# A frame based approach to computing symmetries with non-trivial isotropy
groups

D. D. McNutt<EMAIL_ADDRESS>Department of Mathematics and Statistics,
UiT The Arctic University of Norway, Tromsø, Norway  A. A. Coley
<EMAIL_ADDRESS>Department of Mathematics and Statistics, Dalhousie
University, Halifax, Nova Scotia, Canada, B3H 3J5  R. J. van den Hoogen
<EMAIL_ADDRESS>Department of Mathematics and Statistics, St. Francis Xavier
University, Antigonish, Nova Scotia, Canada, B2G 2W5

###### Abstract

A frame approach to determining the most general solution admitting a desired
symmetry group has been examined previously in Riemannian and teleparallel
geometries with some success. In teleparallel geometries, one must determine
the general form of the frame and spin connection to generate a general
solution admitting the desired symmetry group. Current approaches often rely
on the use of the proper frame, where the spin connection is zero. However
this leads to particular theoretical and practical problems. In this paper we
introduce an entirely general approach to determining the most general
Riemann-Cartan geometries which admit a given symmetry group and apply these
results to teleparallel geometries. To illustrate the approach we determine
the most general geometries, with the minimal number of arbitrary functions,
for particular choices of symmetry groups with dimension one, three, six and
seven. In addition, we rigorously show how the teleparallel analogues of the
Robertson-Walker, de Sitter and Einstein static spacetimes can be determined.

##  I Introduction

There are a wealth of alternative theories of gravity, including analogues of
general relativity (GR), where the torsion tensor and non-metricity tensors
are employed in addition to the curvature tensor. In any theory where the
torsion tensor or non-metricity tensor are non-zero, the Levi-Civita
connection is now a secondary object constructed from the contorsion tensor
and the connection which gives rise to the torsion and non-metricity tensors.
It is often easier to formulate such theories with their metric tensor and the
non Levi-Civita connection as independent quantities. Furthermore, in some
theories, such as teleparallel gravity, it is necessary to work with a frame
basis instead of a metric tensor. In such theories the role of symmetry is no
longer as clearly defined as in pseudo-Riemannian geometry, where the Levi-
Civita connection relates solutions to the Killing equations as symmetries,
and so we can no longer formulate a symmetry as an isometry of the metric.
Instead, for any theory constructed from a metric tensor and a non Levi-Civita
connection, the symmetry must satisfy the following conditions

$\displaystyle\mathcal{L}_{{\bf X}}g_{ab}=0\text{ and }\mathcal{L}_{{\bf
X}}\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=0,$ (1)

where the vector field ${\bf X}$ is the infinitesimal generator of the
symmetry, $g_{ab}$ is the metric tensor,
$\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}$ is the connection one-form
and $\mathcal{L}$ is the Lie derivative. We note that this definition applies
only if the isotropy group of the associated geometry is trivial.

The development of a frame based approach to determining the symmetries of a
spacetime has been explored in four-dimensional (4D) GR [1, 2, 3]. However,
the primary obstacle to these approaches is the potential existence of a non-
trivial linear isotropy group which is a Lie subgroup of Lorentz frame
transformations that leave the associated tensors of the geometry invariant.
If a given spacetime has a non-trivial linear isotropy group, determining the
group of symmetries requires solving a set of inhomogeneous differential
equations with unknown functions, $\lambda^{a}_{~{}b}$ [4];

$\displaystyle\mathcal{L}_{{\bf X}}\mbox{{\bf
h}}^{a}=\lambda^{a}_{~{}b}\mbox{{\bf h}}^{b}\text{ and }\mathcal{L}_{{\bf
X}}\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=\omega^{a}_{~{}bc}\lambda^{c}_{~{}d}\mbox{{\bf
h}}^{d},$ (2)

where, $\mbox{{\bf h}}^{a}$ denotes the orthonormal coframe basis and
$\lambda^{a}_{~{}b}$ is an element of the Lie algebra of Lorentz
transformations. We note that the second condition imposes that the vector
field ${\bf X}$ is an affine collineation $\mathcal{L}_{{\bf
X}}\omega^{a}_{~{}bc}=0$.

If we have a representation of the symmetry group as a set of vector fields,
then solving for the coframe basis and connection is complicated by the
unknown functions in $\lambda^{a}_{~{}b}$. This problem can be avoided by
prolonging to a higher dimensional Lorentzian manifold where the parameters of
the basis of the linear isotropy group are now coordinates. While this
approach is mathematically well-defined, it is unsatisfactory from a physical
perspective where all calculations should be performed on the original
manifold.

In the current paper we introduce a new approach to determine the symmetries
of any geometry based on a frame bases and an independent connection which
admits the torsion tensor and the curvature tensor as geometric objects. In
such theories, the connection will be an independent object. For each example,
we will also specialize to teleparallel gravity by setting the curvature
tensor to zero. We will call any geometry where the non-metricity and
curvature tensors vanish a teleparallel geometry. The approach introduced here
relies on the existence of a particular class of invariantly defined frames
known as symmetry frames, which is amenable to solving the differential
equations arising from (2). In this paper, we have assumed the connection is
metric compatible in order to discuss Riemann-Cartan geometries. However, the
approach outlined in this paper can be readily extended to geometries where
the non-metricity tensor is non-zero, i.e., where the connections are no
longer metric-compatible.

As an alternative to Riemannian geometries which are typically characterized
by the curvature tensor of a Levi-Civita connection calculated from the
metric, teleparallel geometries are characterized by the torsion tensor.
Torsion being a function of the coframe, $\mbox{{\bf h}}^{a}$, derivatives of
the coframe, and a zero curvature and metric compatible spin connection. In
teleparallel geometries the spacetime indexed representation of the
connection, sometimes called the Weitzenböck connection, can be expressed via
derivatives of the frame added to the spin connection,

$\displaystyle\Gamma^{\rho}_{~{}\nu\mu}=h_{c}^{~{}\rho}\left(\partial_{\mu}h^{c}_{~{}\nu}+\omega^{c}_{~{}b\mu}h^{b}_{\nu}\right).$

Assuming an orthonormal frame, the spin connection $\omega^{a}_{~{}bc}$, is
defined in terms of an arbitrary Lorentz transformation, $\Lambda^{a}_{~{}b}$
through the equation,

$\displaystyle\omega^{a}_{~{}bc}=\Lambda^{a}_{~{}d}\mbox{{\bf
h}}_{c}((\Lambda^{-1})^{d}_{~{}b}),$ (3)

where $\mbox{{\bf h}}_{c}((\Lambda^{-1})^{d}_{~{}b})$ denotes a frame vector
acting as a vector field derivation on the components of
$(\Lambda^{-1})^{d}_{~{}b}$.

Teleparallel geometries provide an alternative framework in which to build a
theory of gravity. A variety of teleparallel gravitational theories based on a
Lagrangian can be constructed using various scalars built from the torsion and
functions thereof. One subclass of teleparallel gravitational theories is
dynamically equivalent to GR and is appropriately called the Teleparallel
Equivalent to General Relativity (TEGR) [5]. A particularly interesting
generalization of the TEGR is $F(T)$ teleparallel gravity [6, 7, 8]. In the
covariant approach to $F(T)$ teleparallel gravity [9], the teleparallel
geometry is defined in a gauge invariant manner as a geometry with zero
curvature, having a spin connection that vanishes in a very special class of
frames where all inertial effects are absent, and non-zero in all other frames
[10, 5, 9]. Therefore, the resulting teleparallel gravity theory has Lorentz
covariant field equations and is therefore locally Lorentz invariant [11].

In teleparallel geometries, assuming an orthonormal frame, it is always
possible to transform to a frame where the components of the spin connection
are zero. Such a frame-connection pair is called a proper frame and is
sometimes desirable when analysing the $F(T)$ field equations. However, a
proper frame is not invariantly defined since it is defined in terms of the
connection, which is not a tensorial quantity. There have been attempts to use
the proper frame to determine the symmetries of a teleparallel geometry, and
for a given group of symmetries determine the largest class of teleparallel
geometries which admit this group [12, 13, 14].

There are two significant problems with using the proper frame to determine
symmetries. These problems are related to the fact that the proper frame is
not invariantly defined. The first problem is that in the process of solving
the equations in (2), arbitrary functions of integration will be introduced
and it is not possible to determine when such a function is essential to the
geometry or if it can be removed by an appropriate coordinate transformation.
The second problem is that one cannot determine when two given solutions to
(2) are inequivalent and this can lead to an incomplete list of all geometries
which admit a given symmetry group. Our approach differs by choosing a
particular class of symmetry frames, that we will define later, which are
invariantly defined up to the linear isotropy group and admit a non-zero spin
connection. Then by restricting to a particular coordinate system and imposing
a chosen vielbein form, all of the remaining frame parameters can be fixed,
giving rise to an invariant frame (for this choice of coordinate system). With
this class of frames, all inequivalent geometries admitting a given symmetry
group can be determined along with the minimal number of functions in the
vielbein and spin connection to specify them.

We note as well that the proper frame approach cannot predict the existence or
non-existence of permitted symmetry groups without extensive calculation. As
an example of this, in [15] it was proven that the only maximally symmetric
teleparallel geometry was Minkowski space, and that the (anti-) de Sitter
geometries admit, at most, a seven-dimensional symmetry group. This was later
verified using the proper frame approach by exhaustively proving that the
resulting differential equations from (2) have no solution [16]. In addition,
the proper frame approach is applicable only to teleparallel geometries and
cannot be used for Riemann-Cartan geometries.

The outline of the paper is as follows. In the remainder of the introduction
we briefly specify the notation used in the paper. In section II we introduce
the class of symmetry frames and define affine frame symmetries for any
Riemann-Cartan geometry. In section III we employ symmetry frames to determine
the equations required to determine the most general geometries which admit a
given symmetry group, and specialize this to teleparallel geometries. In
section IV, we provide several examples of this approach by considering the
one-dimensional symmetry group, spherical symmetry, Robertson-Walker (RW)
symmetry and two seven-dimensional symmetry groups which are special subcases
of the RW geometries. In each case, we specialize to teleparallel geometries.
In section V we summarize our results and discuss further applications. Note
that when discussing teleparallel geometries, the associated proper frames
will not be discussed.

###  I.1 Notation

We will denote the coordinate indices by $\mu,\nu,\ldots$ and the tangent
space indices by $a,b,\ldots$. Unless otherwise indicated the spacetime
coordinates will be $x^{\mu}$. The frame fields are denoted as $\mbox{{\bf
h}}_{a}$ and the dual coframe one-forms are $\mbox{{\bf h}}^{a}$. The vielbein
components are $h_{a}^{~{}\mu}$ or $h^{a}_{~{}\mu}$. The spacetime metric will
be denoted as $g_{\mu\nu}$ while the Minkowski tangent space metric is
$\eta_{ab}$.

To denote a local Lorentz transformation leaving $\eta_{ab}$ unchanged, we
write $\Lambda_{a}^{~{}b}(x^{\mu})$. The spin connection one-form
$\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}$, is designated by
$\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=\omega^{a}_{~{}bc}\mbox{{\bf
h}}^{c}$. The curvature and torsion tensors will be denoted, respectively, as
$R^{a}_{~{}bcd}$ and $T^{a}_{~{}bc}$.

Covariant derivatives with respect to a metric-compatible connection will be
denoted using a semi-colon, $T_{abc;e}$. When considering the integral curves
of a vector field ${\bf X}$, with parameter $\tau$, given as $\phi_{\tau}:M\to
M$ we will write the corresponding Lorentz frame transformation as
$\phi_{\tau}^{*}\mbox{{\bf h}}^{a}=\Lambda^{a}_{(\tau)~{}b}\mbox{{\bf
h}}^{b}$.

##  II Affine frame symmetries with isotropy

The starting point for the determination of symmetries of a Riemann-Cartan
geometry, $M$, is the Cartan-Karlhede (CK) algorithm [15, 17]. In this
algorithm, a frame is chosen and the parameters of the Lorentz frame
transformations are fixed by normalizing the components of the curvature
tensor, torsion tensor and their covariant derivatives in an invariant manner.
This algorithm provides a wealth of invariants and these can be used to
uniquely characterize a geometry locally and determine the dimension of the
symmetry group.

Two useful sequences of discrete invariants are particularly important: the
dimension of the linear isotropy group, $dim~{}H_{p}$ at each iteration, $p$,
and the number of functionally independent components of the curvature tensor,
torsion tensor and their covariant derivatives up to order $p$, denoted as
$t_{p}$. The linear isotropy group, $H_{p}$ is a subgroup of the Lorentz frame
transformations that leave the curvature tensor, the torsion tensor and their
covariant derivatives up to $p^{th}$ order invariant. If $H_{p}$ is the
trivial group, we say that the frame is now an invariant frame. The algorithm
ends when $t_{p-1}=t_{p}$ and $dim~{}H_{p-1}=dim~{}H_{p}$, which indicates
that all necessary and sufficient geometric information for the geometry has
been produced.

