J$, since by the considerations
above,
$\mathscr{g}\bullet\\{e\\}=\mathscr{g}\bullet\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)$.
Therefore, unless we indicate otherwise, we will assume that curved (left)
cosets $\mathscr{g}\bullet J$ are _well-defined_ , meaning that
$\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)\leq J$.

For a given $\mathscr{g}\in\mathscr{G}$, $\mathscr{g}\bullet J$ being well-
defined does not necessarily imply that $\mathscr{g^{\prime}}\bullet J$ is
well-defined for $\mathscr{g}^{\prime}\not\in\mathscr{g}\bullet J$. For
$\gamma:[0,1]\to\mathscr{G}$ a path starting at $\mathscr{g}$ such that
$\gamma(1)h^{\prime}=\mathscr{g}^{\prime}$, the curved coset
$\mathscr{g}^{\prime}\bullet J$ is well-defined if and only if $J$ contains
$\operatorname{\mathrm{Hol}}_{\mathscr{g}^{\prime}}(\mathscr{G},\omega)=(\gamma_{{}_{G}}(1)h^{\prime})^{-1}\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)\gamma_{{}_{G}}(1)h^{\prime}$.
In other words, $\mathscr{g}^{\prime}\bullet J$ is well-defined for all
$\mathscr{g}^{\prime}\in\mathscr{G}$ if and only if
$\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)$ is contained
in every subgroup that is conjugate to $J$ by an element of the form
$\gamma_{{}_{G}}(1)h^{\prime}$. Because such conditions on the holonomy can be
quite restrictive, we will also introduce a weaker notion of curved coset.

######  Definition 4.3.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$ and
$\mathscr{g}\in\mathscr{G}$. Let $J\leq G$ be a subgroup such that every path
component of $J$ intersects $H$. Then, the _geometric leaf_ $\mathscr{g}\circ
J$ through $\mathscr{g}$ is given by

$\mathscr{g}\circ J:=\left\\{\gamma(1)h:\leavevmode\nobreak\
\begin{matrix}h\in H\cap J\text{ and }\gamma:[0,1]\to\mathscr{G}\text{
such}\\\ \text{that }\gamma(0)=\mathscr{g}\text{ and
}\gamma_{{}_{G}}([0,1])\subseteq J\end{matrix}\right\\}.$

Similar to the above, we will say that $\mathscr{g}\circ J$ is _well-defined_
if and only if every path component of $J$ intersects $H$ and
$\Omega^{\omega}_{\mathscr{g}^{\prime}}(\mathfrak{j}\wedge\mathfrak{j})\subseteq\mathfrak{j}$
for all $\mathscr{g^{\prime}}\in\mathscr{g}\circ J$, where $\Omega^{\omega}$
is the curvature of $(\mathscr{G},\omega)$. Again, we could do without this
assumption, but there is no real benefit to doing so: for $J^{\prime}$ the
subgroup of $J$ generated by $J\cap H$ and the identity path component
$J^{\circ}$, $\mathscr{g}\circ J=\mathscr{g}\circ J^{\prime}$, and since
$\Omega^{\omega}(X\wedge Y)=[X,Y]-\omega([\omega^{-1}(X),\omega^{-1}(Y)])$,
the condition
$\Omega^{\omega}_{\mathscr{g}^{\prime}}(\mathfrak{j}\wedge\mathfrak{j})\subseteq\mathfrak{j}$
for all $\mathscr{g}^{\prime}\in\mathscr{g}\circ J$ just says that
$\mathscr{g}\circ J$ is a union of integral manifolds for the distribution
$\omega^{-1}(\mathfrak{j})$. Thus, unless we indicate otherwise, we will
assume that our geometric leaves are well-defined whenever we refer to them.

As an example, for every $K\leq H$, $\mathscr{g}\circ
K=\mathscr{g}K=\\{\mathscr{g}h:h\in K\\}$ is always well-defined. Moreover, if
we are working over the Klein geometry $(G,\omega_{{}_{G}})$, then the
geometric leaf $g\circ J$ will be the coset $gJ^{\prime}$, where $J^{\prime}$
is the subgroup generated by the identity path component $J^{\circ}$ and
$J\cap H$, so since we assume $g\circ J$ is well-defined, $J^{\prime}=J$ and
$g\circ J=gJ=\\{gj:j\in J\\}$.

From their definitions, it is clear that $\mathscr{g}\bullet J$ and
$\mathscr{g}\circ J$ are initial submanifolds of $\mathscr{G}$ when they are
well-defined and $J$ is a closed subgroup of $G$, and they inherit the
structure of a Cartan geometry of type $(J,H\cap J)$ over the initial
submanifolds $q_{{}_{H\cap J}}(\mathscr{g}\bullet J)$ and $q_{{}_{H\cap
J}}(\mathscr{g}\circ J)$ of $\mathscr{G}/(H\cap J)$, respectively, by pulling
back by the inclusion map. Moreover, if $\mathscr{g}h\in\mathscr{g}\bullet J$
for some $h\in H$, then by definition developments of $\mathscr{g}h$ from
$\mathscr{g}$ are contained in $J$, so $h\in H\cap J$. Thus, we can naturally
identify $q_{{}_{H\cap J}}(\mathscr{g}\bullet J)$ with
$q_{{}_{H}}(\mathscr{g}\bullet J)$.

We can actually say a bit more about the topology of $\mathscr{g}\bullet J$.

######  Proposition 4.4.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$. Let
$\mathscr{g}\in\mathscr{G}$ and let $J\leq G$ be a closed subgroup containing
$\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)$. Then,
$\mathscr{g}\bullet J$ is an embedded closed submanifold of $\mathscr{G}$.

###### Proof.

We will show that $\mathscr{g}\bullet J$ is closed because $J$ is closed, and
then we will show that, just as for Lie groups, $\mathscr{g}\bullet J$ being
closed implies that $\mathscr{g}\bullet J$ is an embedded submanifold.

For a subset $U\subseteq\mathfrak{g}$, we will use the notation
$\exp(\omega^{-1}(U))\mathscr{g}^{\prime}$ to denote the subset
$\\{\exp(\omega^{-1}(X))\mathscr{g}^{\prime}:X\in U\\}\subseteq\mathscr{G}$.
Note that, for $U$ open and sufficiently small,
$\exp(\omega^{-1}(U))\mathscr{g}^{\prime}$ is open. Suppose that, for every
open $U\subseteq\mathfrak{g}$ containing $0$,
$\exp(\omega^{-1}(U))\mathscr{g}^{\prime}\cap\mathscr{g}\bullet
J\neq\emptyset$. Let $g^{\prime}\in G$ be a development of
$\mathscr{g}^{\prime}$ from $\mathscr{g}$. Then, elements of
$\exp(\omega^{-1}(U))\mathscr{g}^{\prime}$ have developments from
$\mathscr{g}$ in $g^{\prime}\exp(U)$, so since
$\exp(\omega^{-1}(U))\mathscr{g}^{\prime}\cap\mathscr{g}\bullet
J\neq\emptyset$, $g^{\prime}\exp(U)\cap J\neq\emptyset$ for every open
$U\subseteq\mathfrak{g}$ containing $0$. Because $J$ is closed, this implies
$g^{\prime}\in J$, hence $\mathscr{g}^{\prime}\in\mathscr{g}\bullet J$. Thus,
$\mathscr{g}\bullet J$ is closed.

The rest of the proof is essentially the same as the standard proof that
closed subgroups are embedded submanifolds. Suppose, by way of contradiction,
that for some $\mathscr{g}^{\prime}\in\mathscr{g}\bullet J$,
$\exp(\omega^{-1}(U))\mathscr{g}^{\prime}\cap\mathscr{g}\bullet
J\not\subseteq\exp(\omega^{-1}(U\cap\mathfrak{j}))\mathscr{g}^{\prime}$ for
all open neighborhoods $U$ of $0$ in $\mathfrak{g}$. Consider a sequence
$U_{1}\supseteq U_{2}\supseteq\cdots$ of neighborhoods of $0$ in
$\mathfrak{g}$ such that $\cap_{i=0}^{\infty}U_{i}=\\{0\\}$. Then, for each
$i$, we get an
$\mathscr{a}_{i}\in(\exp(\omega^{-1}(U_{i}))\mathscr{g}^{\prime}\cap\mathscr{g}\bullet
J)\setminus\exp(\omega^{-1}(U_{i}\cap\mathfrak{j}))\mathscr{g}^{\prime}$ and,
decomposing $\mathfrak{g}$ (as a vector space) as
$\mathfrak{g}=V\oplus\mathfrak{j}$, we can choose $Z_{i}\in\mathfrak{j}$ and
$v_{i}\in V\setminus\\{0\\}$ such that $Z_{i}\rightarrow 0$, $v_{i}\rightarrow
0$, and
$\mathscr{a}_{i}=\exp(\omega^{-1}(Z_{i}))\exp(\omega^{-1}(v_{i}))\mathscr{g}^{\prime}$
for sufficiently small $U_{i}$. Since $\mathscr{a}_{i}\in\mathscr{g}\bullet
J$, we have that
$\exp(\omega^{-1}(v_{i}))\mathscr{g}^{\prime}=\exp(\omega^{-1}(Z_{i}))^{-1}\mathscr{a}_{i}\in\mathscr{g}\bullet
J$.

Give $V$ a norm $\|\cdot\|$. Since $v_{i}$ is nonzero for each $i$, we can
consider $\frac{1}{\|v_{i}\|}v_{i}$, which subconverges to some $v$ of norm
$1$ in $V$ because the unit sphere in $V$ is compact. Since

$\left\lfloor\frac{t}{\|v_{i}\|}\right\rfloor
v_{i}=\left(\left\lfloor\frac{t}{\|v_{i}\|}\right\rfloor\|v_{i}\|\right)\left(\frac{1}{\|v_{i}\|}v_{i}\right)\rightarrow
tv,$

where $\lfloor\cdot\rfloor$ is the floor function, it follows that
$\exp(\lfloor\frac{t}{\|v_{i}\|}\rfloor\omega^{-1}(v_{i}))\mathscr{g}^{\prime}$
subconverges to $\exp(t\omega^{-1}(v))\mathscr{g}^{\prime}$. As shown above,
$\exp(\omega^{-1}(v_{i}))\mathscr{g}^{\prime}\in\mathscr{g}\bullet J$, hence
$\exp(v_{i})\in J$, so

$\exp\left(\left\lfloor\frac{t}{\|v_{i}\|}\right\rfloor\omega^{-1}(v_{i})\right)\mathscr{g}^{\prime}=\exp(\omega^{-1}(v_{i}))^{\lfloor\frac{t}{\|v_{i}\|}\rfloor}\mathscr{g}^{\prime}\in\mathscr{g}\bullet
J.$

But $\mathscr{g}\bullet J$ is closed, so
$\exp(t\omega^{-1}(v))\mathscr{g}^{\prime}\in\mathscr{g}\bullet J$ for all
$t$, so $v\in V\cap\mathfrak{j}=\\{0\\}$, which contradicts $\|v\|=1$. Thus,
for all $\mathscr{g}^{\prime}\in\mathscr{g}\bullet J$,
$\exp(\omega^{-1}(U))\mathscr{g}^{\prime}\cap\mathscr{g}\bullet
J\subseteq\exp(\omega^{-1}(U\cap\mathfrak{j}))\mathscr{g}^{\prime}$ for some
open neighborhood $U$ of $0$ in $\mathfrak{g}$, so $\mathscr{g}\bullet J$ is
an embedded submanifold.∎

If we fix a point $\mathscr{e}\in\mathscr{G}$ that we think of as the
“identity element”, then both $\mathscr{e}\bullet J$ and $\mathscr{e}\circ J$
can reasonably be thought of as “subgroups” of $(\mathscr{G},\omega)$ when
they are well-defined. Moreover, once we have chosen an “identity element” for
$(\mathscr{G},\omega)$, we can assign developments from that point, and using
this, we can define _curved right cosets_ by analogy to $Jg=g(g^{-1}Jg)$.

######  Definition 4.5.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$. Fix a
point $\mathscr{e}\in\mathscr{G}$, and let $J\leq G$ be a subgroup containing
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$. For $g\in G$
a development of $\mathscr{g}$ from $\mathscr{e}$, the _curved (right) coset_
$J\bullet\mathscr{g}$ through $\mathscr{g}\in\mathscr{G}$ is given by

$J\bullet\mathscr{g}:=\mathscr{g}\bullet(g^{-1}Jg).$

Here, again, we might as well always require that $g^{-1}Jg$ contains
$\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)$, since nothing
is gained by assuming otherwise. However, since
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)=g\operatorname{\mathrm{Hol}}_{\mathscr{g}}(\mathscr{G},\omega)g^{-1}$,
this just means that $J$ has to contain the holonomy group at $\mathscr{e}$,
and if this holds, then curved right cosets through $\mathscr{g}$ are well-
defined independent of any condition on $\mathscr{g}$.

Note that, in order to define these curved right cosets, we must choose a
basepoint $\mathscr{e}\in\mathscr{G}$, so by Proposition 3.3, if $J$ is
closed, then the data needed to define a curved right coset is precisely the
data needed to define a holonomy reduction. In terms of the holonomy reduction
$\sigma:\mathscr{G}\to G/J$ induced by the choice of basepoint, if $g$ is a
development of $\mathscr{g}$, then

$\displaystyle J\bullet\mathscr{g}$
$\displaystyle=\mathscr{g}\bullet(g^{-1}Jg)$
$\displaystyle=\left\\{\beta(1)h:\leavevmode\nobreak\ \begin{matrix}h\in
H\text{ and }\beta:[0,1]\to\mathscr{G}\text{ such that}\\\
\beta(0)=\mathscr{g}\text{ and }\beta_{{}_{G}}(1)h\in
g^{-1}Jg\end{matrix}\right\\}$
$\displaystyle=\left\\{\gamma(1)h:\leavevmode\nobreak\ \begin{matrix}h\in
H\text{ and }\gamma:[0,1]\to\mathscr{G}\text{ such that}\\\
\gamma(0)=\mathscr{e}\text{ and }\gamma_{{}_{G}}(1)h\in
Jg\end{matrix}\right\\}$ $\displaystyle=\sigma^{-1}(g^{-1}J).$

Moreover,
$q_{{}_{H}}(J\bullet\mathscr{g})=q_{{}_{H}}((J\bullet\mathscr{g})H)=q_{{}_{H}}(\sigma^{-1}(Hg^{-1}J))$,
and $q_{{}_{H}}(\sigma^{-1}(Hg^{-1}J))$ is the set of all points on the base
manifold with $H$-type $Hg^{-1}J$, so curved orbits are just the projections
of curved right cosets to the base manifold.

Interestingly, curved cosets are so well-behaved that we can even consider the
corresponding quotient manifolds.

######  Proposition 4.6.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$. Fix
$\mathscr{e}\in\mathscr{G}$ and a closed subgroup $J\leq G$ containing
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$. Consider the
map $\pi_{{}_{J}}:\mathscr{G}\to J\backslash\mathscr{G},\mathscr{g}\mapsto
J\bullet\mathscr{g}$, where $J\backslash\mathscr{G}$ is the space of curved
right cosets under the quotient topology given by $\pi_{{}_{J}}$. Then,
$J\backslash\mathscr{G}$ admits a (necessarily unique) smooth manifold
structure such that $\pi_{{}_{J}}$ is a submersion.

###### Proof.

We will use the notation $J\bullet
N=\pi_{{}_{J}}^{-1}(\pi_{{}_{J}}(N))=\cup_{\mathscr{g}\in
N}J\bullet\mathscr{g}$ for subsets $N\subseteq\mathscr{G}$. Given an open
neighborhood $U$ of $0$ in $\mathfrak{g}$ that is small enough that
$\exp(\omega^{-1}(U))\mathscr{g}$ is open and
$\exp(\omega^{-1}(u))\mathscr{g}$ is well-defined for all $u\in U$, consider
$\mathscr{g}^{\prime}\in J\bullet(\exp(\omega^{-1}(U))\mathscr{g})$. Suppose
$g$ is a development of $\mathscr{g}$ and $g^{\prime}$ is a development of
$\mathscr{g}^{\prime}$. For sufficiently small $U^{\prime}$ around $0$ in
$\mathfrak{g}$ such that $\exp(\omega^{-1}(U^{\prime}))\mathscr{g}^{\prime}$
is open and $g^{\prime}\exp(U^{\prime})\subseteq Jg\exp(U)$, we get
$\exp(\omega^{-1}(U^{\prime}))\mathscr{g}^{\prime}\subseteq
J\bullet(\exp(\omega^{-1}(U))\mathscr{g})$, so $\mathscr{g}^{\prime}$ is an
interior point. This means that
$J\bullet(\exp(\omega^{-1}(U))\mathscr{g})=\pi_{{}_{J}}^{-1}(\pi_{{}_{J}}(\exp(\omega^{-1}(U))\mathscr{g}))$
is open in $\mathscr{G}$, so since $J\backslash\mathscr{G}$ has the quotient
topology, $\pi_{{}_{J}}(\exp(\omega^{-1}(U))\mathscr{g})$ is open in
$J\backslash\mathscr{G}$. Thus, $\pi_{{}_{J}}$ is an open map.

Suppose $g$ is a development of $\mathscr{g}$ from $\mathscr{e}$ and
$g^{\prime}$ is a development of $\mathscr{g}^{\prime}$ from $\mathscr{e}$. If
$J\bullet(\exp(\omega^{-1}(U))\mathscr{g})\cap
J\bullet(\exp(\omega^{-1}(V))\mathscr{g}^{\prime})\neq\emptyset$ for all
neighborhoods $U$ and $V$ of $0$ in $\mathfrak{g}$, then the corresponding
developments $Jg\exp(U)\cap Jg^{\prime}\exp(V)\neq\emptyset$ for all
neighborhoods $U$ and $V$ of $0$ in $\mathfrak{g}$, so since $J$ is closed,
$Jg=Jg^{\prime}$, so $J\bullet\mathscr{g}=J\bullet\mathscr{g}^{\prime}$. Thus,
$J\backslash\mathscr{G}$ is Hausdorff.

The rest of this proof is essentially the same as the standard proof for the
homogeneous space $J\backslash G$. Pick a complementary subspace $V$ to
$\mathfrak{j}$ in $\mathfrak{g}$ and $W$ a small enough neighborhood of $0$ in
$V$ so that, for some neighborhood $U$ of $0$ in $\mathfrak{j}$, $\psi:U\times
W\to\psi(U\times W)\subseteq\mathscr{G},\leavevmode\nobreak\
(u,w)\mapsto\exp(\omega^{-1}(w))\exp(\omega^{-1}(u))\mathscr{e}$ is an
embedding. Thus, $\pi_{{}_{J}}\circ\psi|_{\\{0\\}\times W}$ is bijective,
continuous, and open, so its inverse is a chart around $J\bullet\mathscr{e}$,
and since we can push $\psi$ around all of $\mathscr{G}$ using flows by
$\omega$-constant vector fields and right-translating by elements of $H$, this
gives a smooth manifold structure on $J\backslash\mathscr{G}$.∎

In particular, if $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$,
$\mathscr{e}\in\mathscr{G}$, and $J$ is a closed normal subgroup of $G$
containing $\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$,
then $J\backslash\mathscr{G}$ is a principal $(H\cap J)\backslash H$-bundle
and naturally inherits a Cartan connection $\bar{\omega}$ of type
$(J\backslash G,(H\cap J)\backslash H)$ with trivial holonomy, given by
$\pi_{{}_{J}}^{*}\bar{\omega}=\omega+\mathfrak{j}$.

##  5\. Relation to tractor connections

When we want to study Lie groups, it is often useful to look at their
representations. Naturally, then, when we want to study Cartan geometries, it
might be useful to have a corresponding notion of representation. To that end,
suppose we are given a Cartan geometry $(\mathscr{G},\omega)$ of type $(G,H)$
over a manifold $M$ and a finite-dimensional representation
$\rho:G\to\operatorname{\mathrm{GL}}(V)$.

We would like to find a “representation” for $(\mathscr{G},\omega)$
corresponding to $\rho$. If we fix an “identity element”
$\mathscr{e}\in\mathscr{G}$ and
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)\leq\ker(\rho)$,
then we have a convenient answer: define $\rho(\mathscr{g}):=\rho(g)$ for each
$\mathscr{g}\in\mathscr{G}$, where $g\in G$ is some development of
$\mathscr{g}$ from $\mathscr{e}$. Unfortunately, this is generally not a
useful assumption to make for $(\mathscr{G},\omega)$; if $G$ is a connected
centerless simple Lie group, for example, then the only normal subgroups of
$G$ are $\\{e\\}$ and $G$ itself, so if
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$ is nontrivial,
then this will only get us the trivial representation. Thus, while this is a
good answer when it works, we would like something more general that does not
have such stringent requirements on the holonomy.

Of course, the restriction $\rho|_{H}$ of $\rho$ to $H$ is still a
representation. Thus, identifying $H$ with the “subgroup” $\mathscr{e}\circ
H=\mathscr{e}H$, we can try to form the “(co)induced representation” of
$\rho|_{H}$ to $(\mathscr{G},\omega)$, given by the space
$\Gamma(\mathscr{G}\times_{H}V)$ of sections of the associated bundle
$\mathscr{G}\times_{H}V$ over $M$, which we identify with the space of
$H$-equivariant functions from $\mathscr{G}$ to $V$. In the model case, $G$
acts on $\Gamma(G\times_{H}V)$ by
$g\cdot\sigma:=\sigma\circ\operatorname{L}_{g^{-1}}$, and an isomorphic copy
of the representation $\rho$ is contained in $\Gamma(G\times_{H}V)$, given by
the subspace of sections of the form $\widetilde{v}:G\to
V,g\mapsto\rho(g)^{-1}v$ with $v\in V$. This gives an identification of
$G\times_{H}V$ with the trivial bundle $G/H\times V$, where the copy of $\rho$
in $\Gamma(G\times_{H}V)$ is precisely the subspace that is identified with
the constant sections of $G/H\times V$.

In the general case, we lack a meaningful notion of action of
$(\mathscr{G},\omega)$ on $\Gamma(\mathscr{G}\times_{H}V)$, and we do not
necessarily have an isomorphism between $\mathscr{G}\times_{H}V$ and $M\times
V$. However, under the identification of $G/H\times V$ and $G\times_{H}V$ in
the model case, the exterior derivative for $G/H\times V$ determines a
corresponding differential operator
$\nabla^{\rho}:\Gamma(G\times_{H}V)\to\Gamma(T^{\vee}(G/H)\otimes(G\times_{H}V))$
given by, for $X\in T(G/H)$ and $\sigma\in\Gamma(G\times_{H}V)$,

$(\nabla^{\rho}\sigma)(X)=\nabla^{\rho}_{X}\sigma=\widehat{X}(\sigma)+\rho_{*}(\omega_{{}_{G}}(\widehat{X}))\sigma,$

where $\widehat{X}\in TG$ is an arbitrary lift of $X$ to $TG$ and
$\rho_{*}:\mathfrak{g}\to\mathfrak{gl}(V)$ is the Lie algebra homomorphism
induced by $\rho$, and the kernel of this differential operator is the copy of
$\rho$. We can define a corresponding differential operator
$\nabla^{\rho}:\Gamma(\mathscr{G}\times_{H}V)\to\Gamma(T^{\vee}M\otimes(\mathscr{G}\times_{H}V))$
given by, for $X\in TM$ and $\sigma\in\Gamma(\mathscr{G}\times_{H}V)$,

$(\nabla^{\rho}\sigma)(X)=\nabla^{\rho}_{X}\sigma:=\widehat{X}(\sigma)+\rho_{*}(\omega(\widehat{X}))\sigma,$

where $\widehat{X}\in T\mathscr{G}$ is an arbitrary lift of $X$ to
$T\mathscr{G}$ and $\rho_{*}$ is, again, the Lie algebra homomorphism induced
by $\rho$. This operator $\nabla^{\rho}$ is better known as the _tractor
connection_ for $\rho$.

######  Definition 5.1.

Let $\rho:G\to\operatorname{\mathrm{GL}}(V)$ be a representation of $G$. Given
a Cartan geometry $(\mathscr{G},\omega)$ of type $(G,H)$ over $M$, the
_tractor bundle_ corresponding to $\rho$ is defined to be the associated
bundle $\mathscr{G}\times_{H}V$. Identifying sections $\sigma$ of
$\mathscr{G}\times_{H}V$ with $H$-equivariant functions $\sigma:\mathscr{G}\to
V$, the _tractor connection_ $\nabla^{\rho}$ is defined by

$\nabla^{\rho}_{X}\sigma:=\widehat{X}(\sigma)+\rho_{*}(\omega(\widehat{X}))\sigma,$

where $X\in TM$ and $\widehat{X}$ is some lift of $X$ to $T\mathscr{G}$.

Not surprisingly, tractor connections are valuable tools for working with
Cartan geometries, in much the same way that representation theory is useful
for studying Lie groups. As an example, for (torsion-free) Cartan geometries
of type $(\operatorname{\mathrm{I}}(m),\operatorname{\mathrm{O}}(m))$, which
correspond to Riemannian manifolds, the tractor bundle corresponding to the
standard representation of
$\operatorname{\mathrm{I}}(m)\simeq\mathbb{R}^{m}\rtimes\operatorname{\mathrm{O}}(m)$
on $\mathbb{R}^{m}$ is just the tangent bundle of the base manifold, and the
tractor connection is the Levi-Civita connection.

For parabolic geometries, the topic of tractor connections is covered
extensively in [9], and their holonomy (defined in the usual way for vector
bundle connections) has been thoroughly studied in several cases (see [2],
[3], and [4]). Indeed, when thinking of the holonomy group of a Cartan
geometry in the extrinsic sense, as the holonomy group of a principal
connection on an extended principal bundle, most of the results of this
section are almost certainly considered well-known in some form or another.

We can think of either the tractor connection $\nabla^{\rho}$ itself or its
space of parallel sections as the analogue of $\rho$ for
$(\mathscr{G},\omega)$. Indeed, we can explicitly describe these parallel
sections in a way precisely analogous to the copy of $\rho$ in
$\Gamma(G\times_{H}V)$.