At the conclusion of the algorithm, say at the $q^{th}$ iteration, the linear
isotropy group $H_{q}$ encodes the freedom in specifying an invariantly
defined (co)frame. That is, the CK algorithm produces an invariantly defined
(co)frame, $\mbox{{\bf h}}^{a}$, up to linear isotropy $H_{q}$ such that for
any diffeomorphism, $\phi:M\to M$, the invariantly defined (co)frame
satisfies:

$\displaystyle\phi^{*}\mbox{{\bf h}}^{a}=\Lambda^{a}_{~{}b}\mbox{{\bf
h}}^{b},$ (4)

where $\Lambda^{a}_{~{}b}$ belongs to $H_{q}$ and arises from the coordinate
form of $\phi$, i.e., $y^{\mu}=\phi^{\mu}(x^{\nu})$ [4]. If $\phi=\phi_{\tau}$
is generated by exponentiating the vector field locally representing an
infinitesimal generator of a symmetry, ${\bf X}$, with some parameter $\tau$,
then we can calculate the effect of the Lie derivative of ${\bf X}$ on the
frame as

$\displaystyle\mathcal{L}_{{\bf X}}\mbox{{\bf
h}}^{a}=\lambda^{a}_{~{}b}\mbox{{\bf h}}^{b},$ (5)

where $\lambda^{a}_{~{}b}$ is the Lie algebra generator for
$\Lambda^{a}_{(\tau)~{}b}$. By using an invariantly defined frame, up to the
linear isotropy group, we have reduced the number of unknown functions in
$\lambda^{a}_{~{}b}$ and restricted their functional dependence to only those
coordinates which are affected by $\phi_{\tau}$. To distinguish between those
coordinates which are affected or unaffected by the symmetry generators we
will denote these as $x^{\alpha^{\prime}}$ and $x^{\tilde{\alpha}}$,
respectively.

This is one part of the recipe for determining or fixing a symmetry in a
Riemann-Cartan geometry. Such manifolds are equipped with a metric-compatible
connection one-form
$\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=\omega^{a}_{~{}bc}\mbox{{\bf
h}}^{c}$, where
$\mbox{{\mbox{\boldmath{$\omega$}}}}_{ab}=-\mbox{{\mbox{\boldmath{$\omega$}}}}_{ba}$.
If ${\bf X}$ is an infinitesimal generator of a symmetry for the geometry then
it must also be an affine collineation [18]:

$\displaystyle\mathcal{L}_{{\bf X}}\omega^{a}_{~{}bc}=0,$ (6)

or equivalently, by using the Leibniz rule for the Lie derivative, this
condition can be restated for the connection one-form
$\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=\omega^{a}_{~{}bc}\mbox{{\bf
h}}^{c}$,

$\displaystyle\mathcal{L}_{{\bf
X}}\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=\omega^{a}_{~{}bc}\mathcal{L}_{{\bf
X}}\mbox{{\bf h}}^{c}=\omega^{a}_{~{}bc}\lambda^{c}_{~{}d}\mbox{{\bf h}}^{d}.$
(7)

We note that this condition can be rewritten as equation (40) in [13] using
the identity in equation (18). From the conditions (5) and (6) it follows that
the Lie derivative of the curvature tensor, the torsion tensor and its
covariant derivatives with respect to ${\bf X}$ will vanish. We will call any
vector field satisfying equations (5) and (6) an affine frame symmetry. We
note that if $\lambda^{a}_{~{}b}=0$ then we recover the original definition of
an affine frame symmetry in [15] where the linear isotropy group is trivial
and the frame resulting from the CK algorithm is an invariant frame.

While the above definition is necessary and sufficient to determine a symmetry
of any Riemann-Cartan geometry, in practice it is difficult to implement as
the parameters of the Lorentz transformations must be determined in an
invariant manner for each of the possible canonical forms of the curvature
tensor and torsion tensor. We will expand this definition to a larger class of
frames that are acted on by the Lie algebra generators of the isotropy group
under Lie differentiation with the affine frame symmetry generators.

######  Definition II.1.

Consider a Riemann-Cartan geometry, $({\mbox{{\bf
h}}}^{a},\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b})$, admitting an affine
frame symmetry, ${\bf X}$. The class of symmetry frames, $\mbox{{\bf h}}^{a}$,
satisfy the following condition

$\displaystyle\mathcal{L}_{{\bf X}}\mbox{{\bf
h}}^{a}=f_{X}^{~{}\hat{i}}\lambda^{a}_{\hat{i}~{}b}\mbox{{\bf h}}^{b},$ (8)

where $\lambda^{a}_{\hat{i}~{}b}$ are basis elements of the Lie algebra of the
isotropy group (so that $\lambda_{\hat{i}ab}=-\lambda_{\hat{i}ba}$) and
$\hat{i}$ ranges from 1 to the dimension of the isotropy group. The components
of $f_{X}^{~{}\hat{i}}$ are functions dependent on the coordinates affected by
the symmetry generators $x^{\alpha^{\prime}}$ and have been introduced to
permit the possibility of an arbitrary infinitesimal generator of an element
of the isotropy group. This is a general definition and is not (strictly
speaking) an invariantly defined frame until the components
$f_{X}^{~{}\hat{i}}$ are fixed in some coordinate independent way.

Assuming that equations (6) and (8) are satisfied, so that ${\bf X}$ generates
an affine frame symmetry, we find that for $g_{\mu\nu}=\eta_{ab}\mbox{{\bf
h}}^{a}_{~{}\mu}\mbox{{\bf h}}^{b}_{~{}\nu}$,

$\displaystyle\mathcal{L}_{{\bf X}}(\eta_{ab}\mbox{{\bf h}}^{a}\mbox{{\bf
h}}^{b})=f_{X}^{~{}\hat{i}}[\lambda_{\hat{i}bc}\mbox{{\bf h}}^{b}\mbox{{\bf
h}}^{c}+\lambda_{\hat{i}cb}\mbox{{\bf h}}^{b}\mbox{{\bf h}}^{c}]=0,$ (9)

which ensures that ${\bf X}$ is a Killing vector field as well.

The class of symmetry frames is guaranteed to exist since they are constructed
by taking the class of frames generated from the CK algorithm, which are
invariantly defined up to the linear isotropy group, and acting upon them by
the group of Lorentz transformations that leave the representation of the
isotropy group unchanged (which we have denoted as $\overline{Iso}$). This can
be determined using the chosen Lie algebra basis for the Lorentz group and
picking those Lie algebra generators whose commutator with the Lie algebra
elements of the isotropy group lie in the isotropy group. To be precise, we
can determine the Lie algebra generators of $\overline{Iso}$,
$\tilde{\lambda}$, by solving the $n$ equations,
$[\lambda_{\hat{i}},\tilde{\lambda}]\in~{}\text{span}\\{\lambda_{\hat{1}},\ldots,\lambda_{\hat{n}}\\}$,
and then exponentiating the linearly independent parts of $\tilde{\lambda}$ to
produce $\overline{Iso}$. For example, in the case of spherical symmetry,
there are three generators of the isotropy group, ${\bf X}_{I}$, $I=1,2,3$ and
the group $\overline{Iso}$ consists of all spatial rotations.

Since the components of $f_{X}^{~{}\hat{i}}$ are associated with the Lie
derivative of a symmetry generating vector field, they are tensor quantities
that depend on the frame. Under a change of frame, $\tilde{\mbox{{\bf
h}}}^{a}=\tilde{\Lambda}^{a}_{~{}b}\mbox{{\bf h}}^{b}$, the components
transform as:

$\displaystyle{\bf
X}_{I}(\tilde{\Lambda}^{a}_{~{}b})[\tilde{\Lambda}^{-1}]^{b}_{~{}c}+\tilde{\Lambda}^{a}_{~{}b}f_{I}^{~{}\hat{i}}\lambda^{b}_{\hat{i}~{}d}[\tilde{\Lambda}^{-1}]^{d}_{~{}c}=\tilde{f}_{I}^{~{}\hat{i}}\lambda^{a}_{\hat{i}~{}c}.$
(10)

We can employ them to construct invariantly defined frames up to some minimal
subgroup of $\overline{Iso}$ or invariant frames if this subgroup is trivial.
To be precise, if we employ the frame freedom in $\overline{Iso}$ to fix the
form of the $f_{X}^{~{}\hat{i}}$ as much as possible, so that some elements
are constant or zero, the symmetry frame is then an invariantly defined
symmetry frame up to the isotropy group $\overline{H}_{q}$, where
$\overline{H}_{q}$ is some subgroup of $\overline{Iso}$ which leaves the
chosen form of $f^{~{}\hat{i}}_{X}$ unchanged. Again, using the spherically
symmetric case as an example, a choice of $f^{~{}\hat{i}}_{{X}_{I}}\equiv
f^{~{}\hat{i}}_{I}$, $I=1,2,3$, for each symmetry generator can be made so
that this subgroup, $\overline{H}_{q}$, consists of rotations in the
$\mbox{{\bf h}}^{3}-\mbox{{\bf h}}^{4}$ plane where their Lorentz parameters
are constant. In subsection IV, we will outline the procedure to fix the
components of $f_{X}^{~{}\hat{i}}$ and determine $\overline{H}_{q}$ for the
axially-symmetric, spherically symmetric and spatially homogeneous and
isotropic symmetric geometries, respectively.

We note that once a class of symmetry frames is invariantly defined up to the
group $\overline{H}_{q}$, any additional fixing of the remaining frame
parameters should be maximally fixed, up to the linear isotropy group of the
CK algorithm, $H_{q}$, in an invariant manner. This can be done using the
curvature tensor, the torsion tensor or their respective covariant
derivatives, as in the CK algorithm. While this can be done readily for a
given geometry, or even particular subclasses of geometries, it is difficult
to implement for a general class of geometries. However, since the solution to
equation (8) yields a particular frame ansatz, we can sidestep the problem of
fixing the frame invariantly by restricting the set of permitted
diffeomorphisms of the geometry to the “shape preserving” transformations,
those which preserve the function form of the metric or coframe [19] but alter
the expressions for the metric or vielbein functions. That is, we can make a
choice of vielbein, which provides a way to further fix the frame fully and
construct an invariant frame, relative to the given coordinate system. This is
sufficient for finding inequivalent solutions but leads to difficulties if
they are to be subclassified.

##  III Determining the most general geometry for a given isometry group

Let us assume that we have a Lie group of affine frame symmetries of dimension
$N$, with a non-trivial isotropy group with dimension $n$ $(n<N)$. We choose
coordinates so that the symmetry group is represented as a set of vector
fields, ${\bf X}_{I}$, where $I,J,K\in[1,N]$, with the corresponding Lie
algebra:

$\displaystyle[{\bf X}_{I},{\bf X}_{J}]=C^{K}_{~{}IJ}{\bf X}_{K},$ (11)

where $C^{K}_{~{}IJ}$ are structure constants of the Lie algebra.

Then we can determine the most general Riemann-Cartan geometries admitting
this Lie group as affine symmetries by solving the following equations for an
orthonormal symmetry coframe 111A null, or even complex null frame will work
as well, depending on the desired representation of the isotropy group,
$\mbox{{\bf h}}^{a}$, the associated metric,
$g_{\mu\nu}=\eta_{ab}h^{a}_{~{}\mu}h^{b}_{~{}\nu}$, and with a metric-
compatible connection, $\omega^{a}_{~{}bc}$:

$\displaystyle\mathcal{L}_{{\bf X}_{I}}\mbox{{\bf
h}}^{a}=f_{I}^{~{}\hat{i}}\lambda^{a}_{~{}\hat{i}~{}b}\mbox{{\bf h}}^{b}$ (12)
$\displaystyle\mathcal{L}_{{\bf X}_{I}}\omega^{a}_{~{}bc}=0,$ (13)

where $\hat{i},\hat{j},\hat{k}\in[1,n]$ and the components of
$f_{I}^{~{}\hat{i}}$ are functions of the coordinates, $x^{\alpha^{\prime}}$.
The form of $f_{I}^{~{}\hat{i}}$ can be specified by choosing some components
to be equal to others, to be constant or to be zero. This defines a class of
symmetry frames invariantly up to $\overline{H}_{q}$. For the moment we will
consider an arbitrary symmetry frame. In addition, due to the properties of
the Lie derivative, we have an additional relationship coming from (11):

$\displaystyle[\mathcal{L}_{{\bf X}_{I}},\mathcal{L}_{{\bf X}_{J}}]\mbox{{\bf
h}}^{a}=\mathcal{L}_{[{\bf X}_{I},{\bf X}_{J}]}\mbox{{\bf
h}}^{a}=C^{K}_{~{}IJ}\mathcal{L}_{{\bf X}_{K}}\mbox{{\bf h}}^{a}.$ (14)

We will focus for the moment on equations (12) and (14). These can be expanded
to give the following equations:

$\displaystyle
X_{I}^{~{}\nu}\partial_{\nu}h^{a}_{~{}\mu}+\partial_{\mu}{X}_{I}^{~{}\nu}h^{a}_{~{}\nu}=f_{I}^{~{}\hat{i}}\lambda^{a}_{\hat{i}~{}b}h^{b}_{~{}\mu},$
(15)

and

$\displaystyle 2{\bf
X}_{[I}(f_{J]}^{~{}\hat{k}})-f_{I}^{~{}\hat{i}}f_{J}^{~{}\hat{j}}C^{\hat{k}}_{~{}\hat{i}\hat{j}}=C^{K}_{~{}IJ}f_{K}^{~{}\hat{k}},$
(16)

where the associated Lie algebra of the isotropy group is given as:

$\displaystyle[\lambda_{\hat{i}},\lambda_{\hat{j}}]=\lambda^{a}_{\hat{i}~{}b}\lambda^{b}_{\hat{j}~{}c}-\lambda^{a}_{\hat{j}~{}b}\lambda^{b}_{\hat{i}~{}c}=C^{\hat{k}}_{~{}\hat{i}\hat{j}}\lambda^{a}_{\hat{k}~{}c}.$

These equations determine the most general frame basis which admits a given
symmetry group with a non-trivial isotropy group. To determine the connection
we must solve equation (13) relative to this frame. Using the coordinate basis
expression for the Lie derivative of the connection in [21], the corresponding
frame basis expression is,

$\displaystyle\mathcal{L}_{{\bf X}_{I}}(\omega^{a}_{~{}bc})\mbox{{\bf
h}}_{a}={\bf R}({\bf X},\mbox{{\bf h}}_{c})\mbox{{\bf h}}_{b}-{\bf T}({\bf
X},\nabla_{c}\mbox{{\bf h}}_{b})+\nabla_{c}{\bf T}({\bf X},\mbox{{\bf
h}}_{b})+\nabla_{c}\nabla_{b}{\bf X}-\nabla_{\nabla_{c}\mbox{{\bf h}}_{b}}{\bf
X},$ (17)

where ${\bf R}(\mbox{{\bf h}}_{d},\mbox{{\bf h}}_{c})\mbox{{\bf
h}}_{b}=R^{a}_{~{}bcd}\mbox{{\bf h}}_{a}$ and ${\bf T}(\mbox{{\bf
h}}_{c},\mbox{{\bf h}}_{b})=T^{a}_{~{}bc}\mbox{{\bf h}}_{a}$, respectively,
denote the curvature tensor and torsion tensor of the connection
$\omega^{a}_{~{}bc}$, and $\nabla_{a}=\nabla_{\mbox{{\bf h}}_{a}}$.

We can use equations (12) and (13) to find the following set of equations for
$\omega^{a}_{~{}bc}$:

$\displaystyle\mathcal{L}_{{\bf
X}_{I}}(\omega^{a}_{~{}bc})={X}_{I}^{~{}d}\mbox{{\bf
h}}_{d}(\omega^{a}_{~{}bc})+\omega^{d}_{~{}bc}f_{I}^{~{}\hat{i}}\lambda^{a}_{\hat{i}~{}d}-\omega^{a}_{~{}dc}f_{I}^{~{}\hat{i}}\lambda^{d}_{\hat{i}~{}b}-\omega^{a}_{~{}bd}f_{I}^{~{}\hat{i}}\lambda^{d}_{\hat{i}~{}c}-\mbox{{\bf
h}}_{c}(f_{I}^{~{}\hat{i}})\lambda^{a}_{\hat{i}~{}b}=0$ (18)

where $\mbox{{\bf h}}_{c}(f(x^{\alpha}))$ denotes the frame derivative of a
function $f(x^{\alpha})$ with respect to $\mbox{{\bf h}}_{c}$.

######  Theorem III.1.

The most general Riemann-Cartan geometry admitting a given group of
symmetries, ${\bf X}_{I},~{}I,J,K\in\\{1,\ldots,N\\}$ with a non-trivial
isotropy group of dimension $n$ can be determined by solving for the unknowns
$h^{a}_{~{}\mu}$, $f_{I}^{~{}\hat{i}}$ (with
$\hat{i},\hat{j},\hat{k}\in\\{1,\ldots,n\\}$) and $\omega^{a}_{~{}bc}$ from
the following equations:

$\displaystyle\begin{aligned}
&{X}_{I}^{~{}\nu}\partial_{\nu}h^{a}_{~{}\mu}+\partial_{\mu}{X}_{I}^{~{}\nu}h^{a}_{~{}\nu}=f_{I}^{~{}\hat{i}}\lambda^{a}_{\hat{i}~{}b}h^{b}_{~{}\mu}\\\
&2{\bf
X}_{[I}(f_{J]}^{~{}\hat{k}})-f_{I}^{~{}\hat{i}}f_{J}^{~{}\hat{j}}C^{\hat{k}}_{~{}\hat{i}\hat{j}}=C^{K}_{~{}IJ}f_{K}^{~{}\hat{k}}\\\
&{X}_{I}^{~{}d}\mbox{{\bf
h}}_{d}(\omega^{a}_{~{}bc})+\omega^{d}_{~{}bc}f_{I}^{~{}\hat{i}}\lambda^{a}_{\hat{i}~{}d}-\omega^{a}_{~{}dc}f_{I}^{~{}\hat{i}}\lambda^{d}_{\hat{i}~{}b}-\omega^{a}_{~{}bd}f_{I}^{~{}\hat{i}}\lambda^{d}_{\hat{i}~{}c}-\mbox{{\bf
h}}_{c}(f_{I}^{~{}\hat{i}})\lambda^{a}_{\hat{i}~{}b}=0.\end{aligned}$ (19)

where $\\{\lambda^{a}_{\hat{i}~{}b}\\}_{\hat{i}=1}^{n}$ is a basis of the Lie
algebra of the isotropy group,
$[\lambda_{\hat{i}},\lambda_{\hat{j}}]=C^{\hat{k}}_{~{}\hat{i}\hat{j}}\lambda_{\hat{k}}$,
$[{\bf X}_{I},{\bf X}_{J}]=C^{K}_{~{}IJ}{\bf X}_{K}$.

We remark that while the above equations apply to any symmetry frame, solving
these equations can be more easily attained by using an invariantly defined
symmetry frame up to $\overline{H}_{q}$ where the components of
$f_{I}^{~{}\hat{i}}$ have been invariantly specified. As discussed in the
previous subsection, this entails using the group $\overline{Iso}$ to fix the
form of the matrix $f_{I}^{~{}\hat{i}}$ in an invariant manner, for example by
setting some components to be constant, to be zero or equal to other
components.

To specialize this result to a teleparallel geometry, in addition to the
differential equation coming from equation (6), the connection must be flat
and so the curvature tensor expressed in terms of the spin connection and its
derivatives must vanish. This condition is equivalent to (3) for some choice
of Lorentz transformation $\Lambda^{a}_{~{}b}$. In summary:

######  Theorem III.2.

The most general teleparallel geometry which admits a given group of
symmetries, ${\bf X}_{I},~{}I,J,K\in\\{1,\ldots N\\}$ with a non-trivial
isotropy group of dimension $n$ can be determined by solving for the unknowns
$h^{a}_{~{}\mu}$, $f_{I}^{~{}\hat{i}}$ (with
$\hat{i},\hat{j},\hat{k}\in\\{1,\ldots n\\}$) and $\omega^{a}_{~{}bc}$ from
equations (19) along with an additional condition:

$\displaystyle\begin{aligned}
&R^{a}_{\phantom{a}bcd}=h_{c}^{\mu}\partial_{\mu}\omega^{a}_{\phantom{a}bd}-h_{d}^{\nu}\partial_{\nu}\omega^{a}_{\phantom{a}bc}+\omega^{a}_{\phantom{a}fc}\omega^{f}_{\phantom{a}bd}-\omega^{a}_{\phantom{a}fd}\omega^{f}_{\phantom{a}bc}=0.\end{aligned}$
(20)

##  IV Examples

###  IV.1 One-dimensional symmetry groups

If the symmetry group is one-dimensional, one may always choose a
neighbourhood and rectify the infinitesimal generator, ${\bf X}$, so that it
is of the form ${\bf X}=\partial_{\chi}$ with coordinates
$x^{\mu}=(x^{\tilde{\alpha}},\chi)$ with $\tilde{\alpha}=1,2,3$. We note that
the coordinate $\chi$ can be spacelike, timelike or null, depending on the
nature of the affine frame symmetry. To distinguish the behaviour of the
resulting affine-frame symmetry, we must examine the norm of the vector field
${\bf X}$ and its effect on the symmetry frame:

$\displaystyle\mathcal{L}_{{\bf X}}\mbox{{\bf
h}}^{a}=f(\chi)\lambda^{a}_{~{}b}\mbox{{\bf h}}^{b}.$ (21)

where $f(\chi)$ is an arbitrary function to be determined. If $f$ is
identically zero, the resulting affine frame symmetry is translational, while
if $f$ is non-trivial, the affine frame symmetry is an affine frame isotropy.
By computing the squared norm of $\lambda^{a}_{~{}b}$ we can use global
Lorentz transformations to simplify the form of this anti-symmetric matrix
into four classes: a boost, a null rotation, a loxodromic transformation or a
spatial rotation. Following [22] we will use two-forms to describe the
matrices $\lambda_{ab}=\eta_{ac}\lambda^{c}_{~{}b}$ so that these
transformations are described relative to an orthonormal frame, respectively,
as:

$\displaystyle\lambda_{ab}\in\\{\mbox{{\bf h}}^{1}\wedge\mbox{{\bf
h}}^{4},~{}\mbox{{\bf h}}^{1}\wedge\mbox{{\bf h}}^{2}+\mbox{{\bf
h}}^{2}\wedge\mbox{{\bf h}}^{4},\mbox{{\bf h}}^{1}\wedge\mbox{{\bf
h}}^{4}+a_{0}\mbox{{\bf h}}^{2}\wedge\mbox{{\bf h}}^{4},\mbox{{\bf
h}}^{2}\wedge\mbox{{\bf h}}^{3}\\}.$ (22)

where $a_{0}$ is a real-valued constant acting as a fixed parameter in the
loxodromic transformation. In this general setting, determining
$\overline{Iso}$ and $\overline{H}_{q}$ for each of the possible Lie algebra
generators in this representation is a tedious but straightforward
calculation. We will employ axisymmetry in order to illustrate this approach
in section IV.1.1 .

Without loss of generality, we can use a Lorentz transformation to eliminate
the arbitrary function $f$ using
$\tilde{\Lambda}^{a}_{~{}b}=[e^{\tau{\bf\lambda}}]^{a}_{~{}b}$ and requiring
that the parameter $\tau$ is a function of $\chi$ determined by equation (10).
Explicitly, we will solve the following differential equation:

$\displaystyle
X(\tilde{\Lambda}^{a}_{~{}b})[\tilde{\Lambda}^{-1}]^{b}_{~{}c}+\tilde{\Lambda}^{a}_{~{}b}f\lambda^{b}_{~{}e}[\tilde{\Lambda}^{-1}]^{e}_{~{}c}=0.$
(23)

Since $\tilde{\Lambda}^{a}_{~{}b}$ has the same Lie algebra generator as
$\lambda^{a}_{~{}b}$, $\lambda^{a}_{~{}b}$ in the second term is invariant
under this transformation, and the matrix equation reduces to a simple
differential equation:

$\displaystyle\tau_{,\chi}+f=0.$

Thus by integrating $\tau$ we can always remove $f$ from the Lie derivative of
$\mbox{{\bf h}}^{a}$ and instead involve it in the connection, according to
the transformation rule for the connection one-form:

$\displaystyle\tilde{\mbox{{\mbox{\boldmath{$\omega$}}}}}^{a}_{~{}b}=\Lambda^{a}_{~{}c}\mbox{{\mbox{\boldmath{$\omega$}}}}^{c}_{~{}d}(\Lambda^{-1})^{d}_{~{}b}+\Lambda^{a}_{~{}c}(d\Lambda^{-1})^{c}_{~{}b}.$
(24)

The new equation for the symmetry frame is now

$\displaystyle\mathcal{L}_{{\bf X}}\tilde{\mbox{{\bf h}}}^{a}=0,$ (25)

which implies that the vielbein matrix components are independent of $\chi$,
i.e., $\mbox{{\bf h}}^{a}=h^{a}_{~{}\alpha}(x^{\tilde{\alpha}})dx^{\alpha}$.
This frame is another symmetry frame, however, it is now invariantly defined
up to $\overline{H}_{q}$ since $f_{I}^{~{}\hat{i}}=0$. Imposing the affine
frame symmetry conditions on the new connection gives an additional condition
on the function $f$.