######  Lemma 5.2.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$ over $M$
and $\nabla^{\rho}$ is the tractor connection on $\mathscr{G}\times_{H}V$
associated to a representation $\rho:G\to\operatorname{\mathrm{GL}}(V)$. Fix a
point $(\mathscr{e},v)\in\mathscr{G}\times_{H}V$ and a path $\gamma:[0,1]\to
M$ with $\gamma(0)=q_{{}_{H}}(\mathscr{e})$. For
$\widehat{\gamma}:[0,1]\to\mathscr{G}$ a lift of $\gamma$ to $\mathscr{G}$
with $\widehat{\gamma}(0)=\mathscr{e}$, the parallel transport of
$(\mathscr{e},v)$ along $\gamma$ with respect to $\nabla^{\rho}$ is given by
$\|_{\gamma}(\mathscr{e},v):t\mapsto(\widehat{\gamma}(t),\rho(\widehat{\gamma}_{{}_{G}}(t))^{-1}v)$.

###### Proof.

Using the usual abuses of notation for computing parallel transport,

$\displaystyle\nabla^{\rho}_{\dot{\gamma}}\|_{\gamma}(\mathscr{e},v)$
$\displaystyle=\partial_{t}(\rho(\widehat{\gamma}_{{}_{G}})^{-1}v)+\rho_{*}(\omega(\dot{\widehat{\gamma}}))(\rho(\widehat{\gamma}_{{}_{G}})^{-1}v)$
$\displaystyle=-\rho_{*}(\omega(\dot{\widehat{\gamma}}))(\rho(\widehat{\gamma}_{{}_{G}})^{-1}v)+\rho_{*}(\omega(\dot{\widehat{\gamma}}))(\rho(\widehat{\gamma}_{{}_{G}})^{-1}v)=0.\quad\mbox{\qed}$

######  Corollary 5.3.

A section $\sigma\in\Gamma(\mathscr{G}\times_{H}V)$ is parallel with respect
to $\nabla^{\rho}$ if and only if the corresponding $H$-equivariant function
$\sigma:\mathscr{G}\to V$ is of the form
$\sigma:\mathscr{g}\mapsto\rho(\mathscr{g})^{-1}\sigma(\mathscr{e})$, where
$\rho(\mathscr{g})$ is defined to be $\rho(g)$ for some development $g\in G$
of $\mathscr{g}$ from $\mathscr{e}$.

Using Lemma 5.2, we also get an explicit description of the holonomy of a
tractor connection in terms of the holonomy of the associated Cartan geometry.

######  Corollary 5.4.

The isomorphism between the fiber of $\mathscr{G}\times_{H}V$ over
$q_{{}_{H}}(\mathscr{e})$ and $V$ given by identifying
$(\mathscr{e}h,\rho(h)^{-1}v)=(\mathscr{e},v)$ with $v$ induces an isomorphism

$\operatorname{\mathrm{Hol}}_{q_{{}_{H}}(\mathscr{e})}(\mathscr{G}\times_{H}V,\nabla^{\rho})\simeq\rho(\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)).$

It was observed in [8] that parallel sections of tractor bundles determine
holonomy reductions. In our current formulation, this just amounts to the fact
that, for $\widetilde{v}:\mathscr{g}\mapsto\rho(\mathscr{g})^{-1}v$ to be
well-defined, we only need
$\widetilde{v}_{\mathscr{g}}=\rho(g)^{-1}v=\rho(jg)^{-1}v=\rho(g)^{-1}\rho(j)^{-1}v$
for every $j\in\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$
and $g$ a development of $\mathscr{g}$ from $\mathscr{e}$, so $\widetilde{v}$
is well-defined if and only if
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$ is contained
in the stabilizer $\mathrm{Stab}_{G}(v)$. In particular, if
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)\leq\ker(\rho)$,
then we can recover the full copy of $V$ from the beginning of this section.

##  6\. Interactions with automorphisms

More than being just a principal $H$-bundle on which a Cartan connection of
type $(G,H)$ is defined, in the model case the Lie group $G$ is also the group
of symmetries for the geometric structure, acting on itself and the underlying
homogeneous space by left-translation. For general Cartan geometries,
geometric automorphisms act as analogues of these left-translations. In this
section, we will briefly recall some elementary properties of automorphisms of
Cartan geometries and describe how they interact with curved cosets and
holonomy reductions.

######  Definition 6.1.

Given two Cartan geometries $(\mathscr{G},\omega)$ and
$(\mathscr{G}^{\prime},\omega^{\prime})$ of type $(G,H)$, an $H$-equivariant
map $\varphi:\mathscr{G}\to\mathscr{G}^{\prime}$ is a _geometric map_ if and
only if $\varphi^{*}\omega^{\prime}=\omega$. A _geometric isomorphism_ is a
bijective geometric map, and a _(geometric) automorphism_ is a geometric
isomorphism from a Cartan geometry to itself.

We will denote by $\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$ the group
of automorphisms of $(\mathscr{G},\omega)$. Similar to how $a(gb)=(ag)b$ for
$a,b,g\in G$, for $\varphi\in\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$,
$h\in H$, $X\in\mathfrak{g}$, and $\mathscr{g}\in\mathscr{G}$ we get
$\varphi(\mathscr{g}h)=\varphi(\mathscr{g})h$ by $H$-equivariance and
$\varphi(\exp(t\omega^{-1}(X))\mathscr{g})=\exp(t\omega^{-1}(X))\varphi(\mathscr{g})$
because $\varphi^{*}\omega=\omega$. For a Cartan geometry over a connected
base manifold, if we fix a point $\mathscr{e}\in\mathscr{G}$, then every
element of $\mathscr{G}$ is of the form
$(\exp(\omega^{-1}(X_{k}))\cdots\exp(\omega^{-1}(X_{1}))\mathscr{e})h$ for
some $X_{1},\dots,X_{k}\in\mathfrak{g}$ and $h\in H$, so automorphisms
$\varphi\in\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$ are uniquely
determined by the image $\varphi(\mathscr{e})$ of $\mathscr{e}$. In
particular, $\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$ acts freely on
$\mathscr{G}$. We can further use this analogue of associativity to see that
the orbits of $\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$ are embedded
closed submanifolds, so $\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$
inherits a natural Lie group structure from its orbits on $\mathscr{G}$.

For $\varphi\in\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$ and
$\gamma:[0,1]\to\mathscr{G}$ a path in $\mathscr{G}$,
$(\varphi\circ\gamma)^{*}\omega=\gamma^{*}\varphi^{*}\omega=\gamma^{*}\omega$,
so $\gamma_{{}_{G}}=(\varphi\circ\gamma)_{{}_{G}}$. Therefore, if
$\mathscr{e},\mathscr{g}\in\mathscr{G}$, $g\in G$ is a development of
$\mathscr{g}$ from $\mathscr{e}$, and $a\in G$ is a development of
$\varphi(\mathscr{e})$ from $\mathscr{e}$, then $ag$ is a development of
$\varphi(\mathscr{g})$ from $\mathscr{e}$. Heuristically, this means that
automorphisms look like particular left-translations to the developments.

Using the holonomy group, we can even place fairly strong restrictions on what
left-translations an automorphism can look like in this sense.

######  Proposition 6.2.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$ and
$\mathscr{e}\in\mathscr{G}$. Let
$\varphi\in\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$. Then,
$\varphi(\mathscr{e})\in\mathscr{e}\bullet\mathrm{N}_{G}(\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega))$,
where $\mathrm{N}_{G}(J)$ denotes the normalizer of a subgroup $J\leq G$. In
other words, if $\varphi(\mathscr{e})$ develops to $a\in G$ from
$\mathscr{e}$, then
$a\in\mathrm{N}_{G}(\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega))$.

###### Proof.

If $\varphi\in\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$, then

$\displaystyle\operatorname{\mathrm{Hol}}_{\varphi(\mathscr{e})}(\mathscr{G},\omega)$
$\displaystyle=\left\\{\alpha_{{}_{G}}(1)h\in G:\begin{matrix}h\in H\text{ and
}\alpha:[0,1]\to\mathscr{G}\\\ \text{ such that
}\alpha(0)=\varphi(\mathscr{e})=\alpha(1)h\end{matrix}\right\\}$
$\displaystyle=\left\\{(\varphi(\beta))_{{}_{G}}(1)h\in G:\begin{matrix}h\in
H\text{ and }\beta:[0,1]\to\mathscr{G}\\\ \text{ such that
}\beta(0)=\mathscr{e}=\beta(1)h\end{matrix}\right\\}$
$\displaystyle=\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega).$

Thus, if $\varphi(\mathscr{e})$ develops to $a$ from $\mathscr{e}$, then

$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)=\operatorname{\mathrm{Hol}}_{\varphi(\mathscr{e})}(\mathscr{G},\omega)=a^{-1}\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)a.\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{\qed}$

In particular, if a Cartan geometry has a large automorphism group, then the
normalizer of the holonomy group must also be large. Further, if a Cartan
geometry of type $(G,H)$ is homogeneous, meaning that the automorphism group
acts transitively on the base manifold, then the normalizer of the holonomy
group will have an open orbit in $G/H$.

As we might expect, our curved cosets behave well under automorphisms: for
$\varphi\in\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$,
$\mathscr{e}\in\mathscr{G}$, and $J\leq G$ a subgroup containing
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$,

$\displaystyle\varphi(\mathscr{e}\bullet J)$
$\displaystyle=\left\\{\varphi(\gamma(1)h):\leavevmode\nobreak\
\begin{matrix}h\in H\text{ and }\gamma:[0,1]\to\mathscr{G}\text{ such}\\\
\text{that }\gamma(0)=\mathscr{e}\text{ and }\gamma_{{}_{G}}(1)h\in
J\end{matrix}\right\\}$ $\displaystyle=\left\\{\beta(1)h:\leavevmode\nobreak\
\begin{matrix}h\in H\text{ and }\beta:[0,1]\to\mathscr{G}\text{ such}\\\
\text{that }\beta(0)=\varphi(\mathscr{e})\text{ and }\beta_{{}_{G}}(1)h\in
J\end{matrix}\right\\}$ $\displaystyle=\varphi(\mathscr{e})\bullet J,$

in the same way that $a(eJ)=aJ$. However, analogous to how $a(Jg)$ is not
necessarily equal to $J(ag)$ unless $a\in\mathrm{N}_{G}(J)$,
$\varphi(J\bullet\mathscr{g})$ is not necessarily equal to
$J\bullet\varphi(\mathscr{g})$ unless an analogous condition is met.

######  Theorem 6.3.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$,
$\mathscr{e}\in\mathscr{G}$, and $J\leq G$ is a subgroup containing
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$. For
$\varphi\in\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$, if
$\varphi(\mathscr{e})\in\mathscr{e}\bullet\mathrm{N}_{G}(J)$, then
$\varphi(J\bullet\mathscr{g})=J\bullet\varphi(\mathscr{g})$ for all
$\mathscr{g}\in\mathscr{G}$.

###### Proof.

Suppose $a$ is a development of $\varphi(\mathscr{e})$ from $\mathscr{e}$ and
$g$ is a development of $\mathscr{g}$ from $\mathscr{e}$. Then,
$\varphi(\mathscr{g})$ will have development $ag$, so

$J\bullet\varphi(\mathscr{g})=\varphi(\mathscr{g})\bullet((ag)^{-1}J(ag))=\varphi(\mathscr{g})\bullet(g^{-1}(a^{-1}Ja)g)=\varphi((a^{-1}Ja)\bullet\mathscr{g}).$

Thus, if $a^{-1}Ja=J$, then
$J\bullet\varphi(\mathscr{g})=\varphi(J\bullet\mathscr{g})$. Note that
$a^{-1}Ja$ necessarily contains
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$ by Proposition
6.2, so $a^{-1}Ja\bullet\mathscr{g}$ is well-defined.∎

By Proposition 6.2, the above theorem implies that

$\varphi(\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)\bullet\mathscr{g})=\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)\bullet\varphi(\mathscr{g}).$

Since $a^{-1}Ja$ and $J$ are not disjoint, it is possible for
$a^{-1}Ja\bullet\mathscr{g}=J\bullet\mathscr{g}$, and hence
$\varphi(J\bullet\mathscr{g})=J\bullet\varphi(\mathscr{g})$, without
$\varphi(\mathscr{e})\in\mathscr{e}\bullet\mathrm{N}_{G}(J)$. For example,
this occurs for the Cartan geometry of type
$(\mathbb{R}^{2}\rtimes\operatorname{\mathrm{O}}(2),\operatorname{\mathrm{O}}(2))$
inherited by an open unit disk around the origin in the Euclidean plane: for
$J$ the subgroup generated by a translation that moves the origin outside of
the unit disk, $\varphi$ a rotation around the origin by an angle that is not
an integer multiple of $\pi$, and $\mathscr{e}$ corresponding to the image of
the identity element of $\mathbb{R}^{2}\rtimes\operatorname{\mathrm{O}}(2)$
under the restriction to the disk, we get that
$\varphi(\mathscr{e})=\mathscr{e}A$, where $A$ is the element of
$\operatorname{\mathrm{O}}(2)$ corresponding to the chosen rotation around the
origin, but $A$ does not normalize $J$. However, we still get
$\varphi(J\bullet\mathscr{e})=J\bullet\varphi(\mathscr{e})=\\{\varphi(\mathscr{e})\\}$,
essentially because the geometry does not “see” the translations in $J$ from
$\mathscr{e}$.

In contrast, if $aJg\neq Jg$, then $aJg$ and $Jg$ are disjoint, so the
corresponding condition for curved orbits is necessary and sufficient.

######  Theorem 6.4.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$,
$\mathscr{e}\in\mathscr{G}$, and $J\leq G$ is a subgroup containing
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$. For
$\varphi\in\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$ and
$\mathscr{g}\in\mathscr{G}$,
$\varphi(J\bullet\mathscr{g})=J\bullet\mathscr{g}$ if and only if
$\varphi(\mathscr{e})\in\mathscr{e}\bullet J$.

###### Proof.

As we saw in the proof of Theorem 6.3, with $a\in G$ a development of
$\varphi(\mathscr{e})$ and $g\in G$ a development of $\mathscr{g}$,
$\varphi(J\bullet\mathscr{g})=aJa^{-1}\bullet\varphi(\mathscr{g})$, so if
$a\in J$, then $\varphi(J\bullet\mathscr{g})=J\bullet\mathscr{g}$. Conversely,
if $\varphi(J\bullet\mathscr{g})=J\bullet\mathscr{g}$, then every element
$\mathscr{g}^{\prime}\in\varphi(J\bullet\mathscr{g})=J\bullet\mathscr{g}$ has
developments of the form $ajg$ and $j^{\prime}g$ for some $j,j^{\prime}\in J$,
so
$ajg(j^{\prime}g)^{-1}=aj(j^{\prime})^{-1}\in\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$,
hence $a\in J$.∎

One situation where this idea applies particularly well is when the base
manifold itself is a single curved orbit of $J$.

######  Corollary 6.5.

Suppose $q_{{}_{H}}(J\bullet\mathscr{e})=q_{{}_{H}}(\mathscr{G})$. Then, for
$\mathscr{g}\in\mathscr{G}$ with development $g\in G$, the $H$-equivariant
extensions of the automorphisms of the induced Cartan geometry of type
$(g^{-1}Jg,g^{-1}Jg\cap H)$ on $J\bullet\mathscr{g}$ are automorphisms of
$(\mathscr{G},\omega)$, and an automorphism $\varphi$ of
$(\mathscr{G},\omega)$ restricts to an automorphism of $J\bullet\mathscr{g}$
if and only if $\varphi(\mathscr{e})\in\mathscr{e}\bullet J$.

As we will see below, Corollary 6.5 could be remarkably useful, in some
circumstances, for studying essential automorphisms of parabolic geometries.
However, it is already quite useful on its own. It tells us, for example, that
if $Q\subseteq\mathrm{Fr}(M)$ is a first-order $H$-structure, where
$\mathrm{Fr}(M)$ is the frame bundle of $M$, and $\nabla$ is an affine
connection compatible with $Q$, then an automorphism $\varphi:M\to M$ of the
affine manifold $(M,\nabla)$ restricts to an automorphism of the first-order
$H$-structure if and only if, for some frame $u\in Q$, the derivative map
$\varphi_{*q_{{}_{H}}(u)}$ satisfies $\varphi_{*q_{{}_{H}}(u)}\circ u\in Q$.
In other words, $\varphi$ preserves all of $Q$ if and only if it preserves a
single frame of $Q$.

We can also consider _infinitesimal automorphisms_ of $(\mathscr{G},\omega)$,
which are vector fields $\xi$ on $\mathscr{G}$ whose flows are automorphisms
of $(\mathscr{G},\omega)$. Equivalently, $\xi$ is an infinitesimal
automorphism if and only if it is a complete vector field such that
$[\xi,\omega^{-1}(X)]=0$ for all $X\in\mathfrak{g}$ and
$\operatorname{R}_{h*}\xi=\xi$ for all $h\in H$. We will denote the Lie
algebra of infinitesimal automorphisms of $(\mathscr{G},\omega)$ under the
negative of the Lie bracket by $\mathfrak{aut}(\mathscr{G},\omega)$, since
this is precisely the Lie algebra of
$\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$.

Consider the subalgebra
$\mathfrak{z}_{\mathfrak{g}}(\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega))\leq\mathfrak{g}$
of all
$\mathrm{Ad}_{\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)}$-invariant
elements of $\mathfrak{g}$. For
$Z\in\mathfrak{z}_{\mathfrak{g}}(\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega))$,
we can consider the vector field
$\widetilde{Z}:\mathscr{g}\mapsto\omega_{\mathscr{g}}^{-1}(\mathrm{Ad}_{\mathscr{g}^{-1}}Z)$,
where $\mathrm{Ad}_{\mathscr{g}^{-1}}$ is, again, shorthand for
$\mathrm{Ad}_{g^{-1}}$ for some development $g$ of $\mathscr{g}$. By the
considerations of the previous section, these will naturally correspond to
parallel sections of the adjoint tractor bundle
$\mathscr{G}\times_{H}\mathfrak{g}$ with respect to the tractor connection
$\nabla^{\mathrm{Ad}}$. Because developments of $\mathscr{g}$ are all of the
form $\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)g$ whenever
$g$ is a development of $\mathscr{g}$, $\widetilde{Z}$ is well-defined, since
if $jg$ were another development of $\mathscr{g}$, then
$\mathrm{Ad}_{(jg)^{-1}}Z=\mathrm{Ad}_{g^{-1}}(\mathrm{Ad}_{j^{-1}}Z)=\mathrm{Ad}_{g^{-1}}Z$.
On the model geometry $(G,H)$, these are precisely the right-invariant vector
fields, so we will call the vector fields $\widetilde{Z}$ _right-invariant
vector fields_ on $\mathscr{G}$.

Since they coincide in the model geometry, we would like infinitesimal
automorphisms and right-invariant vector fields to be the same thing. We will
soon see that they often are not, but as we should expect, the curvature and
holonomy will tell us exactly how they can differ and constrain how badly they
can fail to coincide.

Given a vector field $\eta$ on $\mathscr{G}$ and $X\in\mathfrak{g}$,

$\displaystyle\Omega(\eta\wedge\omega^{-1}(X))$
$\displaystyle=\mathrm{d}\omega(\eta\wedge\omega^{-1}(X))+[\omega(\eta),X]$
$\displaystyle=\eta\omega(\omega^{-1}(X))-\omega^{-1}(X)\omega(\eta)-\omega([\eta,\omega^{-1}(X)])+[\omega(\eta),X]$
$\displaystyle=-\omega^{-1}(X)\omega(\eta)-\omega([\eta,\omega^{-1}(X)])-[X,\omega(\eta)],$

so

$\omega^{-1}(X)\omega(\eta)=-\left(\mathrm{ad}_{X}\omega(\eta)+\Omega(\eta\wedge\omega^{-1}(X))+\omega([\eta,\omega^{-1}(X)])\right).$

In particular, for $\xi\in\mathfrak{aut}(\mathscr{G},\omega)$,
$[\xi,\omega^{-1}(X)]=0$, so

$\omega^{-1}(X)\omega(\xi)=-(\mathrm{ad}_{X}\omega(\xi)+\Omega(\xi\wedge\omega^{-1}(X))).$

Thus, for $\gamma:[0,1]\to\mathscr{G}$ a path with $\gamma(0)=\mathscr{e}$,
$\omega(\xi_{\gamma(t)})=\mathrm{Ad}_{\gamma_{{}_{G}}(t)^{-1}}\omega(\xi_{\mathscr{e}})$
for all $t\in[0,1]$ if and only if
$\Omega(\xi_{\gamma(t)}\wedge\dot{\gamma}(t))=0$ for all $t\in[0,1]$, so $\xi$
is a right-invariant vector field if and only if
$\Omega(\xi\wedge\omega^{-1}(X))=0$ for all $X\in\mathfrak{g}$.

Similarly, for
$Z\in\mathfrak{z}_{\mathfrak{g}}(\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega))$,
we know that
$\omega^{-1}(X)\omega(\widetilde{Z})=-\mathrm{ad}_{X}\omega(\widetilde{Z})$,
so
$\Omega(\widetilde{Z}\wedge\omega^{-1}(X))+\omega([\widetilde{Z},\omega^{-1}(X)])=0$,
hence $[\widetilde{Z},\omega^{-1}(X)]=0$ if and only if
$\Omega(\widetilde{Z}\wedge\omega^{-1}(X))=0$. Because
$\operatorname{R}_{h*}\widetilde{Z}=\widetilde{Z}$ for all $h\in H$, this
means that $\widetilde{Z}$ is an infinitesimal automorphism if and only if it
is a complete vector field and $\Omega(\widetilde{Z}\wedge\omega^{-1}(X))=0$
for all $X\in\mathfrak{g}$.

Thinking of right-invariant vector fields as parallel sections of the adjoint
tractor connection, this equivalence of infinitesimal automorphisms and right-
invariant vector fields modulo curvature is essentially a consequence of Lemma
1.5.12 in [9].

We can also constrain how far $\omega(\xi_{\mathscr{g}})$ can be from
$\mathrm{Ad}_{\mathscr{g}^{-1}}\omega(\xi_{\mathscr{e}})$ in terms of the
holonomy.

######  Proposition 6.6.

Suppose $\xi\in\mathfrak{aut}(\mathscr{G},\omega)$. Let
$\mathfrak{hol}_{\mathscr{e}}(\mathscr{G},\omega)$ denote the Lie subalgebra
of $\mathfrak{g}$ whose one-parameter subgroups are contained in
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$. Then, if $g$
is a development of $\mathscr{g}$ from $\mathscr{e}$, then

$\omega(\xi_{\mathscr{g}})\in\mathrm{Ad}_{g^{-1}}\left(\omega(\xi_{\mathscr{e}})+\mathfrak{hol}_{\mathscr{e}}(\mathscr{G},\omega)\right).$

###### Proof.

Let $\gamma:\mathbb{R}\to\mathscr{G}$ be the curve
$t\mapsto\exp(t\xi)\mathscr{g}$. We want to show that
$\dot{\gamma}_{{}_{G}}(0)=\omega(\dot{\gamma}(0))\in\mathrm{Ad}_{g^{-1}}(\omega(\xi_{\mathscr{e}})+\mathfrak{hol}_{\mathscr{e}}(\mathscr{G},\omega))$.
Because $\xi$ is an infinitesimal automorphism, we know that
$g\gamma_{{}_{G}}(t)\in\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)\exp(t\omega(\xi_{\mathscr{e}}))g$,
so we get a curve $\gamma_{\operatorname{\mathrm{Hol}}}:t\mapsto
g\gamma_{{}_{G}}(t)g^{-1}\exp(t\omega(\xi_{\mathscr{e}}))^{-1}\in\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)$
such that $\gamma_{\operatorname{\mathrm{Hol}}}(0)=e$ and
$\gamma_{{}_{G}}(t)=g^{-1}(\gamma_{\operatorname{\mathrm{Hol}}}(t)\exp(t\omega(\xi_{\mathscr{e}})))g$,
hence

$\dot{\gamma}_{{}_{G}}(0)=\mathrm{Ad}_{g^{-1}}(\dot{\gamma}_{\operatorname{\mathrm{Hol}}}(0)+\omega(\xi_{\mathscr{e}}))\in\mathrm{Ad}_{g^{-1}}\left(\mathfrak{hol}_{\mathscr{e}}(\mathscr{G},\omega)+\omega(\xi_{\mathscr{e}})\right).\quad\mbox{\qed}$

##  7\. Applications

###  7.1. Product decompositions

We can use the ideas above to generalize the de Rham decomposition theorem for
Riemannian manifolds. To get there, we will need a couple of definitions.

######  Definition 7.1.

Suppose $(G,H)$ and $(Q,K)$ are model geometries. Let $i:H\to K$ be a
homomorphism and $\lambda:\mathfrak{g}\to\mathfrak{q}$ be a linear map such
that $\lambda\circ\mathrm{Ad}_{h}=\mathrm{Ad}_{i(h)}\circ\lambda$ for all
$h\in H$, $\lambda|_{\mathfrak{h}}=i_{*}$, and
$\bar{\lambda}:\mathfrak{g}/\mathfrak{h}\to\mathfrak{q}/\mathfrak{k}$ is a
linear isomorphism. Then, the _extension functor_ corresponding to
$(i,\lambda)$ takes Cartan geometries of type $(G,H)$ to Cartan geometries of
type $(Q,K)$ by

$(\mathscr{G},\omega)\longmapsto(\mathscr{G}\times_{i}K,\omega_{\lambda}),$

where $\omega_{\lambda}$ is uniquely characterized by
$j^{*}\omega_{\lambda}=\lambda(\omega)$, with
$j:\mathscr{g}\mapsto(\mathscr{g},e)$.

Using inclusion maps for both $i$ and $\lambda$ gives a way to reverse a
holonomy reduction on a single curved orbit. That is, if
$(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$, and we make a
holonomy reduction with basepoint $\mathscr{e}\in\mathscr{G}$ and subgroup $J$
such that $q_{{}_{H}}(J\bullet\mathscr{e})=q_{{}_{H}}(\mathscr{G})$, then
$J\bullet\mathscr{e}$ has the structure of a Cartan geometry of type $(J,H\cap
J)$, and the extension functor using inclusion maps recovers
$(\mathscr{G},\omega)$ from $J\bullet\mathscr{e}$.

We can, then, define product geometries as extensions of Cartesian products of
two given geometries.