In this coordinate system, with the appropriate symmetry frame determined we
can impose the second condition for an affine frame symmetry:

$\displaystyle\mathcal{L}_{{\bf X}}\omega^{a}_{~{}bc}={\bf
X}(\omega^{a}_{~{}bc})=0.$ (26)

This is satisfied if and only if the connection coefficients,
$\omega^{a}_{~{}bc}$, are independent of $\chi$ and so any choice of
$\omega^{a}_{~{}bc}=W^{a}_{~{}bc}(x^{\tilde{\alpha}})$ is sufficient to define
the connection. Using the transformation rule for the connection one-form with
the Lorentz transformation $\tilde{\Lambda}^{a}_{~{}b}$ with parameter $\tau$
such that $\partial_{\chi}\tau=-f$, Lie differentiation with respect to $\chi$
of this new connection one-form must be zero, and this implies that
$\partial_{\chi}^{2}\tau=0$. Since $\partial_{\chi}\tau=-f$, this function is
independent of $\chi$, and hence is a constant, $\epsilon$. With the function
$f(\chi)$ specified, we have the following result.

######  Proposition IV.1.

The most general Riemann-Cartan geometry admitting a one-dimensional symmetry,
${\bf X}=\partial_{\chi}$ is given by a coframe and a connection of the form

$\displaystyle\mbox{{\bf
h}}^{a}=h^{a}_{~{}\alpha}(x^{\tilde{\alpha}})dx^{\alpha},\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=W^{a}_{~{}bc}(x^{\tilde{\alpha}})\mbox{{\bf
h}}^{c}-\mbox{{\bf h}}_{c}(\epsilon\chi)\lambda^{a}_{~{}b}\mbox{{\bf h}}^{c}$
(27)

where $\mbox{{\bf h}}_{c}(\epsilon\chi)$ denotes the derivation of the linear
function $\epsilon\chi$ by the frame element $\mbox{{\bf h}}_{c}$ and
$\epsilon=0,1$.

The magnitude of the infinitesimal generator $|{\bf X}|$ determines whether
the affine frame symmetry is timelike, null or spacelike at each point, while
the Lie algebra element $\lambda^{a}_{~{}b}$ indicates whether the symmetry is
translational, or it admits a fixed point and hence is an isotropy.

We remark here that, that the second term in the connection one-form in
equation (27) can always be absorbed into the first term. However, where our
intention is to construct the most general class of Riemann-Cartan geometries
which admit one affine frame symmetry, we have left this second term explicit,
in order to emphasise the role of $\lambda^{a}_{~{}b}$ in choosing the affine
frame symmetry.

We also note that the frame in equation (27) is invariantly defined (up to
$\overline{H}_{q}$), as this example is very general, we cannot explicitly fix
the frame parameters further but point out that it is always possible to
construct an invariantly defined frame when the symmetry group is one
dimensional.

In the case of teleparallel geometries, there exists a local Lorentz
transformation, $\hat{\Lambda}^{a}_{~{}b}$, that yields the spin connection
through the equation

$\displaystyle\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=\hat{\Lambda}^{a}_{~{}c}(d\hat{\Lambda}^{-1})^{c}_{~{}b}.$
(28)

In this case, it is easiest to formulate the most general frame and connection
admitting a single affine frame symmetry by switching to a proper coframe
first, i.e.,

$\displaystyle{\bf\hat{h}}^{a}=(\hat{\Lambda}^{-1})^{a}_{~{}b}\mbox{{\bf
h}}^{b},~{}\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}={\bf 0}$

Relative to the proper frame, the symmetry generator acts as

$\displaystyle\mathcal{L}_{{\bf
X}}{\bf\hat{h}}^{a}=\lambda^{a}_{~{}b}{\bf\hat{h}}^{a}$

where $\lambda^{a}_{~{}b}$ is some arbitrary element of the Lie algebra of the
Lorentz group.

In previous papers it has been shown that $\lambda^{a}_{~{}b}$ must have
constant components [12]. Due to this, it is tempting to use the proper frame
as the preferred frame to determine symmetries. However, the determination of
proper frames is not invariantly defined using tensor quantities, and so it is
difficult to determine when an arbitrary function in a given solution is
essential, or only a coordinate artefact. Whereas in our approach, the
arbitrary functions in the vielbein describe the minimal number of functions
required to be a teleparallel geometry admitting one affine frame symmetry.

If $\lambda^{a}_{~{}b}$ vanishes, then the symmetry generator is
translational. Otherwise, this constant matrix can be specified using global
Lorentz transformations depending on the squared norm of $\lambda^{a}_{~{}b}$.
If it is negative, positive or zero, the matrix $\lambda^{a}_{~{}b}$ can be
chosen as a simple generator of a boost, rotation or null rotation,
respectively.

Once this invariant is determined, the above process can be repeated for a
teleparallel geometry. Using the fact that we can always pick the proper frame
in equation (21), we can employ global Lorentz transformations to bring
$\lambda^{a}_{~{}b}$ into one of the canonical representations for a boost,
rotation or a null rotation. Then by applying the Lorentz transformation
associated with $\lambda^{a}_{~{}b}$ and solving a simple differential
equation, it is possible to shift this term to the spin connection so that the
Lie derivative of the frame basis with respect to ${\bf X}$ is zero. The role
of $\lambda^{a}_{~{}b}$ is even more apparent in the coordinate basis as only
the $d\chi$ term appears in the connection one-form:

$\displaystyle-\mbox{{\bf h}}_{c}(\epsilon\chi)\lambda^{a}_{~{}b}\mbox{{\bf
h}}^{c}=-\epsilon\lambda^{a}_{~{}b}d\chi.$ (29)

Summarizing these observations, we have the following result.

######  Proposition IV.2.

The most general teleparallel geometry admitting a one-dimensional symmetry,
${\bf X}=\partial_{\chi}$ is given by a frame and a connection of the form

$\displaystyle{\mbox{{\bf
h}}}^{a}={h}^{a}_{~{}\alpha}(x^{\tilde{\alpha}})dx^{\alpha},\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=-\epsilon\lambda^{a}_{~{}b}d\chi.$
(30)

where $\mbox{{\bf h}}_{c}(\epsilon\chi)$ denotes the derivation of the linear
function $\epsilon\chi$ by the frame element $\mbox{{\bf h}}_{c}$ and
$\epsilon=0,1$.

The magnitude of the infinitesimal generator $|{\bf X}|$ determines whether
the affine frame symmetry is timelike, null or spacelike at each point, while
the Lie algebra element $\lambda^{a}_{~{}b}$ indicates whether the symmetry is
translational, or it admits a fixed point and hence is an isotropy.

As in the Riemann-Cartan case, the frame is invariantly defined up to
$\overline{H}_{q}$. In each example of a one dimensional symmetry group, an
invariantly defined symmetry frame can be found.

The frame and spin connection given here is the most general as the majority
of the spin connection has been absorbed into the frame which consists of
fully arbitrary components dependent on $x^{\tilde{\alpha}}$. The remaining
non-zero part of the spin connection can only be removed using a Lorentz
transformation with $\chi$-dependent parameters and the resulting symmetry
frame is no longer invariantly defined.

####  IV.1.1 An application: Axisymmetric geometries

As an example that will be relevant in the remaining examples, we will
consider the class of geometries which admit rotational isotropy. These have
been discussed previously for teleparallel solutions, either by using the
axially symmetric field equations [23] or by using the proper frame approach
[13].

In [13], two solutions were found to admit axial symmetry. One choice relied
on a non-trivial homomorphism from $SO(2)$ into $SO(1,3)$, while the second
arose from the trivial homomorphism from $SO(2)$ into $SO(1,3)$. The solution
corresponding to the trivial homomorphism is not axisymmetric, in the sense of
admitting a symmetry in the isotropy group, but instead admits a translational
affine frame symmetry, which can be seen by noting that the proper frame is
already a symmetry frame with $\lambda^{a}_{~{}b}=0$.

Our approach recovers both cases as solutions where, relative to the
appropriate symmetry frame, the connection one-form in equation (29) is either
non-trivial or identically zero, respectively. This provides an explanation
for why the second class of frames corresponding to the trivial homomorphism
in [13] does not have a proper limit to spherically symmetric geometries. In
order for this to occur, the affine-symmetry generator $\partial_{\phi}$ must
lie in the isotropy group in the context of spherical symmetry but the
vanishing of the connection one-form, relative to the invariantly defined
symmetry frame, implies that $\partial_{\phi}$ does not belong to the isotropy
subgroup and hence cannot be expanded into the spherically symmetric group.

For these axisymmetric geometries we will work in the coordinate system
$x^{\mu}=(t,r,\theta,\phi)$ and treat the vector field ${\bf
X}=\partial_{\phi}$ as the generator of an axisymmetric symmetry. The Lie
algebra generator is then

$\displaystyle\lambda^{a}_{~{}b}=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\end{array}\right]$ (35)

which corresponds to the representation $\lambda=\mbox{{\bf
h}}^{3}\wedge\mbox{{\bf h}}^{4}$.

With this choice, we can determine the frame transformations that leave this
form invariant by solving for $\tilde{\lambda}$ from
$[\lambda,\tilde{\lambda}]=\lambda$ and exponentiating $\tilde{\lambda}$ to
determine the elements of the Lorentz frame transformation group.
$\overline{Iso}$ consists of rotations in the $\mbox{{\bf h}}^{3}-\mbox{{\bf
h}}^{4}$ plane, and boosts in the $\mbox{{\bf h}}^{1}-\mbox{{\bf h}}^{2}$
subspace. As in the general case of a Riemann-Cartan geometry, the function
$f(\phi)$ can be set to zero, which fixes the frame invariantly. To determine
$\overline{H}_{q}$, we solve equation (10) with
$f^{\hat{i}}_{I}=\tilde{f}^{\hat{i}}_{I}=0$ for $\tilde{\Lambda}^{a}_{~{}b}$
in $\overline{Iso}$. This implies that

$\displaystyle{\bf
X}_{I}(\tilde{\Lambda}^{a}_{~{}b})=0=\partial_{\phi}(\tilde{\Lambda}^{a}_{~{}b})=0,$

and so $\overline{H}_{q}$ consists of elements of $\overline{Iso}$ but with
constant parameters.

The remaining frame freedom can be fixed by either using the curvature and
torsion tensor or by employing shape-preserving transformations to simplify
the vielbein. Due to the large degree of generality in the vielbein, we will
omit this step, as further frame fixing will not reduce the number of
arbitrary functions. However, we note that it is always possible to construct
an invariantly defined frame using either approach for a given solution.

Relative to a generic coframe, where the vielbein are arbitrary but
independent of $\phi$:

$\displaystyle\mbox{{\bf
h}}^{a}=h^{a}_{~{}\beta}(x^{\tilde{\alpha}})dx^{\beta},$ (36)

with $x^{\tilde{\alpha}}=(t,r,\theta)$. To ensure the traditional axisymmetric
geometry, the norm of ${\bf X}$ must be spatial and this gives an inequality
for the vielbein,

$\displaystyle
g_{\alpha\beta}X^{\alpha}X^{\beta}=\eta_{ab}h^{a}_{~{}\alpha}h^{b}_{\beta}X^{\alpha}X^{\beta}=h^{a}_{~{}\phi}h_{a\phi}>0,$
(37)

in a neighbourhood of the axis, and zero only at points on the axis. The
connection one-form for a Riemann-Cartan geometry then takes the form

$\displaystyle\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=W^{a}_{~{}bc}(x^{\tilde{\alpha}})\mbox{{\bf
h}}^{c}-\mbox{{\bf h}}_{c}(\epsilon\phi)\lambda^{a}_{~{}b}\mbox{{\bf h}}^{c},$
(38)

where $\epsilon=0,1$

In the case of a teleparallel geometry, we can employ the same argument
leading to Proposition IV.2 to describe an axisymmetric teleparallel geometry
in terms of the following coframe and connection one-form:

$\displaystyle\mbox{{\bf
h}}^{a}=h^{a}_{~{}\beta}(x^{\tilde{\alpha}})dx^{\beta}\text{ and
}\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}=-\mbox{{\bf
h}}_{c}(\epsilon\phi)\lambda^{a}_{~{}b}\mbox{{\bf
h}}^{c}=-\epsilon\lambda^{a}_{~{}b}d\phi,$ (39)

where $\lambda^{a}_{~{}b}$ is given in equation (35) and the vielbein must
satisfy equation (37).

Within the class of axisymmetric geometries defined in Riemann-Cartan
geometries with non-zero torsion, the stationary axisymmetric solutions are
particularly important from a physical perspective as these will contain
solutions similar to the Kerr metric. To the authors’ knowledge these
geometries have not been investigated using symmetry methods [23]. Relative to
this coordinate system where the symmetry generator takes the form, ${\bf
X}=\partial_{\phi}$, stationary axisymmetric geometries can be generated by
the inclusion of a second affine frame symmetry, ${\bf Y}$ with $Y^{a}Y_{a}<0$
locally, which commutes with ${\bf X}$; i.e., $[{\bf X},{\bf Y}]=0$. A new
coordinate system can be chosen so that ${\bf X}=\partial_{\phi}$ and ${\bf
Y}=\partial_{t}$. The timelike condition on ${\bf Y}$ imposes an additional
condition on the vielbein.

In a similar manner to the stationary axisymmetric geometries, a rigorous
investigation of the static axisymmetric geometries in Riemann-Cartan
geometries with non-zero torsion has not been performed. However, within the
current framework, it is not difficult to formulate these geometries. Static
axisymmetric geometries arise by requiring that ${\bf Y}$ is hypersurface
orthogonal and hence satisfies

$\displaystyle{\bf Y}\wedge d{\bf Y}=0.$ (40)

The larger symmetry group introduces an additional function,
$f_{I}^{~{}\hat{i}}$, into equation (12) and this can be determined using
equation (14). The analysis of these equations is not difficult and lead to
the condition that the vielbein and the components of the connection one-form
must be independent of the coordinates $t$ and $\phi$.

###  IV.2 A three-dimensional symmetry group: spherical symmetry

Working in coordinates $x^{\mu}=(t,r,\theta,\phi)$, the affine frame symmetry
generators associated with spherical symmetry are,

$\displaystyle\begin{aligned} &{\bf X}_{z}=\partial_{\phi},~{}{\bf
X}_{y}=-\cos\phi\partial_{\theta}+\frac{\sin\phi}{\tan\theta}\partial_{\phi},{\bf
X}_{x}=\sin\phi\partial_{\theta}+\frac{\cos\phi}{\tan\theta}\partial_{\phi}.\end{aligned}$
(41)

Writing $\\{{\bf X}_{I}\\}_{I=1}^{3}=\\{{\bf X}_{x},{\bf X}_{y},{\bf
X}_{z}\\}$ the non-zero commutator constants, $C^{I}_{~{}JK}$ are

$\displaystyle
C^{{3}}_{~{}{1}{2}}=-1,~{}C^{{2}}_{~{}{1}{3}}=1,~{}C^{{1}}_{~{}{2}{3}}=-1$
(42)

The basis for the isotropy group is chosen to be:

$\displaystyle\lambda_{\hat{1}}=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\end{array}\right],~{}\lambda_{\hat{2}}=-\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&1&0\\\ 0&-1&0&0\\\
0&0&0&0\end{array}\right],~{}\lambda_{\hat{3}}=-\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&1\\\ 0&0&0&0\\\ 0&-1&0&0\end{array}\right]$ (55)

with the corresponding commutator constants,

$\displaystyle
C^{\hat{3}}_{~{}\hat{1}\hat{2}}=-1,~{}C^{\hat{2}}_{~{}\hat{1}\hat{3}}=1,~{}C^{\hat{1}}_{~{}\hat{2}\hat{3}}=-1$
(56)

To determine the group, $\overline{Iso}$ for this choice of isotropy
generators, we again consider an arbitrary element of the Lie algebra of
Lorentz frame transformations, $\tilde{\lambda}$ and impose that
$[\lambda_{\hat{i}},\tilde{\lambda}]\in~{}\text{span}\\{\lambda_{\hat{1}},\lambda_{\hat{2}},\lambda_{\hat{3}}\\}$.
The result of this calculation implies that
$\tilde{\lambda}\in~{}\text{span}\\{\lambda_{\hat{1}},\lambda_{\hat{2}},\lambda_{\hat{3}}\\}$
and so exponentiating each of the generators
$\lambda_{\hat{i}},~{}\hat{i}=1,2,3$ shows that $\overline{Iso}$ consists of
all spatial rotations.

We will exploit the freedom of choice in the components of
$f_{I}^{~{}\hat{i}}(x^{\alpha^{\prime}})$ using the isotropy group, where the
isotropy group affects a change to these components through equation (10).
Since ${\bf X}_{3}={\bf X}_{z}$ is a generator of a spatial rotation, we will
choose our frame so that ${\bf X}_{3}$ acts as a rotation on the basis
elements $\mbox{{\bf h}}^{3}$ and $\mbox{{\bf h}}^{4}$, i.e.,

$\displaystyle
f_{3}^{~{}\hat{i}}\lambda_{\hat{i}}=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&f_{3}^{~{}\hat{1}}\\\
0&0&-f_{3}^{~{}\hat{1}}&0\end{array}\right],$ (61)

then by applying a rotation about $\mbox{{\bf h}}^{3}$ and $\mbox{{\bf
h}}^{4}$, the remaining component, $f_{3}^{~{}\hat{1}}$, can be set to zero
using equation (10):

$\displaystyle{\bf
X}_{3}(\tilde{\Lambda}^{a}_{~{}b})[\tilde{\Lambda}^{-1}]^{b}_{~{}c}+\tilde{\Lambda}^{a}_{~{}b}f_{3}^{~{}\hat{i}}\lambda^{b}_{\hat{i}~{}e}[\tilde{\Lambda}^{-1}]^{e}_{~{}c}=0$
(62)

Choosing $f_{3}^{~{}\hat{i}}$ in this way, we can consider the combinations:
${\bf X}_{1}$ with ${\bf X}_{3}$, and ${\bf X}_{2}$ with ${\bf X}_{3}$ in
equation (14). These combinations yield the following non-trivial equations:

$\displaystyle\begin{aligned}
&\partial_{\phi}f_{1}^{~{}\hat{1}}+f_{2}^{~{}\hat{1}}=0,\partial_{\phi}f_{1}^{~{}\hat{2}}+f_{2}^{~{}\hat{2}}=0,\partial_{\phi}f_{1}^{~{}\hat{3}}+f_{2}^{~{}\hat{3}}=0.\\\
&\partial_{\phi}f_{2}^{~{}\hat{1}}-f_{1}^{~{}\hat{1}}=0,\partial_{\phi}f_{2}^{~{}\hat{2}}-f_{1}^{~{}\hat{2}}=0,\partial_{\phi}f_{2}^{~{}\hat{3}}-f_{1}^{~{}\hat{3}}=0,\end{aligned}$
(63)

which leads to the following solutions,

$\displaystyle\begin{aligned}
&f_{2}^{~{}\hat{i}}=f_{2}^{~{}\hat{i}}(\theta)\cos\phi+g_{2}^{~{}\hat{i}}(\theta)\sin\phi,\\\
&f_{1}^{~{}\hat{i}}=\partial_{\phi}f_{2}^{~{}\hat{i}}.\end{aligned}$ (64)

If we use Lorentz transformations that are dependent on $\theta$ alone, i.e.,
$\partial_{\phi}\tilde{\Lambda}^{a}_{~{}b}=0$, then we can, without loss of
generality, apply a rotation to set the coefficients of $\cos\phi$ in
$f_{2}^{~{}\hat{i}}$ to zero using equation (10):

$\displaystyle{\bf
X}_{2}(\tilde{\Lambda}^{a}_{~{}b})[\tilde{\Lambda}^{-1}]^{b}_{~{}c}+\tilde{\Lambda}^{a}_{~{}b}f_{2}^{~{}\hat{i}}\lambda^{b}_{\hat{i}~{}e}[\tilde{\Lambda}^{-1}]^{e}_{~{}c}=-f_{2}^{~{}\hat{i}}\lambda^{a}_{~{}\hat{i}~{}c}\cos\phi.$
(65)

Doing so, equation (14) with ${\bf X}_{2}$ and ${\bf X}_{1}$ gives:

$\displaystyle\partial_{\theta}g_{2}^{~{}\hat{i}}+\tan(\theta)g_{2}^{~{}\hat{i}}=0,$
(66)

which has a simple solution:

$\displaystyle g_{2}^{~{}\hat{i}}=\frac{C_{2}^{~{}\hat{i}}}{\sin(\theta)}.$
(67)

This fully determines $f_{1}^{~{}\hat{i}}$ and $f_{2}^{~{}\hat{i}}$, as the
first is the $\phi$-derivative of the other. Applying a rotation with constant
parameters, we can set $f_{2}^{~{}\hat{2}}=f_{2}^{~{}\hat{3}}=0$, without loss
of generality we will choose the first component to be non-zero and set the
arbitrary constant to one, and the solutions for the remaining components are:

$\displaystyle\begin{aligned}
f_{2}^{~{}\hat{1}}=\frac{\sin\phi}{\sin\theta},~{}f_{1}^{~{}\hat{1}}=\frac{\cos\phi}{\sin\theta}.\end{aligned}$
(68)

This gives the following matrix representation of the coefficients:

$\displaystyle
f_{I}^{~{}\hat{i}}=\left[\begin{array}[]{ccc}\frac{\cos(\phi)}{\sin(\theta)}&0&0\\\
\frac{\sin(\phi)}{\sin(\theta)}&0&0\\\ 0&0&0\end{array}\right].$ (72)

By making this choice, we have effectively fixed the frame up to
$\overline{H}_{q}$. To determine this subgroup of $\overline{Iso}$, we will
determine the form of $\tilde{\Lambda}$ which satisfy equation (10) with
$f_{I}^{~{}\hat{i}}=\tilde{f}_{I}^{~{}\hat{i}}$ taking the form of equation
(72). We have the following three matrix equations:

$\displaystyle\begin{aligned} &{\bf
X}_{1}(\tilde{\Lambda}^{a}_{~{}b})[\tilde{\Lambda}^{-1}]^{b}_{~{}c}+f_{1}^{~{}\hat{1}}[\tilde{\Lambda}^{a}_{~{}b}\lambda^{b}_{\hat{1}~{}d}[\tilde{\Lambda}^{-1}]^{d}_{~{}c}-\lambda^{a}_{\hat{1}~{}c}]=0\\\
&{\bf
X}_{2}(\tilde{\Lambda}^{a}_{~{}b})[\tilde{\Lambda}^{-1}]^{b}_{~{}c}+f_{2}^{~{}\hat{1}}[\tilde{\Lambda}^{a}_{~{}b}\lambda^{b}_{\hat{1}~{}d}[\tilde{\Lambda}^{-1}]^{d}_{~{}c}-\lambda^{a}_{\hat{1}~{}c}]=0\\\
&{\bf
X}_{3}(\tilde{\Lambda}^{a}_{~{}b})[\tilde{\Lambda}^{-1}]^{b}_{~{}c}=0.\end{aligned}$
(73)

The first equation implies the parameters of $\tilde{\Lambda}^{a}_{~{}b}$ must
be independent of $\phi$. Furthermore, using the chain rule, the first term
can rewritten as, ${\bf
X}_{I}(\tilde{\Lambda}^{a}_{~{}b})[\tilde{\Lambda}^{-1}]^{b}_{~{}c}={\bf
X}_{I}(g^{\hat{i}}))\lambda^{a}_{\hat{i}~{}c}$ where $g^{\hat{i}}$ are the
parameters of $\tilde{\Lambda}^{a}_{~{}b}$. Lastly, we can write

$\displaystyle\tilde{\Lambda}^{a}_{~{}b}\lambda^{b}_{\hat{1}~{}d}[\tilde{\Lambda}^{-1}]^{d}_{~{}c}=\alpha^{\hat{j}}\lambda^{a}_{\hat{j}~{}c},$

where the $\alpha^{\hat{j}}$ are expressions in terms of the parameters
$g^{\hat{i}}$. The remaining equations can be summarized in a single equation:

$\displaystyle\begin{aligned}
&\sin\phi\partial_{\theta}g^{\hat{i}}\lambda^{a}_{\hat{i}~{}c}-\frac{\cos\phi}{\sin\theta}[\alpha^{\hat{j}}\lambda^{a}_{\hat{j}~{}c}-\lambda^{a}_{\hat{1}~{}c}]=0.\\\
\end{aligned}$ (74)

The only solution, up to rescaling, of this differential equation is
$g^{\hat{1}}=1$ and $g^{\hat{2}}=g^{\hat{3}}=0$. This implies that the
subgroup $\overline{H}_{q}$ consists of rotations in the $\mbox{{\bf
h}}^{3}-\mbox{{\bf h}}^{4}$ plane with constant parameters.

Relative to the representation for the isotropy group in equation (55), we can
solve the first equation in (19),

$\displaystyle{X}_{I}^{~{}\nu}\partial_{\nu}h^{a}_{~{}\mu}+\partial_{\mu}{X}_{I}^{~{}\nu}h^{a}_{~{}\nu}=f_{I}^{~{}\hat{i}}\lambda^{a}_{\hat{i}~{}b}h^{b}_{~{}\mu}$
(75)

to determine the most general frame admitting this symmetry group:

$\displaystyle
h^{a}_{~{}\mu}=\left[\begin{array}[]{cccc}A_{1}(t,r)&A_{2}(t,r)&0&0\\\
A_{3}(t,r)&A_{4}(t,r)&0&0\\\ 0&0&A_{5}(t,r)&-A_{6}(t,r)\sin(\theta)\\\
0&0&A_{6}(t,r)&A_{5}(t,r)\sin(\theta)\end{array}\right].$ (80)

We can choose a new coordinate system where the top block is diagonal and
apply a spatial rotation to diagonalize the second block. The effect of these
choices will be absorbed into the connection, which can be determined relative
to this frame. The resulting vielbein is:

$\displaystyle h^{a}_{~{}\mu}=\left[\begin{array}[]{cccc}A_{1}(t,r)&0&0&0\\\
0&A_{2}(t,r)&0&0\\\ 0&0&A_{3}(t,r)&0\\\
0&0&0&A_{3}(t,r)\sin(\theta)\end{array}\right].$ (85)

As the form of the $f_{I}^{~{}\hat{i}}$ in equation (72) has been defined
invariantly, the frame in equation (80) is invariantly defined up to a linear
isotropy subgroup $\overline{H}_{q}$ consisting of rotations in the
$\mbox{{\bf h}}^{3}-\mbox{{\bf h}}^{4}$ plane with constant parameters.
Relative to this coordinate system, the linear isotropy group can be further
restricted by imposing a particular form on the vielbein as in equation (85).
With this choice the frame in equation (85) is an invariant symmetry frame,
since the linear isotropy group is trivial.