######  Definition 7.2.

Suppose $(A,P)$, $(B,K)$, and $(G,H)$ are model geometries such that
$i:A\times B\hookrightarrow G$ determines an extension functor from $(A\times
B,P\times K)$ to $(G,H)$. Let $(\mathscr{A},\alpha)$ be a Cartan geometry of
type $(A,P)$ and $(\mathscr{B},\beta)$ be a Cartan geometry of type $(B,K)$.
Then, a Cartan geometry $(\mathscr{G},\omega)$ is the _product of type
$(G,H)$_ of $(\mathscr{A},\alpha)$ and $(\mathscr{B},\beta)$ if and only if it
is geometrically isomorphic to the extension of
$(\mathscr{A}\times\mathscr{B},\alpha+\beta)$ to type $(G,H)$.

Equivalently, we could say that $(\mathscr{G},\omega)$ is the product of type
$(G,H)$ of $(\mathscr{A},\alpha)$ and $(\mathscr{B},\beta)$ if and only if,
for some $\mathscr{e}\in\mathscr{G}$, $\mathscr{e}\bullet AB$ is well-defined,
$q_{{}_{H}}(\mathscr{e}\bullet AB)=q_{{}_{H}}(\mathscr{G})$, and the induced
geometric structure on $\mathscr{e}\bullet AB$ is geometrically isomorphic to
$(\mathscr{A}\times\mathscr{B},\alpha+\beta)$.

For example, the models
$(\operatorname{\mathrm{I}}(m),\operatorname{\mathrm{O}}(m))$ and
$(\operatorname{\mathrm{I}}(n),\operatorname{\mathrm{O}}(n))$ for Riemannian
geometries in dimension $m$ and $n$, respectively, fit together, so that the
inclusion map
$\operatorname{\mathrm{I}}(m)\times\operatorname{\mathrm{I}}(n)\hookrightarrow\operatorname{\mathrm{I}}(m+n)$
gives an extension functor from
$(\operatorname{\mathrm{I}}(m)\times\operatorname{\mathrm{I}}(n),\operatorname{\mathrm{O}}(m)\times\operatorname{\mathrm{O}}(n))$
to $(\operatorname{\mathrm{I}}(m+n),\operatorname{\mathrm{O}}(m+n))$. Thus,
given two Riemannian geometries, we can take their product and extend to a new
Riemannian geometry, which is just the standard Riemannian product.

By vague analogy to how a connected Lie group $G$ with closed subgroups $A$
and $B$ such that $\mathfrak{a}+\mathfrak{b}=\mathfrak{g}$ is isomorphic to
the product $A\times B$ if and only if $A\cap B=[A,B]=\\{e\\}$, the following
theorem gives necessary and sufficient conditions for a Cartan geometry to be
a product.

######  Theorem 7.3.

Suppose $(A,P)$, $(B,K)$, and $(G,H)$ are model geometries such that $A\times
B\hookrightarrow G$ determines an extension functor from $(A\times B,P\times
K)$ to $(G,H)$. Let $(\mathscr{G},\omega)$ be a Cartan geometry of type
$(G,H)$ over a connected smooth manifold $M$, with quotient map
$q_{{}_{H}}:\mathscr{G}\to M$, and fix $\mathscr{e}\in\mathscr{G}$ such that
$\mathscr{e}\circ A$, $\mathscr{e}\circ B$, and $\mathscr{e}\circ AB$ are
well-defined and $q_{{}_{H}}(\mathscr{e}\circ AB)=M$. Then,
$(\mathscr{G},\omega)$ is the product of type $(G,H)$ of $\mathscr{e}\circ A$
and $\mathscr{e}\circ B$ if and only if the following hold:

  * •

$\mathscr{e}\circ A\cap\mathscr{e}\circ B=\\{\mathscr{e}\\}$;

  * •

$\exp(\omega^{-1}(X))\exp(\omega^{-1}(Y))\mathscr{g}=\exp(\omega^{-1}(Y))\exp(\omega^{-1}(X))\mathscr{g}$
for every $X\in\mathfrak{a}$, $Y\in\mathfrak{b}$, and
$\mathscr{g}\in\mathscr{e}\circ AB$ such that either side is well-defined;

  * •

if $\mathscr{g}\in\mathscr{e}\circ AB$ and $\mathscr{g}h\in\mathscr{e}\circ
AB$ for some $h\in H$, then $h\in PK$.

###### Proof.

The forward implication is clear, so we just need to prove that the conditions
are sufficient to give the product structure. To do this, we will define a
geometric map that is clearly an isomorphism if well-defined, then prove that
it is well-defined. For the purpose of clean presentation, we will use the
notation $\exp(\omega^{-1}(\overline{X}))$ to denote a composition
$\exp(\omega^{-1}(X_{k}))\cdots\exp(\omega^{-1}(X_{1}))$ throughout this
proof.

Suppose the conditions given above are satisfied. Since
$q_{{}_{H}}(\mathscr{e}\circ AB)=M$, every $\mathscr{g}\in\mathscr{G}$ is of
the form $\mathscr{g}^{\prime}h$ for some
$\mathscr{g}^{\prime}\in\mathscr{e}\circ AB$ and $h\in H$, and by the
condition on the flows, every $\mathscr{g}^{\prime}\in\mathscr{e}\circ AB$ is
of the form
$(\exp(\omega^{-1}(\overline{X}))\exp(\omega^{-1}(\overline{Y}))\mathscr{e})pk$,
where $p\in P$, $k\in K$, and each $X_{i}\in\mathfrak{a}$ and
$Y_{i}\in\mathfrak{b}$. Therefore, every $\mathscr{g}\in\mathscr{G}$ is of the
form
$(\exp(\omega^{-1}(\overline{X}))\exp(\omega^{-1}(\overline{Y}))\mathscr{e})h$
for some $X_{i}\in\mathfrak{a}$, $Y_{i}\in\mathfrak{b}$, and $h\in H$, so we
can consider a map $\varphi:\mathscr{G}\to(\mathscr{e}\circ
A\times\mathscr{e}\circ B)\times_{PK}H$ given by

$(\exp(\omega^{-1}(\overline{X}))\exp(\omega^{-1}(\overline{Y}))\mathscr{e})h\mapsto(\exp(\omega^{-1}(\overline{X}))\mathscr{e},\exp(\omega^{-1}(\overline{Y}))\mathscr{e},h).$

Let us suppose that $\varphi$ is well-defined. Then, it is $H$-equivariant,
and since the Cartan connections on the geometric leaves are given by taking
the Cartan connection on $\mathscr{G}$ and pulling back by inclusion, it
follows that $\varphi$ is geometric. Moreover, since
$(\mathscr{a},\mathscr{b},h)=(\mathscr{a}^{\prime},\mathscr{b}^{\prime},h^{\prime})$
implies that there are $p\in P$ and $k\in K$ such that $pkh=h^{\prime}$,
$\mathscr{a}p=\mathscr{a}^{\prime}$, and $\mathscr{b}k=\mathscr{b}^{\prime}$,
$\varphi$ is injective, hence bijective. Thus, if we prove $\varphi$ is well-
defined, then it will give us the desired geometric isomorphism.

Suppose

$(\exp(\omega^{-1}(\overline{X}))\exp(\omega^{-1}(\overline{Y}))\mathscr{e})h=(\exp(\omega^{-1}(\overline{X^{\prime}}))\exp(\omega^{-1}(\overline{Y^{\prime}}))\mathscr{e})h^{\prime}.$

Then,

$\exp(\omega^{-1}(\overline{X}))\exp(\omega^{-1}(\overline{Y}))\mathscr{e}=(\exp(\omega^{-1}(\overline{X^{\prime}}))\exp(\omega^{-1}(\overline{Y^{\prime}}))\mathscr{e})h^{\prime}h^{-1},$

so $h^{\prime}h^{-1}\in PK$. Writing $h^{\prime}h^{-1}=pk$ for $p\in P$ and
$k\in K$, we get that

$\displaystyle(\exp(\omega^{-1}(\overline{X}))\exp(\omega^{-1}(\overline{Y}))\mathscr{e})k^{-1}$
$\displaystyle=\exp(\omega^{-1}(\overline{X}))(\exp(\omega^{-1}(\overline{Y}))\mathscr{e})k^{-1}$
$\displaystyle=(\exp(\omega^{-1}(\overline{X^{\prime}}))\exp(\omega^{-1}(\overline{Y^{\prime}}))\mathscr{e})p$
$\displaystyle=\exp(\omega^{-1}(\overline{Y^{\prime}}))(\exp(\omega^{-1}(\overline{X^{\prime}}))\mathscr{e})p,$

so

$\exp(\omega^{-1}(\overline{Y^{\prime}}))^{-1}\left((\exp(\omega^{-1}(\overline{Y}))\mathscr{e})k^{-1}\right)=\exp(\omega^{-1}(\overline{X}))^{-1}\left((\exp(\omega^{-1}(\overline{X^{\prime}}))\mathscr{e})p\right).$

Thus, since

$\exp(\omega^{-1}(\overline{Y^{\prime}}))^{-1}\left((\exp(\omega^{-1}(\overline{Y}))\mathscr{e})k^{-1}\right)\in\mathscr{e}\circ
B$

and

$\exp(\omega^{-1}(\overline{X}))^{-1}\left((\exp(\omega^{-1}(\overline{X^{\prime}}))\mathscr{e})p\right)\in\mathscr{e}\circ
A,$

both are equal to $\mathscr{e}$ because we are assuming $\mathscr{e}\circ
A\cap\mathscr{e}\circ B=\\{\mathscr{e}\\}$. It follows that
$\exp(\omega^{-1}(\overline{X}))\mathscr{e}=(\exp(\omega^{-1}(\overline{X^{\prime}}))\mathscr{e})p$
and
$\exp(\omega^{-1}(\overline{Y}))\mathscr{e}=(\exp(\omega^{-1}(\overline{Y^{\prime}}))\mathscr{e})k$,
hence

$\displaystyle\varphi\left((\exp(\omega^{-1}(\overline{X}))\exp(\omega^{-1}(\overline{Y}))\mathscr{e})h\right)$
$\displaystyle=(\exp(\omega^{-1}(\overline{X}))\mathscr{e},\exp(\omega^{-1}(\overline{Y}))\mathscr{e},h)$
$\displaystyle=((\exp(\omega^{-1}(\overline{X^{\prime}}))\mathscr{e})p,(\exp(\omega^{-1}(\overline{Y^{\prime}}))\mathscr{e})k,h)$
$\displaystyle=(\exp(\omega^{-1}(\overline{X^{\prime}}))\mathscr{e},\exp(\omega^{-1}(\overline{Y^{\prime}}))\mathscr{e},pkh)$
$\displaystyle=(\exp(\omega^{-1}(\overline{X^{\prime}}))\mathscr{e},\exp(\omega^{-1}(\overline{Y^{\prime}}))\mathscr{e},h^{\prime})$
$\displaystyle=\varphi\left((\exp(\omega^{-1}(\overline{X^{\prime}}))\exp(\omega^{-1}(\overline{Y^{\prime}}))\mathscr{e})h^{\prime}\right)\\!.\leavevmode\nobreak\
\mbox{\qed}$

Of the above three conditions, the one that can be conceivably strenuous to
verify is that the flows commute whenever either composition makes sense. As
such, additional constraints on a geometry that can simplify this condition
can be helpful in practice. Toward this end, note that there are two parts to
verifying that the flow condition holds: that
$\exp(\omega^{-1}(X))\exp(\omega^{-1}(Y))$ and
$\exp(\omega^{-1}(Y))\exp(\omega^{-1}(X))$ are equal on $\mathscr{e}\circ AB$
wherever both are defined and that if either of them is well-defined at some
point of $\mathscr{e}\circ AB$, then so is the other.

The first part is fairly straightforward to get around: given two vector
fields $V$ and $W$, $\exp(sV)\exp(tW)=\exp(tW)\exp(sV)$ for all
$s,t\in\mathbb{R}$ such that both sides are well-defined if and only if
$[V,W]=0$. In our case,
$[\omega^{-1}(X),\omega^{-1}(Y)]=\omega^{-1}([X,Y]-\Omega^{\omega}(X\wedge
Y))$, so since $[X,Y]=0$ for $X\in\mathfrak{a}$ and $Y\in\mathfrak{b}$, the
first part of the flow condition holds if and only if
$\Omega^{\omega}_{\mathscr{g}}(X\wedge Y)=0$ for all $X\in\mathfrak{a}$,
$Y\in\mathfrak{b}$, and $\mathscr{g}\in\mathscr{e}\circ AB$.

This condition on the curvature is sufficient to apply Theorem 7.3 locally. In
other words, if $\mathscr{e}\circ AB$ is well-defined and
$\Omega^{\omega}_{\mathscr{g}}(X\wedge Y)=0$ for all $X\in\mathfrak{a}$,
$Y\in\mathfrak{b}$, and $\mathscr{g}\in\mathscr{e}\circ AB$, then for each
$\mathscr{g}\in\mathscr{e}\circ AB$ such that $\mathscr{g}\circ A$ and
$\mathscr{g}\circ B$ are well-defined, there exists a neighborhood of $U$ of
$q_{{}_{H}}(\mathscr{g})\in M$ such that the restricted geometry
$(q_{{}_{H}}^{-1}(U),\omega)$ satisfies the conditions of the theorem when we
take the basepoint to be $\mathscr{g}$. We can get such a $U$ by taking
sufficiently small open sets $U_{\mathfrak{a}}\subset\mathfrak{a}$ and
$U_{\mathfrak{b}}\subset\mathfrak{b}$ and setting
$U=q_{{}_{H}}(\exp(\omega^{-1}(U_{\mathfrak{a}}))\exp(\omega^{-1}(U_{\mathfrak{b}}))\mathscr{g})$.