With the chosen coframe, we can take the third equation in (19) and solve for
the most general connection. This is essentially a large set of differential
equations for the connection coefficients and can be solved in a
straightforward, albeit tedious manner.

######  Proposition IV.3.

The most general Riemann-Cartan geometry admitting spherical symmetry is given
by the vielbein (85) and the metric compatible connection with components,

$\displaystyle\begin{aligned}
&\omega_{341}=W_{1}(t,r),~{}\omega_{342}=W_{2}(t,r),\\\
&\omega_{233}=\omega_{244}=W_{3}(t,r),~{}\omega_{234}=-\omega_{243}=W_{4}(t,r),\\\
&\omega_{121}=W_{5}(t,r),~{}\omega_{122}=W_{6}(t,r),\\\
&\omega_{133}=\omega_{144}=W_{7}(t,r),~{}\omega_{134}=-\omega_{143}=W_{8}(t,r),\\\
&\omega_{344}=-\frac{\cos(\theta)}{A_{3}\sin(\theta)}.\end{aligned}$ (86)

To determine the most general connection for a teleparallel geometry we next
impose the flatness condition in (20). The resulting equations are large but
determining the solution is straightforward albeit tedious.

######  Proposition IV.4.

Any spherically symmetric teleparallel geometries is defined by the three
arbitrary functions in the vielbein (85) along with the following spin
connection components.

$\displaystyle\begin{aligned}
W_{1}&=-\frac{\partial_{t}\chi}{A_{1}},W_{2}=-\frac{\partial_{r}\chi}{A_{2}},\\\
W_{3}&=\frac{\cosh(\psi)\cos(\chi)}{A_{3}},W_{4}=\frac{\cosh(\psi)\sin(\chi)}{A_{3}},\\\
W_{5}&=-\frac{\partial_{t}\psi}{A_{1}},W_{6}=-\frac{\partial_{r}\psi}{A_{2}},\\\
W_{7}&=\frac{\sinh(\psi)\cos(\chi)}{A_{3}},W_{8}=\frac{\sinh(\psi)\sin(\chi)}{A_{3}},\end{aligned}$
(87)

where $\chi$ and $\psi$ are arbitrary functions of the coordinates $t$ and
$r$.

Any choice of the arbitrary functions $\psi$ and $\chi$, picks out a unique
teleparallel geometry, as any change in the form of the spin connection which
could affect the form of $\psi$ or $\chi$ leads to a change in the form of the
vielbein. For a given pair of functions, the invariantly defined frame up to
the linear isotropy group $H_{q}$ arising from the CK algorithm could be
computed to provide further sub-classification. We note that there are only
five arbitrary functions required to specify a geometry:
$A_{1},A_{2},A_{3},\psi$ and $\chi$. This characterization of spherically
symmetric geometries in terms of $\psi$ and $\chi$ has not been presented
before in the literature and provides a formulation of spherically symmetric
teleparallel geometries using the smallest number of arbitrary functions. With
this, we can determine new families of geometries by imposing conditions on
the arbitrary functions where these conditions are related to the inertial
effects of the geometries.

While these geometries have been investigated in the teleparallel context
before using the proper frame approach [13, 12, 24], a thorough investigation
in the context of Riemann-Cartan geometries with non-vanishing torsion has not
been performed. We note that subclasses of these geometries have been studied
earlier in teleparallel gravity [25] using the Killing equations for an
arbitrary spherically symmetric metric. However, such an approach does not
consider the sensitivity of the frame basis to Lorentz transformations and
does not pick out the most general classes of spherically symmetric
teleparallel geometries as the parameters of the Lorentz transformations do
not appear at all in their formulation.

In the context of teleparallel gravity, our approach determines the minimal
number of arbitrary functions, at most five, in the vielbein and spin
connection to define the geometries. It also readily distinguishes four
inequivalent branches of spherically symmetric teleparallel geometries,
depending on whether $\chi$ or $\psi$ vanish, as the frame in (85) has been
fully fixed by restricting the vielbein to a particular form. In comparison,
the proper frame approach requires at most six arbitrary functions and the
inequivalence of two spherically symmetric teleparallel geometries will
require the full machinery of the CK algorithm [13, 12].

We will conclude this section with a brief discussion of stationary
spherically symmetric geometries. To the authors’ knowledge, very little is
known about the stationary spherically symmetric Riemann-Cartan geometries
with non-vanishing torsion. Stationary or static spherically symmetric
teleparallel geometries have been discussed in applications [25, 26, 24]. In
these papers, the stationarity or static condition is not rigorously derived
but instead arises from restricting the arbitrary functions to be dependent on
the radial coordinate alone. Although, such a choice intuitively makes sense,
it does not necessarily constitute the most general class of stationary or
static spherically symmetric geometries since it does not consider the
inequivalent subclasses of spherically symmetric metrics.

In a similar manner to the axisymmetric geometries, stationary spherically
symmetric solutions can be produced by including an additional affine frame
symmetry, ${\bf Y}$ with $Y^{a}Y_{a}<0$, locally. This vector field must
commute with the original frame symmetries, $[{\bf X}_{I},{\bf Y}]=0$ for
$I\in\\{1,\ldots,3\\}$. Therefore, a new coordinate system can be chosen so
that ${\bf Y}=\partial_{t}$, while preserving the vielbein form in (80)
instead of the more specialized vielbein in (85). The timelike condition on
${\bf Y}$ imposes an additional condition on the vielbein functions $A_{i}$,
$i\in\\{1,\ldots,6\\}$ in equation (80). Static spherically symmetric
geometries are produced by requiring that ${\bf Y}$ is hypersurface
orthogonal:

$\displaystyle{\bf Y}\wedge d{\bf Y}=0.$ (88)

In the case of static solutions, the vielbein form (85) is applicable. The
expansion of the symmetry group introduces three additional functions into
(12) and this can be determined using (14). As the commutator constants are
zero, the analysis of these equations is straightforward and imply that the
vielbein and the components of the connection one-form must be dependent on
the coordinate $r$ alone. As no other conditions are imposed, this proves that
the static spherically symmetric solutions can be found by requiring that the
functions $A_{i},~{}i=1,2,3$, $\psi$ and $\chi$ are functions of the $r$
coordinate.

We emphasize that we have not attempted to address the anti-symmetric field
equations for $F(T)$ theories. While this has been considered previously using
the proper frame approach [12] for particular choices of the arbitrary
functions, a general analysis of the field equations has not been carried out
[16]. However, the phenomenology of spherically symmetric static and
stationary solutions in $F(T)$ theories have been explored in [27, 28, 29,
30]. In our approach, the minimal number of arbitrary functions for each
inequivalent subclass can potentially allow for a general analysis of the
field equations in $F(T)$ theories.

###  IV.3 Robertson-Walker geometries

Working in the coordinate system $(t,r,\theta,\phi)$, the affine frame
symmetry generators associated with the RW metric are

$\displaystyle\begin{aligned} &{\bf
X}_{1}=\zeta\sin\theta\cos\phi\partial_{r}+\frac{\zeta}{r}\cos\theta\cos\phi\partial_{\theta}-\frac{\zeta\sin\phi}{r\sin\theta}\partial_{\phi},\\\
&{\bf
X}_{2}=\zeta\sin\theta\sin\phi\partial_{r}+\frac{\zeta}{r}\cos\theta\sin\phi\partial_{\theta}+\frac{\zeta\cos\phi}{r\sin\theta}\partial_{\phi},\\\
&{\bf
X}_{3}=\zeta\cos\theta\partial_{r}-\frac{\zeta}{r}\sin\theta\partial_{\theta},\\\
&{\bf X}_{z}=\partial_{\phi},~{}{\bf
X}_{y}=-\cos\phi\partial_{\theta}+\frac{\sin\phi}{\tan\theta}\partial_{\phi},~{}{\bf
X}_{x}=\sin\phi\partial_{\theta}+\frac{\cos\phi}{\tan\theta}\partial_{\phi},\end{aligned}$
(89)

where $\zeta=\sqrt{1-kr^{2}}$ and $k\in\mathbb{R}$. Writing $\\{{\bf
X}_{I}\\}_{I=1}^{6}=\\{{\bf X}_{1},{\bf X}_{2},{\bf X}_{3},{\bf X}_{x},{\bf
X}_{y},{\bf X}_{z}\\}$ the non-zero commutator constants, $C^{I}_{~{}JK}$ are

$\displaystyle\begin{aligned}
&C^{6}_{~{}12}=-k,C^{5}_{~{}13}=-k,C^{3}_{~{}15}=1\\\
&C^{2}_{~{}16}=1,~{}C^{4}_{~{}23}=k,~{}C^{3}_{~{}24}=-1,\\\
&C^{1}_{~{}26}=-1,~{}C^{2}_{~{}34}=1,~{}C^{1}_{~{}35}=-1,\\\
&C^{6}_{~{}45}=-1,~{}C^{5}_{~{}46}=1,~{}C^{4}_{~{}56}=-1.\end{aligned}$ (90)

The basis for the isotropy group is of the form

$\displaystyle\lambda_{\hat{1}}=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\end{array}\right],~{}\lambda_{\hat{2}}=-\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&1&0\\\ 0&-1&0&0\\\
0&0&0&0\end{array}\right],~{}\lambda_{\hat{3}}=-\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&1\\\ 0&0&0&0\\\ 0&-1&0&0\end{array}\right]$ (103)

with the corresponding commutator constants,

$\displaystyle
C^{\hat{3}}_{~{}\hat{1}\hat{2}}=-1,~{}C^{\hat{2}}_{~{}\hat{1}\hat{3}}=1,~{}C^{\hat{1}}_{~{}\hat{2}\hat{3}}=-1.$
(104)

The form of $\overline{Iso}$ can be determined in the same manner as in case
of spherical symmetry since the isotropy group is unchanged. Thus,
$\overline{Iso}$ consists of spatial rotations.

With these quantities, we can substitute them into the first two equations in
(19), first by solving the second equation to determine the form of the
functions in $f_{I}^{~{}\hat{i}}$.

$\displaystyle 2{\bf
X}_{[I}(f_{J]}^{~{}\hat{k}})-f_{I}^{~{}\hat{i}}f_{J}^{~{}\hat{j}}C^{\hat{k}}_{~{}\hat{i}\hat{j}}=C^{K}_{~{}IJ}f_{K}^{~{}\hat{k}}.$
(105)

We will exploit the freedom of choice in the components of
$f_{I}^{~{}\hat{i}}$ using the isotropy group to transform the components with
equation (10). Since ${\bf X}_{z}$ is a generator of a spatial rotation, we
will choose our frame representation so that it acts as a rotation on the
basis elements $\mbox{{\bf h}}^{3}$ and $\mbox{{\bf h}}^{4}$, i.e.,

$\displaystyle
f_{6}^{~{}\hat{i}}\lambda_{\hat{i}}=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&f_{6}^{~{}\hat{1}}\\\
0&0&-f_{6}^{~{}\hat{1}}&0\end{array}\right],$ (110)

then by applying a rotation about $\mbox{{\bf h}}^{3}$ and $\mbox{{\bf
h}}^{4}$, the remaining component, $f_{6}^{~{}\hat{1}}$, can be set to zero
using

$\displaystyle{\bf
X}_{6}(\tilde{\Lambda}^{a}_{~{}b})[\tilde{\Lambda}^{-1}]^{b}_{~{}c}+\tilde{\Lambda}^{a}_{~{}b}f_{6}^{~{}\hat{i}}\lambda^{b}_{\hat{i}~{}e}[\tilde{\Lambda}^{-1}]^{e}_{~{}c}=0$
(111)

From the second equation in (19), using ${\bf X}_{6}$ paired with ${\bf
X}_{\overline{I}}$, $\overline{I}\in[1,5]$, we have the following differential
equations:

$\displaystyle\begin{aligned}
&\partial_{\phi}f_{1}^{~{}\hat{1}}+f_{2}^{~{}\hat{1}}=0,\partial_{\phi}f_{1}^{~{}\hat{2}}+f_{2}^{~{}\hat{2}}=0,\partial_{\phi}f_{1}^{~{}\hat{3}}+f_{3}^{~{}\hat{1}}=0,\\\
&\partial_{\phi}f_{2}^{~{}\hat{1}}+f_{1}^{~{}\hat{1}}=0,\partial_{\phi}f_{2}^{~{}\hat{2}}+f_{1}^{~{}\hat{2}}=0,\partial_{\phi}f_{2}^{~{}\hat{3}}+f_{1}^{~{}\hat{3}}=0,\\\
&\partial_{\phi}f_{3}^{~{}\hat{1}}=0,\partial_{\phi}f_{3}^{~{}\hat{2}}=0,\partial_{\phi}f_{3}^{~{}\hat{3}}=0,\\\
&\partial_{\phi}f_{4}^{~{}\hat{1}}+f_{5}^{~{}\hat{1}}=0,\partial_{\phi}f_{4}^{~{}\hat{2}}+f_{5}^{~{}\hat{2}}=0,\partial_{\phi}f_{4}^{~{}\hat{3}}+f_{5}^{~{}\hat{3}}=0\\\
&\partial_{\phi}f_{5}^{~{}\hat{1}}+f_{4}^{~{}\hat{1}}=0,\partial_{\phi}f_{5}^{~{}\hat{2}}+f_{4}^{~{}\hat{2}}=0,\partial_{\phi}f_{5}^{~{}\hat{3}}+f_{4}^{~{}\hat{3}}=0\end{aligned}$
(112)

Leading to the following solutions,

$\displaystyle\begin{aligned}
&f_{5}^{~{}\hat{i}}=f_{5}^{~{}\hat{i}}(r,\theta)\cos\phi+g_{5}^{~{}\hat{i}}(r,\theta)\sin\phi,\\\
&f_{4}^{~{}\hat{i}}=\partial_{\phi}f_{5}^{~{}\hat{i}},\\\
&f_{3}^{~{}\hat{i}}=f_{3}^{~{}\hat{i}}(r,\theta),\\\
&f_{2}^{~{}\hat{i}}=f_{2}^{~{}\hat{i}}(r,\theta)\cos\phi+g_{2}^{~{}\hat{i}}(r,\theta)\sin\phi,\\\
&f_{1}^{~{}\hat{i}}=\partial_{\phi}f_{2}^{~{}\hat{i}}.\end{aligned}$ (113)

If Lorentz transformations are restricted to transformations that are
dependent on $r$ and $\theta$, we can apply a rotation to set the coefficients
of $\cos\phi$ in $f_{5}^{~{}\hat{i}}$ to zero using equation (10) which takes
the form:

$\displaystyle{\bf
X}_{5}(\tilde{\Lambda}^{a}_{~{}b})[\tilde{\Lambda}^{-1}]^{b}_{~{}c}+\tilde{\Lambda}^{a}_{~{}b}f_{5}^{~{}\hat{i}}\lambda^{b}_{\hat{i}~{}e}[\tilde{\Lambda}^{-1}]^{e}_{~{}c}=-f_{5}^{~{}\hat{i}}\lambda^{a}_{~{}\hat{i}~{}c}\cos\phi.$
(114)

Doing so, the second equation in (19) with ${\bf X}_{5}$ and ${\bf X}_{4}$
gives

$\displaystyle\partial_{\theta}g_{5}^{~{}\hat{i}}+\cos(\theta)g_{5}^{~{}\hat{i}}=0,$
(115)

with the resulting solution:

$\displaystyle g_{5}^{~{}\hat{i}}=\frac{G_{5}^{~{}\hat{i}}(r)}{\sin(\theta)}.$
(116)

This fully determines $f_{5}^{~{}\hat{i}}$ and $f_{4}^{~{}\hat{i}}$ up to
functions of $r$, as the second is the $\phi$-derivative of the first.
Applying a rotation dependent on only the coordinates $r$ we can fix some of
the components in $f_{5}^{~{}\hat{i}}$ to be zero. Without loss of generality,
we will choose the first component to be non-zero:

$\displaystyle\begin{aligned}
&f_{5}^{~{}\hat{1}}=\frac{\sin\phi}{\sin\theta},\\\
&f_{4}^{~{}\hat{1}}=\frac{\cos\phi}{\sin\theta}.\end{aligned}$ (117)

The remaining equations in (105) can now be solved by looking at the
coefficients of $\cos\phi$, and $\sin\phi$. With these choices,
$f_{I}^{~{}\hat{i}}$ takes the form:

$\displaystyle
f_{I}^{~{}\hat{i}}=\left[\begin{array}[]{ccc}-\frac{\sqrt{1-kr^{2}}\sin(\phi)\cos(\theta)}{r\sin(\theta)}&-\frac{\cos(\theta)\cos(\phi)}{r}&\frac{\sin(\phi)}{r}\\\
\frac{\sqrt{1-kr^{2}}\cos(\phi)\cos(\theta)}{r\sin(\theta)}&-\frac{\cos(\theta)\sin(\phi)}{r}&-\frac{\cos(\phi)}{r}\\\
0&\frac{\sin(\theta)}{r}&0\\\ \frac{\cos(\phi)}{\sin(\theta)}&0&0\\\
\frac{\sin(\phi)}{\sin(\theta)}&0&0\\\ 0&0&0\end{array}\right]$ (124)

The form of $f_{I}^{~{}\hat{i}}$ determines the parameters of Lorentz
transformations in $\overline{Iso}$ up to $\overline{H}_{q}$.

To determine $\overline{H}_{q}$, we employ equation (10) with
$f_{I}^{~{}\hat{i}}=\tilde{f}_{I}^{~{}\hat{i}}$ defined as in equation (124).
The analysis proceeds in the same manner as the spherically symmetric case,
except that the parameters of $\tilde{\Lambda}^{a}_{~{}b}$ now depend on
$r,\theta$ and $\phi$. Considering the Lie algebra generator of
$\tilde{\Lambda}^{a}_{~{}b}$ as
$\tilde{\lambda}^{a}_{~{}c}=g^{\hat{i}}\lambda^{a}_{\hat{i}~{}b}$, then using
${\bf X}_{4},{\bf X}_{5}$ and ${\bf X}_{6}$, equation (10) implies that
$g^{\hat{1}}=g(r)$ and $g^{\hat{2}}=g^{\hat{3}}=0$. Finally equation (10) with
${\bf X}_{3}$ gives

$\displaystyle\zeta\cos\theta\partial_{r}f\lambda^{a}_{\hat{1}~{}c}+\frac{\sin\theta}{r}[(\cos
f)\lambda^{a}_{\hat{2}~{}c}+(\sin
f)\lambda^{a}_{\hat{3}~{}c}-\lambda^{a}_{\hat{2}~{}c}]=0$ (125)

The only solution to this equation is $f=0$, implying that
$\tilde{\Lambda}^{a}_{~{}b}$ is the identity Lorentz transformation. Thus, the
subgroup $\overline{H}_{q}$ is the trivial subgroup.

Using the representation of the isotropy group in (103), and setting the
components of $f_{I}^{~{}\hat{i}}$ into the first equation in (19) yields the
following general solution for the vielbein:

$\displaystyle h^{a}_{~{}\mu}=\left[\begin{array}[]{cccc}A_{1}(t)&0&0&0\\\
0&\frac{A_{2}(t)}{\sqrt{1-kr^{2}}}&0&0\\\ 0&0&A_{2}(t)r&0\\\
0&0&0&A_{2}(t)r\sin(\theta)\end{array}\right].$ (130)

Imposing the vielbein form in equation (130), since $\overline{H}_{q}$ is the
trivial group, this frame is now an invariant frame. In general, a coordinate
transformation can be made to set $A_{1}=1$ and $A_{2}=a(t)$. Doing so, we
will work with the coframe associated with

$\displaystyle h^{a}_{~{}\mu}=\left[\begin{array}[]{cccc}1&0&0&0\\\
0&\frac{a(t)}{\sqrt{1-kr^{2}}}&0&0\\\ 0&0&a(t)r&0\\\
0&0&0&a(t)r\sin(\theta)\end{array}\right].$ (135)

Using the coframe associated to this matrix, we can readily solve the
resulting equations coming from equations (6) for each of the affine frame
symmetries. The resulting differential equations can be solved readily by hand
or by using the frame and connection ansatz from the spherically symmetric
case.

######  Proposition IV.5.

The most general Riemann-Cartan geometry which admits a RW symmetry group
consists of the vielbein (135) along with the metric-compatible connection
with components:

$\displaystyle\begin{aligned}
&\omega_{122}=\omega_{133}=\omega_{144}=W_{1}(t),\\\
&\omega_{234}=-\omega_{243}=\omega_{342}=W_{2}(t),\\\
&\omega_{233}=\omega_{244}=-\frac{\sqrt{1-kr^{2}}}{a(t)r},\\\
&\omega_{344}=-\frac{\cos(\theta)}{a(t)r\sin(\theta)}.\end{aligned}$ (136)

The class of connections that describe a teleparallel geometry can be
determined by solving the flatness condition in equation (20). This gives the
following differential constraints on $W_{1}$ and $W_{2}$:

$\displaystyle\begin{aligned} &a\partial_{t}W_{1}+W_{1}\partial_{t}a=0,\\\
&W_{1}W_{2}=0,\\\ &a\partial_{t}W_{2}+W_{2}\partial_{t}a=0,\\\
&W_{2}^{2}a^{2}-W_{1}^{2}a^{2}-k\end{aligned}$ (137)

These equations have four distinct solutions, leading to the following
proposition

######  Proposition IV.6.

The most general teleparallel geometry admitting the RW symmetry group are
specified by the arbitrary function in the vielbein (135) along with the
choice of a metric-compatible connection (136) with the functions $W_{1}$ and
$W_{2}$ given as

  1. 1.

$W_{1}=0$ and $W_{2}=-\frac{\sqrt{k}}{a}$.

  2. 2.

$W_{1}=0$ and $W_{2}=\frac{\sqrt{k}}{a}$.

  3. 3.

$W_{1}=-\frac{\sqrt{-k}}{a}$ and $W_{2}=0$.

  4. 4.

$W_{1}=\frac{\sqrt{-k}}{a}$ and $W_{2}=0$.

Each choice of $W_{1}$ and $W_{2}$ yields an inequivalent teleparallel
geometry.

Imposing the reasonable condition that the torsion tensors must be real-
valued, this naturally distinguishes two cases for $k>0$ and two cases for
$k<0$. When $k$ vanishes, these four inequivalent cases coincide. Without loss
of generality, the coordinate $r$ can be rescaled to set $k=-1,0$ or $1$. Note
that the antisymmetric field equations are automatically satisfied in all
cases. In [31], the proper frames for these teleparallel geometries are given
and the $F(T)$ field equations are analysed for each inequivalent case.

Alternatively, the connection coefficients for this case can be determined
using the connection coefficients of the spherically symmetric case in
equation (87) by setting

$\displaystyle\begin{aligned} k<0&:\chi=0,\text{or
}\pi,~{}\psi=\cosh^{-1}(-\cos(\chi)\sqrt{1-kr^{2}}),\\\ k\geq
0&:\chi=\cos^{-1}(-\sqrt{1-kr^{2}}),\psi=0.\end{aligned}$ (138)

This choice of representation of the arbitrary functions in the connection
coefficients shows that the sign difference in the case of $k>0$ is purely a
choice of representation and hence these subcases are equivalent. We note that
the sign difference for the $k<0$ case does indeed give two inequivalent
subcases.

While the RW Riemann-Cartan geometries with non-vanishing torsion have not
been explored at all, the teleparallel RW geometries have been discussed
previously. Using a metric ansatz and the Killing equations a very limited
subclass were determined in [25], and the flat case has been explored by many
authors [32]. The first attempt at using a frame approach to determine TRW
geometries with $k\neq 0$ were in [6, 7]. However, the solutions required a
complex orthonormal frame which was undesirable from a physical standpoint.

The general class was determined using the proper frame approach in [14]. We
note that the field equations studied in [31] agree with the field equations
found in [14]. However, there are two significant problems with the approach
in [14]: certain branches of solutions require complex orthonormal proper
frames [12], and the proper frames contain an additional arbitrary function of
$t$ which is purely a coordinate artefact. Despite the appearance of complex
orthonormal proper frames, these have been used to study the RW models with
non-zero $k$ in [33, 34].

Many of the applications in the literature of these geometries have
concentrated on the analysis of the flat ($k=0$) TRW cosmological models [16,
32]. In addition, particular forms for $F(T)$ have been considered. For
example, employing reconstruction methods, a functional form for the solution,
in terms of the scale factor, is assumed and the corresponding $F(T)$ theory
is then constructed. However, geometries with non-zero $k$ have been recently
studied in bounce models [35] and in inflation where the use of a complex
orthonormal frame potentially restricts to the $k=1$ case [36]. Lastly, the
TRW solutions have been used to study perturbations in non-flat cosmology
[37]. More generally, the TRW solutions with non-zero $k$ have been employed
in $f(T,B)$ theories, where $B$ is a boundary term [38].