The second part of the flow condition, that if either of the compositions
$\exp(\omega^{-1}(X))\exp(\omega^{-1}(Y))\mathscr{g}$ or
$\exp(\omega^{-1}(Y))\exp(\omega^{-1}(X))\mathscr{g}$ is well-defined for
$\mathscr{g}\in\mathscr{e}\circ AB$ then so is the other, is slightly
trickier. However, we can bypass it by simply guaranteeing that the flows are
always well-defined. This constraint is called completeness.

######  Definition 7.4.

A Cartan geometry $(\mathscr{G},\omega)$ of type $(G,H)$ is _complete_ if and
only if $\omega^{-1}(X)$ is a complete vector field whenever
$X\in\mathfrak{g}$.

Thus, if $(\mathscr{G},\omega)$ is complete and
$\Omega^{\omega}_{\mathscr{g}}(X\wedge Y)=0$ for all $X\in\mathfrak{a}$,
$Y\in\mathfrak{b}$, and $\mathscr{g}\in\mathscr{e}\circ AB$, then the flow
condition in Theorem 7.3 is automatically satisfied. If we also assume that
the base manifold $M$ is simply connected, then we can further show that all
three conditions are satisfied.

######  Corollary 7.5.

Suppose $(A,P)$, $(B,K)$, and $(G,H)$ are model geometries such that
$i:A\times B\hookrightarrow G$ determines an extension functor from $(A\times
B,P\times K)$ to $(G,H)$. Let $(\mathscr{G},\omega)$ be a complete Cartan
geometry of type $(G,H)$ over a simply connected smooth manifold $M$, with
quotient map $q_{{}_{H}}:\mathscr{G}\to M$. Fix $\mathscr{e}\in\mathscr{G}$
such that $\mathscr{e}\circ A$, $\mathscr{e}\circ B$, and $\mathscr{e}\circ
AB$ are well-defined and $q_{{}_{H}}(\mathscr{e}\circ AB)=M$. Then,
$(\mathscr{G},\omega)$ is the product of type $(G,H)$ of $\mathscr{e}\circ A$
and $\mathscr{e}\circ B$ if and only if $\Omega^{\omega}_{\mathscr{g}}(X\wedge
Y)=0$ whenever $X\in\mathfrak{a}$, $Y\in\mathfrak{b}$, and
$\mathscr{g}\in\mathscr{e}\circ AB$.

###### Proof.

As we said above, the condition on the curvature together with completeness
are sufficient to guarantee the flow condition in Theorem 7.3 is satisfied.
Moreover, since $q_{{}_{PK}}|_{\mathscr{e}\circ AB}:\mathscr{e}\circ
AB\to(\mathscr{e}\circ AB)/PK$ is a fiber bundle map and
$\bar{q}_{{}_{H}}:(\mathscr{e}\circ AB)/PK\to M$ is a local diffeomorphism,
$q_{{}_{H}}|_{\mathscr{e}\circ AB}=\bar{q}_{{}_{H}}\circ
q_{{}_{PK}}|_{\mathscr{e}\circ AB}$ is a fibration. Because $M$ is simply
connected, there is a homotopy rel endpoints $\gamma$ from each loop
$\gamma_{0}:[0,1]\to M$ based at $q_{{}_{H}}(\mathscr{g})$ to the constant
loop $\gamma_{1}:t\mapsto q_{{}_{H}}(\mathscr{g})$, so since
$q_{{}_{H}}|_{\mathscr{e}\circ AB}$ is a fibration, if $\widehat{\gamma_{0}}$
is a lift of $\gamma_{0}$ to $\mathscr{e}\circ AB$ with
$\widehat{\gamma_{0}}(0)=\mathscr{g}$, then there is a lift $\widehat{\gamma}$
to $\mathscr{e}\circ AB$ of the homotopy $\gamma$ with
$(\widehat{\gamma})_{0}=\widehat{\gamma_{0}}$ and
$(\widehat{\gamma})_{s}(0)=\mathscr{g}$ for each $s\in[0,1]$. Since
$\widehat{\gamma}$ is a lift of a homotopy rel endpoints and $\gamma_{1}$ is
the constant loop, $(\widehat{\gamma})_{s}(1)\in(\mathscr{e}\circ
AB)\cap\mathscr{g}H$ for each $s\in[0,1]$ and
$(\widehat{\gamma})_{1}(t)\in\mathscr{g}PK$ for each $t\in[0,1]$, so
$(\widehat{\gamma})_{s}(1)\in\mathscr{e}PK$ for each $s\in[0,1]$. In
particular, if $\mathscr{g}$ and $\mathscr{g}h$ are in $\mathscr{e}\circ AB$,
then $h\in PK$. Thus, it only remains to show that $\mathscr{e}\circ
A\cap\mathscr{e}\circ B=\\{\mathscr{e}\\}$.

Suppose $\mathscr{g}\in\mathscr{e}\circ A\cap\mathscr{e}\circ B$. Then, there
exist paths $\alpha:[0,1]\to\mathscr{e}\circ A$ and
$\beta:[0,1]\to\mathscr{e}\circ B$, $p\in P$, and $k\in K$ such that
$\alpha(0)=\beta(0)=\mathscr{e}$ and $\alpha(1)p=\beta(1)k=\mathscr{g}$. Again
using that $q_{{}_{H}}|_{\mathscr{e}\circ AB}$ is a fibration and $M$ is
simply connected, there exists a homotopy
$\widehat{\gamma}:[0,1]^{2}\to\mathscr{e}\circ AB$ such that
$\widehat{\gamma}_{0}=\alpha$ and $\widehat{\gamma}_{1}=\beta\zeta$, where
$\zeta:[0,1]\to PK$. Moreover, by right-translating by a path in $PK$
parametrized by $s$ if necessary, we may assume that
$\widehat{\gamma}_{s}(0)=\mathscr{e}$ for all $s\in[0,1]$.

Let $\pi:A\times B\to A$ be the natural quotient map. For
$\sigma:[0,1]\to\mathscr{e}\circ AB$ with $\sigma(0)=\mathscr{e}$, define
$\pi_{\mathscr{e}}(\sigma):[0,1]\to\mathscr{e}\circ A$ to be the unique path
such that $\pi_{\mathscr{e}}(\sigma)(0)=\mathscr{e}$ and
$\pi_{\mathscr{e}}(\sigma)_{G}=\pi(\sigma_{G})$.

######  Lemma 7.6.

Suppose $\widehat{\gamma}:[0,1]^{2}\to\mathscr{e}\circ AB$ is a homotopy
between two paths $\widehat{\gamma}_{0}$ and $\widehat{\gamma}_{1}$ such that,
for every $s\in[0,1]$, $\widehat{\gamma}_{s}(0)=\mathscr{e}$ and
$\widehat{\gamma}_{s}(1)=\widehat{\gamma}_{0}(1)h_{s}$ for some $h_{s}\in PK$.
Then,
$\pi_{\mathscr{e}}(\widehat{\gamma}_{s})(1)=\pi_{\mathscr{e}}(\widehat{\gamma}_{0})(1)\pi(h_{s})$.

###### Proof.

We will restrict to the case where, for all $s\in[0,1]$,
$q_{{}_{H}}(\widehat{\gamma}_{s}(t))=q_{{}_{H}}(\widehat{\gamma}_{0}(t))$ for
all $t$ outside of a small interval $[t_{1},t_{2}]$ for which
$q_{{}_{H}}(\widehat{\gamma}([0,1]\times[t_{1},t_{2}]))$ is contained in an
open set $U\subseteq M$ such that, for some $\mathscr{g}^{\prime}\in
q_{{}_{H}}^{-1}(U)\cap\mathscr{e}\circ AB$, there is a geometric isomorphism

$\varphi:q_{{}_{H}}^{-1}(U)\cap\mathscr{e}\circ
AB\to(\mathscr{g}^{\prime}\circ
A)_{q_{{}_{H}}^{-1}(U)}\times(\mathscr{g}^{\prime}\circ
B)_{q_{{}_{H}}^{-1}(U)},$

where $(\mathscr{g}^{\prime}\circ A)_{q_{{}_{H}}^{-1}(U)}$ and
$(\mathscr{g}^{\prime}\circ B)_{q_{{}_{H}}^{-1}(U)}$ are the geometric leaves
for $A$ and $B$ through $\mathscr{g}^{\prime}$ taken in the restricted
geometry $(q_{{}_{H}}^{-1}(U),\omega)$, as opposed to in
$(\mathscr{G},\omega)$ where, a priori, $\mathscr{g}^{\prime}\circ A$ might
intersect $\mathscr{g}^{\prime}\circ B$ at a point other than
$\mathscr{g}^{\prime}$. Note that, because
$\Omega^{\omega}_{\mathscr{g}}(X\wedge Y)=0$ for all
$\mathscr{g}\in\mathscr{e}\circ AB$ and $(\mathscr{G},\omega)$ is complete, if

$\mathscr{g}^{\prime}=(\exp(\omega^{-1}(X_{1}))\cdots\exp(\omega^{-1}(X_{\ell}))\exp(\omega^{-1}(Y_{1}))\cdots\exp(\omega^{-1}(Y_{\ell^{\prime}}))\mathscr{e})pk$

for $X_{i}\in\mathfrak{a}$, $Y_{i}\in\mathfrak{b}$, $p\in P$, and $k\in K$,
then

$R_{k}\circ\exp(\omega^{-1}(Y_{1}))\cdots\exp(\omega^{-1}(Y_{\ell^{\prime}}))$

gives a geometric isomorphism between $\mathscr{e}\circ A$ and
$\mathscr{g}^{\prime}\circ A$, so in particular $\mathscr{g}^{\prime}\circ A$
is well-defined, and similarly for $\mathscr{g}^{\prime}\circ B$. Moreover, as
was noted above, the curvature condition guarantees that each
$\mathscr{g}^{\prime}\in\mathscr{e}\circ AB$ has an open set $U$ in $M$ such
that $(q_{{}_{H}}^{-1}(U),\omega)$ splits as a product of type $(G,H)$ of the
restricted geometric leaves if they are well-defined. Thus, since $[0,1]^{2}$
is compact, we may always piece together a homotopy of the type described in
the lemma from finitely many of these homotopies where the projected curves on
$M$ only differ in an open set $U$ of the desired type, so no generality is
lost in making this restriction. This is essentially the same trick used in
Lemma 6 of Section IV.6 in [14].

By assumption, for all $t\not\in[t_{1},t_{2}]$,
$\widehat{\gamma}_{s}(t)=\widehat{\gamma}_{0}(t)\sigma_{s}(t)$ for some
$\sigma:[0,1]\times([0,t_{1})\cup(t_{2},1])\to PK$. For $t\in[0,t_{1})$, this
means
$(\widehat{\gamma}_{s})_{G}(t)=(\widehat{\gamma}_{0})_{G}(t)\sigma_{s}(t)$, so
$\pi_{\mathscr{e}}(\widehat{\gamma}_{s})(t)=\pi_{\mathscr{e}}(\widehat{\gamma}_{0})(t)\pi(\sigma_{s}(t))$.

Let $t_{0},t_{3}\in[0,1]$ such that $t_{0}<t_{1}<t_{2}<t_{3}$ and
$q_{{}_{H}}(\widehat{\gamma}([0,1]\times[t_{0},t_{3}]))\subseteq U$. Denoting
by $\mathrm{pr}_{A}$ and $\mathrm{pr}_{B}$ the natural projections from
$(\mathscr{g}^{\prime}\circ
A)_{q_{{}_{H}}^{-1}(U)}\times(\mathscr{g}^{\prime}\circ
B)_{q_{{}_{H}}^{-1}(U)}$ to $(\mathscr{g}^{\prime}\circ
A)_{q_{{}_{H}}^{-1}(U)}$ and $(\mathscr{g}^{\prime}\circ
B)_{q_{{}_{H}}^{-1}(U)}$, respectively, and by
$\widehat{\gamma}_{s}(\mathrm{id}+t_{0})$ the path
$\widehat{\gamma}_{s}(\mathrm{id}+t_{0}):[0,t_{3}-t_{0}]\to
q_{{}_{H}}^{-1}(U)\cap\mathscr{e}\circ AB$ with
$\widehat{\gamma}_{s}(\mathrm{id}+t_{0})(t)=\widehat{\gamma}_{s}(t+t_{0})$, we
get

$\widehat{\gamma}_{s}(\mathrm{id}+t_{0})_{G}=\mathrm{pr}_{A}(\varphi(\widehat{\gamma}_{s}(\mathrm{id}+t_{0})))_{G}\mathrm{pr}_{B}(\varphi(\widehat{\gamma}_{s}(\mathrm{id}+t_{0})))_{G}.$

By composing flows of vector fields $\omega^{-1}(Y)$ for $Y\in\mathfrak{b}$
and right-translations $R_{k}$ with $k\in K$, we can get a geometric
isomorphism $\lambda:\mathscr{g}^{\prime}\circ A\to\mathscr{e}\circ A$ such
that
$\lambda(\mathrm{pr}_{A}(\varphi(\widehat{\gamma}_{0}(t_{0}))))=\pi_{\mathscr{e}}(\widehat{\gamma}_{0})(t_{0})$.
In particular,

$\displaystyle\lambda(\mathrm{pr}_{A}(\varphi(\widehat{\gamma}_{s}(t_{0}))))$
$\displaystyle=\lambda(\mathrm{pr}_{A}(\varphi(\widehat{\gamma}_{0}(t_{0})\sigma_{s}(t_{0}))))=\lambda(\mathrm{pr}_{A}(\varphi(\widehat{\gamma}_{0}(t_{0}))))\pi(\sigma_{s}(t_{0}))$
$\displaystyle=\pi_{\mathscr{e}}(\widehat{\gamma}_{0})(t_{0})\pi(\sigma_{s}(t_{0}))=\pi_{\mathscr{e}}(\widehat{\gamma}_{s})(t_{0}).$

Thus,
$\pi_{\mathscr{e}}(\widehat{\gamma}_{s})(t)=\lambda(\mathrm{pr}_{A}(\varphi(\widehat{\gamma}_{s}(t))))$
for all $t\in[t_{0},t_{3}]$, since
$\lambda(\mathrm{pr}_{A}(\varphi(\widehat{\gamma}_{s}(\mathrm{id}+t_{0}))))_{G}=\pi(\widehat{\gamma}_{s}(\mathrm{id}+t_{0})_{G})$.
Moreover, since $t_{3}\in(t_{2},1]$, we see that
$\pi_{\mathscr{e}}(\widehat{\gamma}_{s})(t_{3})=\pi_{\mathscr{e}}(\widehat{\gamma}_{0})(t_{3})\pi(\sigma_{s}(t_{3}))$.