###  IV.4 de Sitter and Einstein Static geometries

In GR and Lorentzian geometry, in general, the de Sitter solution is a special
case of the RW solutions, where the six-dimensional group is expanded to a
ten-dimensional group, namely $SO(1,4)$. It has been proven that the only
maximally symmetric geometry which admits a non-vanishing torsion tensor is
Minkowski space and that the largest symmetry group permitted in such a
geometry is at most seven [15, 16]. In addition, it follows from the CK
algorithm that if a geometry admits a seven-dimensional symmetry group then it
must have constant Cartan invariants relative to some class of invariantly
defined frames [15].

To determine the possible special cases of the TRW geometries which admit
admit a seven-dimensional symmetry group, we will first consider the general
case of the Riemann-Cartan geometries and require that their components are
constant. The components of the curvature tensor and the torsion tensor are,
respectively,

$\displaystyle\begin{aligned}
&R_{1212}=R_{1313}=R_{1414}=\frac{W_{2,t}a+a_{,t}W_{2}}{a},\\\
&R_{1234}=R_{1324}=R_{1423}=2W_{1}W_{2},\\\
&R_{2314}=R_{2413}=R_{3412}=\frac{W_{1,t}a+a_{,t}W_{1}}{a},\\\
&R_{2323}=R_{2424}=R_{3434}=\frac{W_{1}^{2}a^{2}-W_{2}^{2}a^{2}-k}{a^{2}},\end{aligned}$
(139)

and

$\displaystyle\begin{aligned}
&T_{212}=T_{313}=T_{414}=\frac{W_{2}a+a_{,t}}{a},\\\
&T_{234}=T_{324}=T_{432}=2W_{1}.\end{aligned}$ (140)

Requiring that these components are constant give two distinct cases.

######  Proposition IV.7.

The subclass of RW Riemann-Cartan geometries admitting a seventh symmetry are
given by the vielbein (135) and the connection (136) where the functions
$W_{1}$, $W_{2}$ and $a(t)$ take the form:

$\displaystyle\begin{aligned} &\text{ Case 1:
}W_{1}=C_{1},W_{2}=C_{2},k=0,\text{ and }a(t)=C_{3}e^{C_{4}t},\\\ &\text{ Case
2: }W_{1}=C_{1},W_{2}=C_{2},\text{ k arbitrary, and }a(t)=C_{3}.\end{aligned}$
(141)

Requiring the curvature tensor vanishes leads to strong conditions on $W_{1}$
and $W_{2}$. In the remainder of this subsection, we will only discuss the
teleparallel geometries admitting a 7-dimensional symmetry group, $G_{7}$.

######  Proposition IV.8.

The subclass of RW teleparallel geometries admitting a seventh symmetry are
given by the vielbein (135) and the connection (136) with

  1. 1.

$W_{1}=0$ and $W_{2}=-\frac{\sqrt{k}}{a}$.

  2. 2.

$W_{1}=0$ and $W_{2}=\frac{\sqrt{k}}{a}$.

  3. 3.

$W_{1}=-\frac{\sqrt{-k}}{a}$ and $W_{2}=0$.

  4. 4.

$W_{1}=\frac{\sqrt{-k}}{a}$ and $W_{2}=0$.

where either

$\displaystyle\begin{aligned} &\text{ Case 1: }a=C_{3}e^{C_{4}t}\text{ and
}k=0,\\\ &\text{ Case 2: }a=C_{3}\text{ and k is $\pm 1$}.\end{aligned}$ (142)

Using (8) and (6) we can determine the form of the seventh symmetry as

$\displaystyle\begin{aligned} &\text{ Case 1: }{\bf
X}_{7}=\frac{1}{C_{3}C_{4}}\partial_{t}+r\partial_{r},\\\ &\text{ Case 2:
}{\bf X}_{7}=\partial_{t}.\end{aligned}$ (143)

In the first case, the resulting Lie algebra of $\\{{\bf
X}_{I}\\}_{I=1}^{7}=\\{{\bf X}_{1},{\bf X}_{2},{\bf X}_{3},{\bf X}_{x},{\bf
X}_{y},{\bf X}_{z},{\bf X}_{7}\\}$ is given by the following non-zero Lie
brackets:

$\begin{array}[]{lll}{}[{\bf X}_{1},{\bf X}_{5}]={\bf X}_{3},&[{\bf
X}_{1},{\bf X}_{6}]={\bf X}_{2},&[{\bf X}_{1},{\bf X}_{7}]={\bf X}_{1},\\\
{}[{\bf X}_{2},{\bf X}_{4}]=-{\bf X}_{3},&[{\bf X}_{2},{\bf X}_{6}]=-{\bf
X}_{1},&[{\bf X}_{2},{\bf X}_{7}]={\bf X}_{2},\\\ {}[{\bf X}_{3},{\bf
X}_{4}]={\bf X}_{2},&[{\bf X}_{3},{\bf X}_{5}]=-{\bf X}_{1},&[{\bf X}_{3},{\bf
X}_{7}]={\bf X}_{3},\\\ {}[{\bf X}_{4},{\bf X}_{5}]=-{\bf X}_{6},&[{\bf
X}_{4},{\bf X}_{6}]={\bf X}_{5},&[{\bf X}_{5},{\bf X}_{6}]=-{\bf
X}_{4}.\end{array}$ (144)

By inspection, this is a subalgebra of the Lie algebra for the group of metric
(Killing) symmetries of de Sitter spacetime. We therefore propose the
following definition.

######  Definition IV.9.

The teleparallel de Sitter geometry (TdS) is a teleparallel geometry with the
$G_{7}$ Lie group of affine symmetries, $\mathcal{R}\rtimes E(3)$ which is a
subgroup of $O(1,4)$.

We note in this geometry the covariant derivative of the torsion tensor is
zero.

In the second case, $a(t)=C_{3}$ a constant, $k=\pm 1$. These geometries
correspond to the direct product $\mathbb{R}\times M_{3}$ where $M_{3}$ is a
locally homogeneous and isotropic Riemannian manifold. This is reflected in
the Lie algebra structure of the affine frame symmetries $\\{{\bf
X}_{I}\\}_{I=1}^{7}=\\{{\bf X}_{1},{\bf X}_{2},{\bf X}_{3},{\bf X}_{x},{\bf
X}_{y},{\bf X}_{z},{\bf X}_{7}\\}$ where ${\bf X}_{7}=\partial_{t}$ satisfies
$[{\bf X}_{i},{\bf X}_{7}]=0,i\in[1,6]$ and the remaining non-zero Lie
brackets are:

$\begin{array}[]{lll}{}[{\bf X}_{1},{\bf X}_{5}]={\bf X}_{3},&[{\bf
X}_{1},{\bf X}_{6}]={\bf X}_{2},&[{\bf X}_{2},{\bf X}_{4}]=-{\bf X}_{3},\\\
{}[{\bf X}_{2},{\bf X}_{6}]=-{\bf X}_{1},&[{\bf X}_{3},{\bf X}_{4}]={\bf
X}_{2},&[{\bf X}_{3},{\bf X}_{5}]=-{\bf X}_{1},\\\ {}[{\bf X}_{4},{\bf
X}_{5}]=-{\bf X}_{6},&[{\bf X}_{4},{\bf X}_{6}]={\bf X}_{5},&[{\bf X}_{5},{\bf
X}_{6}]=-{\bf X}_{4}.\end{array}$ (145)

Since ${\bf X}_{7}=\partial_{t}$ is the additional affine frame symmetry, this
geometry is necessarily static due to the form of the frame basis. This
geometry can be considered as the analogue of the Einstein static geometry in
GR which we call the Teleparallel Einstein Static (TES) geometry. We therefore
propose the following definition.

######  Definition IV.10.

The teleparallel Einstein Static geometry (TES) is a teleparallel geometry
with the $G_{7}$ Lie group of affine symmetries, $\mathcal{R}\times E(3)$
which is a subgroup of $O(1,4)$.

By working with the RW geometries, which are a subclass of the spherically
symmetric geometries, admitting six affine frame symmetries we have determined
two subclasses of Riemann-Cartan geometries which admit two distinct seven-
dimensional groups of affine frame symmetries, respectively. In the case of
teleparallel geometries, we have explicitly determined the analogues of the de
Sitter (TdS) and Einstein static (TES) geometries. We note that the proper
frames and the $F(T)$ field equations for the TdS and TES geometries were
presented in [39]. These are only two examples of geometries admitting a
seven-dimensional group of affine frame symmetries, and we anticipate a much
larger set of geometries admitting a six-dimensional group of affine frame
symmetries with different six-dimensional groups. In the case of geometries
admitting a seven-dimensional group of affine frame symmetries, these will
necessarily belong to the class of locally homogeneous Riemann-Cartan
geometries [40].

##  V Discussion

We have introduced an invariant definition for symmetries in frame based
theories of gravity. This is a generalization of the Killing equations in
pseudo-Riemannian geometry to the more general class of Riemann-Cartan
geometries. As an illustration of our approach, we have provided examples from
teleparallel geometries where the connection is chosen so that the torsion
tensor and its covariant derivatives are the only non-vanishing geometric
quantities. This definition relies on a geometrically preferred class of
frames, called symmetry frames, which are a generalization of the class of
invariantly defined frames, or invariant frames used in the Cartan-Karlhede
(CK) algorithm. Relative to the symmetry frames, the equations for an affine
frame symmetry (2) are put into a simple form given in equations (12) and (13)
which can be solved readily for a coframe $\mbox{{\bf h}}^{a}$ connection one-
form $\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}$ and Lie algebra
generator of the isotropy group, $\lambda^{a}_{~{}b}$. In general, an
invariantly defined frame can then be constructed by fixing the form of
$f_{I}^{~{}\hat{i}}$ in equation (12) in a coordinate independent manner.
Furthermore, an invariant frame, where all parameters of the Lorentz
transformations are fixed, can be found by restricting the form of the
vielbein. This permits the inequivalent subclasses of geometries to be readily
distinguished.

In addition to this definition, we have introduced an approach to determine
the most general Riemann-Cartan geometries which admit a specified symmetry
group. Once a particular coordinate system and representation of the symmetry
group as vector fields are given, the equations for an affine frame symmetry,
(12) and (13), along with the affine frame symmetry commutator condition (14)
can be solved for the coframe $\mbox{{\bf h}}^{a}$ and connection one-form
$\mbox{{\mbox{\boldmath{$\omega$}}}}^{a}_{~{}b}$. In order to restrict to
teleparallel geometries, additional conditions must be imposed on the
connection one-form, namely the vanishing of the curvature tensor. We show
that for the connection one-form that satisfies (13), imposing the flatness
condition can be achieved by solving a system of differential equations.

The geometrical approach in the proper frame approach [12, 13, 14] is similar
to that given here. The definitions are not identical, and we are careful only
to discuss affine symmetries and avoid any complex quantities. Since we have
made no assumptions a priori, our results are general. A significant
difference between the approaches is that, in general, the invariantly defined
symmetry frame formulation requires a non-vanishing spin connection. While
this may complicate the presentation of the geometries which admit a given
symmetry group, the inclusion of the spin connection permits a complete
classification of solutions of equations (12) and (13).

While we have determined two distinct classes of Riemann-Cartan geometries
admitting two particular seven-dimensional groups of symmetries, it is known
that that there are other Riemann-Cartan geometries which admit a group of
seven affine frame symmetries. Within the framework of the symmetry frame
approach, it is possible to explicitly find all possible Riemann-Cartan
geometries admitting a seven dimensional symmetry group, and in future work we
will present an exhaustive list of these. We will also investigate the class
of geometries which permit a six-dimensional group of symmetries in order to
determine the teleparallel analogue of the locally homogeneous pp-waves. The
class of all Riemann-Cartan geometries admitting six-dimensional groups of
affine frame symmetries are more difficult to determine as these can permit
two-dimensional linear isotropy groups in addition to three-dimensional linear
isotropy groups.

We note that the approach discussed in this paper can be extended to
connections that are not metric compatible and hence admit a non-vanishing
non-metricity tensor $Q_{abc}=\nabla_{\mbox{{\bf h}}_{c}}g_{ab}$ and more
general groups of frame transformations such as $SO(p,q)$. The authors believe
that this can provide a helpful tool for exploring more exotic geometrics or
more general pseudo-Riemannian geometries with indefinite signature and give
an insight into the alignment classification [41].

## Acknowledgments

AAC and RvdH are supported by the Natural Sciences and Engineering Research
Council of Canada. RvdH is supported by the St Francis Xavier University
Council on Research. DDM is supported by the Norwegian Financial Mechanism
2014-2021 (project registration number 2019/34/H/ST1/00636). The authors thank
Alexandre Landry for comments and discussion.

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