Finally, for $t\in(t_{2},1]$, we still have
$\widehat{\gamma}_{s}(t)=\widehat{\gamma}_{0}(t)\sigma_{s}(t)$, so we still
get
$\pi_{\mathscr{e}}(\widehat{\gamma}_{s})(t)=\pi_{\mathscr{e}}(\widehat{\gamma}_{0})(t)\pi(\sigma_{s}(t))$,
hence

$\pi_{\mathscr{e}}(\widehat{\gamma}_{s})(1)=\pi_{\mathscr{e}}(\widehat{\gamma}_{0})(1)\pi(\sigma_{s}(1))=\pi_{\mathscr{e}}(\widehat{\gamma}_{0})(1)\pi(h_{s}).\quad\mbox{\qed}$

For our homotopy $\widehat{\gamma}$ from $\alpha$ to $\beta\zeta$, we have
that $\pi_{\mathscr{e}}(\widehat{\gamma}_{0})=\alpha$ and
$\pi_{\mathscr{e}}(\widehat{\gamma}_{1})=\mathscr{e}\pi(\zeta)$, so by the
lemma,

$\mathscr{e}\pi(\zeta(1))=\alpha(1)\pi(pk^{-1}\zeta(1))=\alpha(1)p\pi(\zeta(1))=\mathscr{g}\pi(\zeta(1)),$

hence $\mathscr{g}=\mathscr{e}$.∎

Corollary 7.5 generalizes the de Rham decomposition for Riemannian manifolds.
Unlike in the usual de Rham decomposition, we cannot always get a product
decomposition of the geometry from a product decomposition of a subgroup
containing the holonomy group when the geometry is complete and the base
manifold is simply connected. Indeed, by identifying $\mathfrak{g}$ with
$\mathbb{R}^{\dim(\mathfrak{g})}$, the Maurer-Cartan form on a Lie group $G$
gives a complete Cartan connection of type
$(\mathbb{R}^{\dim(\mathfrak{g})},\\{0\\})$, but not all simply connected Lie
groups decompose into a product of 1-dimensional submanifolds. Thus, in this
more general case, we need to use the additional finesse of the curvature,
rather than just the holonomy group.

We should remark that an alternative proof of Corollary 7.5 can be obtained by
using the generalized Cartan-Ambrose-Hicks theorem from [6]. This would be
similar to the approach from [17], except it would apply to this more general
case.

For reductive222We say that a Cartan geometry $(\mathscr{G},\omega)$ of type
$(G,H)$ is _reductive_ if and only if there exists an
$\mathrm{Ad}_{H}$-invariant subspace $\mathfrak{m}\subseteq\mathfrak{g}$ such
that $\mathfrak{g}\approx\mathfrak{m}\oplus\mathfrak{h}$ as an
$\mathrm{Ad}_{H}$-module. Cartan geometries, such as first-order
$H$-structures with a choice of connection, completeness is not a particularly
strong constraint. Indeed, it is well-known that reductive Cartan geometries
are complete if and only if they are geodesically complete with respect to the
induced covariant derivative. However, it should be noted that, in general,
the completeness hypothesis of Corollary 7.5 is an exceptionally fanciful
assumption. Completeness remains rather mysterious for most Cartan geometries,
but it is known in many cases to be quite restrictive. For example, it was
proved in [15] that every complete complex parabolic geometry is necessarily
flat. On the other hand, we only needed to use that $\mathscr{e}\circ AB$ was
complete in the proof of Corollary 7.5, and there are many Cartan geometries
that have holonomy reductions to complete reductive geometries, so the result
still applies quite generally.

Moreover, we can still get a local decomposition without completeness, and
this can lead to interesting results—even for parabolic geometries—when paired
with an appropriate holonomy reduction. For example, it is not too difficult
to imagine how useful results like Theorem 4.5 of [2] might be gleaned from
Theorem 7.3 above.

###  7.2. Inessential automorphism groups of parabolic geometries

For this section, consider the following two trivial consequences of Corollary
6.5.

######  Proposition 7.7.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$ over $M$
with a holonomy reduction to $J$ based at $\mathscr{e}\in\mathscr{G}$ such
that $q_{{}_{H}}(J\bullet\mathscr{e})=q_{{}_{H}}(\mathscr{G})=M$. Then, for
arbitrary $\mathscr{g}\in\mathscr{G}$, the action of
$K\leq\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$ on $\mathscr{G}$
restricts to an action by automorphisms on $J\bullet\mathscr{g}$ if and only
if $K\mathscr{e}=\\{\varphi(\mathscr{e}):\varphi\in
K\\}\subseteq\mathscr{e}\bullet J$.

######  Proposition 7.8.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,H)$ over $M$
with a holonomy reduction to $J$ based at $\mathscr{e}\in\mathscr{G}$ such
that $q_{{}_{H}}(J\bullet\mathscr{e})=q_{{}_{H}}(\mathscr{G})=M$. Then, if
$\xi$ is an infinitesimal automorphism of $(\mathscr{G},\omega)$ such that
$q_{{}_{H}*}(\xi)$ vanishes at some $x\in M$, then $\xi$ restricts to an
infinitesimal automorphism of $J\bullet\mathscr{g}$ for some $\mathscr{g}\in
q_{{}_{H}}^{-1}(x)$ if and only if
$\omega_{\mathscr{g}}(\xi)\in\mathrm{Ad}_{\mathscr{g}^{-1}}(\mathfrak{j})\cap\mathfrak{h}$.

Since $q_{{}_{H}}(J\bullet\mathscr{e})=q_{{}_{H}}(\mathscr{G})$ in the above
proposition, every $\mathscr{g}\in\mathscr{G}$ has a development of the form
$g=jh$ for some $j\in J$ and $h\in H$, in which case we get
$\mathrm{Ad}_{g^{-1}}(\mathfrak{j})\cap\mathfrak{h}=\mathrm{Ad}_{(jh)^{-1}}(\mathfrak{j})\cap\mathfrak{h}=\mathrm{Ad}_{h^{-1}}(\mathfrak{j}\cap\mathfrak{h})$.
In particular, we can always choose $\mathscr{g}\in J\bullet\mathscr{e}$ and
$\omega_{\mathscr{g}}(\xi)\in\mathfrak{j}\cap\mathfrak{h}$ in the situation of
Proposition 7.8.

Both of these propositions allow us to use a relatively small amount of
information to determine global properties as long as we know the holonomy
group. In the first case, it tells us that a subgroup of automorphisms of a
geometric structure restricts to a subgroup of automorphisms of an underlying
structure if and only if the subgroup’s orbit from the chosen basepoint for
the holonomy is contained in the curved coset corresponding to the holonomy
reduction for the underlying structure. In the second case, we can say whether
an infinitesimal automorphism of a geometric structure with a fixed point on
the base manifold restricts to an automorphism of an underlying structure by
just checking its value at a single point in the fiber above the fixed point.

These results would be particularly useful for exploring inessential
automorphisms of parabolic geometries, which are automorphisms of an
infinitesimal flag structure that restrict to automorphisms of a particular
underlying geometry. Indeed, a local analogue of Proposition 7.8, not relying
on holonomy, has appeared before in [1], generalizing results from [13], for
precisely this purpose. However, there are obstructions that may prevent the
above propositions from giving us global information about the underlying
infinitesimal flag structures, which are often the focus of research for
parabolic geometries. We will discuss these below. For an introduction to
parabolic geometries, we highly recommend [9]; for brevity, we will assume
some familiarity with the subject for the rest of this section.

Given a model parabolic geometry $(G,P)$, we get an associated filtration
$\mathfrak{g}=\mathfrak{g}^{-k}\supset\cdots\supset\mathfrak{g}^{0}=\mathfrak{p}\supset\cdots\supset\mathfrak{g}^{k}\supset\\{0\\}$
and grading
$\mathfrak{g}=\mathfrak{g}_{-k}+\cdots+\mathfrak{g}_{0}+\cdots+\mathfrak{g}_{k}$.
We will denote by $\mathfrak{p}_{+}$ the nilradical
$\mathfrak{g}_{1}+\cdots+\mathfrak{g}_{k}$ of $\mathfrak{p}$ and by
$\mathfrak{g}_{-}$ the nilradical $\mathfrak{g}_{-k}+\cdots+\mathfrak{g}_{-1}$
of the opposite parabolic of $\mathfrak{p}$. We also get closed subgroups
$G_{0}$, $G_{-}$, and $P_{+}$ with corresponding Lie subalgebras
$\mathfrak{g}_{0}$, $\mathfrak{g}_{-}$, and $\mathfrak{p}_{+}$, respectively,
and $P\simeq G_{0}\ltimes P_{+}$.

The subgroup $G_{0}$ is reductive, so it splits as
$\mathfrak{z}(\mathfrak{g}_{0})\oplus\mathfrak{g}_{0}^{\text{ss}}$, where
$\mathfrak{g}_{0}^{\text{ss}}$ is semisimple. If is the Killing form on
$\mathfrak{g}$, then the restriction of to $\mathfrak{g}_{0}$ will still be
nondegenerate and respect this splitting. Thus, given a homomorphism
$\lambda:G_{0}\to\mathbb{R}_{+}$, there will be a unique element
$\lambda_{*}\in\mathfrak{z}(\mathfrak{g}_{0})$ such that the homomorphism
$\lambda_{*}:\mathfrak{g}_{0}\to\mathbb{R}$ satisfies
$\lambda_{*}(Z)=\kgf(\lambda_{*},Z)$ for all $Z\in\mathfrak{g}_{0}$. Following
[10], we call $\lambda_{*}$ a _scaling element_ if and only if the restriction
of $\mathrm{ad}_{\lambda_{*}}$ to each $G_{0}$-irreducible component of
$\mathfrak{p}_{+}$ is given by multiplication by a nonzero real scalar. We
imagine that this was the intended definition in [1], since all of the results
from that paper seem to work otherwise verbatim with this definition, but as
written it was assumed that $\lambda_{*}$ acts by scalar multiplication on
each grading component $\mathfrak{g}_{i}$, in which case it is not hard to see
that $\lambda_{*}$ must be a scalar multiple of the grading element.

Recall that a parabolic geometry of type $(G,P)$ naturally induces an
infinitesimal flag structure of type $(G,P)$, and that we can identify this
infinitesimal flag structure with the equivalence class of parabolic
geometries inducing it.

######  Definition 7.9.

Given an infinitesimal flag structure of type $(G,P)$ and a homomorphism
$\lambda:G_{0}\to\mathbb{R}_{+}$ with $\lambda_{*}$ a scaling element, we say
a _subordinate $\lambda$-exact geometry_ is a choice of holonomy reduction to
$G_{-}\ker(\lambda)$ on some $(\mathscr{G},\omega)$ of type $(G,P)$ inducing
that infinitesimal flag structure such that
$q_{{}_{P}}(G_{-}\ker(\lambda)\bullet\mathscr{e})=q_{{}_{P}}(\mathscr{G})$.

######  Lemma 7.10.

Suppose $(\mathscr{G},\omega)$ is a Cartan geometry of type $(G,P)$ over $M$
inducing a given infinitesimal flag structure and
$\lambda:G_{0}\to\mathbb{R}_{+}$ is a homomorphism with $\lambda_{*}$ a
scaling element. If $\mathscr{e}\in\mathscr{G}$ is such that
$\operatorname{\mathrm{Hol}}_{\mathscr{e}}(\mathscr{G},\omega)\leq
G_{-}\ker(\lambda)$ and $q_{{}_{P}}(G_{-}\ker(\lambda)\bullet\mathscr{e})=M$,
then there is an exact (with respect to $\lambda$) Weyl structure
$\sigma:\mathscr{G}/P_{+}\to\mathscr{G}$ induced by the subordinate
$\lambda$-exact geometry $G_{-}\ker(\lambda)\bullet\mathscr{e}$. Conversely,
every exact Weyl structure for $(\mathscr{G},\omega)$ induces a subordinate
$\lambda$-exact geometry on some Cartan geometry
$(\mathscr{G},\omega^{\prime})$ inducing the same infinitesimal flag structure
as $(\mathscr{G},\omega)$.

###### Proof.

For a Weyl structure $\sigma:\mathscr{G}/P_{+}\to\mathscr{G}$, recall that
$\sigma^{*}\omega$ is a $G_{0}$-equivariant $\mathfrak{g}$-valued 1-form on
$\mathscr{G}/P_{+}$, so we may define $G_{0}$-equivariant
$\mathfrak{g}_{i}$-valued 1-forms $\sigma^{*}\omega_{i}$ for each graded
component $\mathfrak{g}_{i}$. From this, we get a Cartan geometry
$(\mathscr{G}/P_{+},\upsilon)$ of type $(G_{-}G_{0},G_{0})$ over $M$, where
$\upsilon=\sigma^{*}\omega_{-k}+\cdots+\sigma^{*}\omega_{0}$. Using the
extension functor from $(G_{-}G_{0},G_{0})$ to $(G,P)$ given by inclusion
maps, we also get a Cartan geometry $(\mathscr{G},\omega^{\prime})$ of type
$(G,P)$ from $(\mathscr{G}/P_{+},\upsilon)$, where $\omega^{\prime}$ is
uniquely characterized by $\sigma^{*}\omega^{\prime}=\upsilon$. Moreover,
$\omega-\omega^{\prime}$ takes $\omega^{-1}(\mathfrak{g}^{i})$ into
$\mathfrak{p}$ for each filtration component $\mathfrak{g}^{i}$, and it takes
$\omega^{-1}(\mathfrak{p})$ to $\\{0\\}$, so in particular $\omega^{\prime}$
induces the same infinitesimal flag structure as $\omega$.

Given a subordinate $\lambda$-exact geometry
$G_{-}\ker(\lambda)\bullet\mathscr{e}$ in $(\mathscr{G},\omega)$, we get a
corresponding Weyl structure $\sigma:\mathscr{G}/P_{+}\to\mathscr{G}$ by
defining $\sigma(\mathscr{g}P_{+})=\mathscr{g}$ for $\mathscr{g}\in
G_{-}\ker(\lambda)\bullet\mathscr{e}$ and extending by $G_{0}$-equivariance.
In that case, for $X\in\mathfrak{g}_{-}+\ker(\lambda_{*})$, we see that
$\lambda_{*}(\sigma^{*}\omega_{0}(\upsilon^{-1}(X+t\lambda_{*})))=t\lambda_{*}(\lambda_{*})$,
so since $\lambda_{*}(\sigma^{*}\omega_{0})$ descends to the induced principal
$\mathbb{R}_{+}$-connection on
$\mathscr{G}/P_{+}\times_{\lambda}\mathbb{R}_{+}$ and every element of
$\mathscr{G}/P_{+}$ is uniquely of the form
$\sigma^{-1}(\mathscr{g})\exp(t\lambda_{*})$ for some $\mathscr{g}\in
G_{-}\ker(\lambda)\bullet\mathscr{e}$ and $t\in\mathbb{R}$, the principal
$\mathbb{R}_{+}$-connection on
$\mathscr{G}/P_{+}\times_{\lambda}\mathbb{R}_{+}$ corresponds to the one
induced by the global section $\mathscr{g}P\mapsto(\mathscr{g}P_{+},1)$ for
$\mathscr{g}\in G_{-}\ker(\lambda)\bullet\mathscr{e}$, hence $\sigma$ is
exact.

Conversely, suppose $\sigma:\mathscr{G}/P_{+}\to\mathscr{G}$ is an exact Weyl
structure for $(\mathscr{G},\omega)$. Consider a global section
$\sigma_{0}:M\to\mathscr{G}/\ker(\lambda)P_{+}\cong\mathscr{G}/P_{+}\times_{\lambda}\mathbb{R}_{+}$
that induces the same principal $\mathbb{R}_{+}$-connection as the one induced
by $\sigma^{*}\omega_{0}$. Given a loop $\gamma:[0,1]\to M$, let
$\widehat{\gamma}$ be a lift of $\sigma_{0}(\gamma)$ to $\mathscr{G}/P_{+}$.
Then, $\sigma(\widehat{\gamma})$ is a lift of $\gamma$ to $\mathscr{G}$, and
$\sigma(\gamma(1))\in\sigma(\gamma(0))\ker(\lambda)$, so since
$\sigma(\widehat{\gamma})^{*}\omega^{\prime}=\widehat{\gamma}^{*}\sigma^{*}\omega^{\prime}=\widehat{\gamma}^{*}\upsilon$,
and $\widehat{\gamma}^{*}\upsilon$ takes values in
$\mathfrak{g}_{-}+\ker(\lambda_{*})$ by construction, the holonomy based at
$\mathscr{e}=\sigma(\widehat{\gamma}(0))$ of the loop $\gamma$ is contained in
$G_{-}\ker(\lambda)$. Similarly, if $\gamma:[0,1]\to M$ is a path between two
points of $M$ with $\gamma(0)=q_{{}_{P}}(\mathscr{e})$, then we can lift it to
a path $\sigma(\widehat{\gamma})$ with
$\sigma(\widehat{\gamma}(0))=\mathscr{e}$ and
$\sigma(\widehat{\gamma})^{*}\omega^{\prime}$ taking values in
$\mathfrak{g}_{-}+\ker(\lambda_{*})$, so $q_{{}_{P}}(\mathscr{e}\bullet
G_{-}\ker(\lambda))=M$.∎

Before defining inessential and essential automorphism groups in this setting,
we need to remark upon the first of two complications: while automorphisms of
a parabolic geometry necessarily induce automorphisms of the underlying
infinitesimal flag structure, automorphisms of the infinitesimal flag
structure do not necessarily lift to automorphisms of an arbitrary overlying
parabolic geometry. Thus, the automorphisms of two parabolic geometries might
be completely different even if they induce the same infinitesimal flag
structure. Thankfully, if we put certain mild constraints on the curvature,
namely regularity and normality, then modulo some usually negligible Lie
algebra cohomological restrictions on the model geometry (see Theorem 3.1.14
in [9]), this is not an issue: automorphisms of a regular infinitesimal flag
structure lift to automorphisms of the (unique) overlying normal regular
parabolic geometry. Thus, in a loose sense, normality is a “maximal symmetry”
condition on regular parabolic geometries overlying some infinitesimal flag
structure.

######  Definition 7.11.

Suppose $(\mathscr{G},\omega^{\text{norm}})$ is a regular normal parabolic
geometry of type $(G,P)$ and $\lambda:G_{0}\to\mathbb{R}_{+}$ is a
homomorphism with $\lambda_{*}$ a scaling element. A subgroup
$K\leq\operatorname{\mathrm{Aut}}(\mathscr{G},\omega^{\text{norm}})$ is said
to be _$\lambda$ -inessential_ if and only if the action restricts to an
action by automorphisms on some subordinate $\lambda$-exact geometry for the
infinitesimal flag structure induced by $(\mathscr{G},\omega^{\text{norm}})$.
Otherwise, we say that $K$ is _$\lambda$ -essential_. We say that $K$ is
_essential_ if and only if it is $\lambda$-essential for each
$\lambda:G_{0}\to\mathbb{R}_{+}$ with $\lambda_{*}$ a scaling element.

An automorphism
$\varphi\in\operatorname{\mathrm{Aut}}(\mathscr{G},\omega^{\text{norm}})$ is
called $\lambda$-essential or $\lambda$-inessential according to whether the
subgroup generated by $\varphi$ is $\lambda$-essential or
$\lambda$-inessential.

Similarly, subalgebras of $\mathfrak{aut}(\mathscr{G},\omega^{\text{norm}})$
are called $\lambda$-essential or $\lambda$-inessential according to whether
the subgroups generated by them are $\lambda$-essential or
$\lambda$-inessential, and an element of
$\mathfrak{aut}(\mathscr{G},\omega^{\text{norm}})$ is said to be
$\lambda$-essential or $\lambda$-inessential according to whether the
subalgebra generated by it is $\lambda$-essential or $\lambda$-inessential.

In light of Lemma 7.10, we see that these definitions of essential and
inessential are equivalent to the ones in [1], modulo the extended choice of
scaling elements from [10].

Unfortunately, there is a second complication that prevents us from directly
applying Propositions 7.7 and 7.8: just like with automorphism groups, the
holonomy groups of different parabolic geometries inducing the same
infinitesimal flag structure are not required to be the same, and as far as
the author is currently aware, there is no known method of using the holonomy
groups of regular normal parabolic geometries to determine the holonomy groups
of other parabolic geometries inducing the same infinitesimal flag structure.
Thus, unlike in the case of automorphism groups, we cannot just use the
holonomy group of the overlying regular normal parabolic geometry. Moreover,
even if we get a holonomy reduction to $G_{-}\ker(\lambda)$ at $\mathscr{e}$,
$q_{{}_{P}}(G_{-}\ker(\lambda)\bullet\mathscr{e})$ is not necessarily the
whole base manifold. For most cases, this makes our definitions significantly
less graceful to use than the ones from [1].

In lieu of elegance, a clumsy global analogue of Proposition 7.8 for
infinitesimal flag structures would be along the lines of the following.

######  Proposition 7.12.

Suppose $(\mathscr{G},\omega^{\text{norm}})$ is a regular normal parabolic
geometry of type $(G,P)$ over $M$, $\lambda:G_{0}\to\mathbb{R}_{+}$ is a
homomorphism with $\lambda_{*}$ a scaling element, and
$\xi\in\mathfrak{aut}(\mathscr{G},\omega^{\text{norm}})$ such that
$q_{{}_{P}*}(\xi)$ vanishes at $x\in M$. Then, $\xi$ is $\lambda$-inessential
if and only if there exists an $\mathscr{e}\in q_{{}_{P}}^{-1}(x)$ such that
$\omega^{\text{norm}}_{\mathscr{e}}(\xi)\in\ker(\lambda_{*})$ and a
$(\mathscr{G},\omega)$ inducing the same infinitesimal flag structure as
$(\mathscr{G},\omega^{\text{norm}})$ on which there is a subordinate
$\lambda$-exact geometry based at $\mathscr{e}$ and such that $\xi$ is also an
infinitesimal automorphism of $(\mathscr{G},\omega)$.

The only aspect of Proposition 7.12 that does not follow immediately from
Proposition 7.8 and Definition 7.11 is that we can use
$\omega^{\text{norm}}(\xi)$ instead of $\omega(\xi)$, but this just follows
because they differ by a horizontal 1-form.

While Proposition 7.12 does, more or less immediately, recover Theorem 1.2
from [1], and it does tell us that $\xi$ is $\lambda$-essential if
$\omega_{\mathscr{e}}^{\text{norm}}(\xi)\not\in\ker(\lambda_{*})$, the
considerations above severely limit its ability to obtain more global results
about $\lambda$-inessential infinitesimal automorphisms. However, if we are in
a situation where we already know a subordinate $\lambda$-exact geometry, then
Proposition 7.12 is much more useful. For example, when constructing a
parabolic geometry, we often start with a particular Cartan geometry
$(Q,\upsilon)$ of type $(G_{-}\ker(\lambda),\ker(\lambda))$ and get a Cartan
geometry $(\mathscr{G},\omega)$ of type $(G,P)$ by using the extension functor
induced by the inclusion maps. We then get a normal Cartan geometry
$(\mathscr{G},\omega^{\text{norm}})$ by normalizing. In this case, Proposition
7.12 tells us that
$\xi\in\mathfrak{aut}(\mathscr{G},\omega)\leq\mathfrak{aut}(\mathscr{G},\omega^{\text{norm}})$
is $\lambda$-inessential if there is some $\mathscr{e}\in Q\subset\mathscr{G}$
such that $\omega_{\mathscr{e}}^{\text{norm}}(\xi)\in\ker(\lambda_{*})$.

Similarly, we can give a clumsy version of Proposition 7.7 for infinitesimal
flag structures.

######  Proposition 7.13.

Suppose $(\mathscr{G},\omega^{\text{norm}})$ is a regular normal parabolic
geometry of type $(G,P)$ over $M$, $\lambda:G_{0}\to\mathbb{R}_{+}$ is a
homomorphism with $\lambda_{*}$ a scaling element, and
$K\leq\operatorname{\mathrm{Aut}}(\mathscr{G},\omega^{\text{norm}})$. Then,
$K$ is $\lambda$-inessential if and only if there exists a
$(\mathscr{G},\omega)$ inducing the same infinitesimal flag structure as
$(\mathscr{G},\omega^{\text{norm}})$ such that
$K\leq\operatorname{\mathrm{Aut}}(\mathscr{G},\omega)$ and on which there is a
subordinate $\lambda$-exact geometry based at $\mathscr{e}\in\mathscr{G}$ such
that $K\mathscr{e}\subseteq G_{-}\ker(\lambda)\bullet\mathscr{e}$.

Again, while this can be cumbersome, if we have foreknowledge of which
parabolic geometries underlying a regular infinitesimal flag structure support
a subordinate $\lambda$-exact geometry, then finding whether a subgroup is
$\lambda$-essential becomes significantly less complicated.

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