# Integrability Conditions for Killing-Yano Tensors and
Maximally Symmetric Spaces in the Presence of Torsion

Carlos Batista
Departamento de Física
Universidade Federal de Pernambuco
50670-901 Recife-PE, Brazil
<EMAIL_ADDRESS>
###### Abstract

The integrability conditions for the existence of Killing-Yano tensors or,
equivalently, covariantly closed conformal Killing-Yano tensors, in the
presence of torsion are worked out. As an application, all metrics and
torsions compatible with the existence of a Killing-Yano tensor of order $n-1$
are obtained. Finally, the issue of defining a maximally symmetric space with
respect to connections with torsion is addressed. (Keywords: Torsion,
Integrability conditions, Conformal Killing-Yano tensors, Maximally symmetric
spaces, General relativity)

##  1 Introduction

In the standard model, the electromagnetic, weak, and strong interactions are
all described by gauge theories for appropriate gauge groups. Therefore, it
would be desirable to fit the fourth fundamental interaction, the gravitation,
in this beautiful scheme. Although not obvious from the usual presentation, it
turns out that general relativity can also be described as a gauge theory,
with the Lorentz group being its structure group [1]. Since the symmetry under
spacetime translations is of fundamental physical relevance, it is natural to
extend general relativity and look for the gauge theory whose structure is the
Poincaré group. The simplest gravitational theory with such a property is the
so-called Einstein-Cartan theory [2, 3]. The latter theory reduces to general
relativity in vacuum. However, in the presence of particles with spin,
Einstein-Cartan theory yields that the connection must be endowed with
torsion,111If the matter field does not couple to the connection then no
torsion is generated. For instance, the Lagrangian of the electromagnetic
field is defined independently of the connection. So, in spite of the fact
that such a field has spin one, it does not generate torsion. departing from
the formalism of general relativity. Since intrinsic spins are of quantum
nature, it follows that gravitational theories with torsion may be of great
relevance for the quantization of gravity. Particularly, in superstring
theory, the field strength of the Kalb-Ramond field is generally interpreted
as the torsion, as illustrated in [4]. In addition, torsion have also been
considered in the context of AdS/CFT correspondence [5], which can lead to
applications in condensed matter physics.

One of the areas in which the physical implications of torsion have been
exploited the most is cosmology. For instance, the richness of the theories
with torsion may be of relevance to explain dark matter and dark energy [6,
7]. In addition, there are several works studying the possible connections
between torsion and inflation [8, 6]. The torsion can also be used to change
the apparent value of the cosmological constant, which can be valuable for a
conciliation between supersymmetry and experimental observations [7] as well
as for the solution of the so-called cosmological constant problem. A great
drawback towards the acceptance of torsion as a useful tool to model our world
is that it is very difficult to measure the effects of a torsion field.
Indeed, it turns out that torsion couples just to particles with intrinsic
spin [9]. So, for instance, experiments like Gravity Probe B, in which the
measuring devices are gyroscopes with macroscopic orbital angular momentum but
with no net intrinsic spin, are not able to detect the possible existence of
torsion [10, 11]. In spite of such a difficulty, it has been pointed out that
it is possible to put constraints in the torsion by means of the data from
experiments of Lorentz violation [12]. Actually, quite recently, an experiment
using neutrons in liquid Helium have just been put forward with the aim of
constraining the torsion field [13].

The hidden symmetries represented by Killing-Yano (KY) tensors have proved to
be of fundamental relevance to the development of four-dimensional general
relativity. Indeed, it was due to the existence of a KY tensor of order two
that the geodesic, Klein-Gordon, and Dirac equations could be analytically
integrated in Kerr background [14, 15, 16, 17]. More generally, it has been
proved that Kerr-NUT-(A)dS spacetimes of arbitrary dimension admit a closed
conformal Killing-Yano tensor of rank two from which one can construct a tower
of KY tensors [18] that provides just enough conserved charges to enable the
explicit integration of the geodesic [19], the Klein-Gordon [20], and the
Dirac equations [21]. Furthermore, such hidden symmetries are also related to
the separability of gravitational perturbations in Kerr-NUT-(A)dS spacetimes
[22].

The present article aims to work out the integrability conditions for the
existence of KY tensors, of arbitrary order, in the presence of torsion. To
the best of author’s knowledge, this has not been done before. These
integrability conditions can be of great help in the involved task of
integrating the KY equation, since they constrain the algebraic form of the KY
tensor, which, in turn, eliminates several degrees of freedom in the general
ansatz of a KY tensor, as illustrates [23] for the torsion-free case. The
integrability conditions for KY tensors in the absence of torsion have already
been obtained before in Refs. [24, 25]. Regarding works considering hidden
symmetries in the presence of torsion, in Ref. [26] the metrics that allow the
existence of a non-degenerate closed conformal Killing-Yano tensors of rank
two in the presence of skew-symmetric torsion have been investigated, in Ref.
[27] it has been shown that, in general, a KY tensor does not lead to an
operator that commutes with the Dirac operator if the connection has skew-
symmetric torsion, whereas in Ref. [4] some spacetimes in which the torsion
play an important role have been investigated. The interplay between KY
tensors and torsion is also of relevance for the study of spaces with special
holonomy [28, 29].

This article is organized as follows. Section 2 provides a review of
connections with torsion. Particularly, the algebraic identities satisfied by
the curvature tensor are displayed and a discussion about the ambiguity in the
definition of geodesics and Killing vectors, due to the existence torsion, is
made. In Sec. 3, we define KY tensors in the presence of torsion and show that
they can be equivalently described as covariantly closed conformal Killing-
Yano tensors. Then, the integrability conditions for the latter objects are
obtained. As an application, in Sec. 4, all the metrics and antisymmetric
torsions admitting a KY tensor of order $n-1$, with $n$ being the dimension of
the space, are explicitly obtained. Finally, in Sec. 5, the issue of defining
a maximally symmetric space in the presence of torsion is investigated. In
particular, it is shown that, for a general non-vanishing torsion, a manifold
with the maximal number of Killing vectors does not admit the maximum number
of KY tensors. For the sake of generality, most of the calculations performed
here assume no constraint in the torsion such as skew-symmetry.

##  2 Reviewing Connections with Torsion

The intent of the present section is to provide a review of some basic aspects
of the connections with torsion as well as to set the notation convention
adopted in the forthcoming sections. A nice review on the role of torsion in
theories of gravitation is available in [3]. Let $(M,\boldsymbol{g})$ be a
differential manifold of dimension $n$ endowed with a metric tensor
$\boldsymbol{g}$. Then, if $\boldsymbol{V}$ is a vector field in such
manifold, its covariant derivative shall be written as

$\nabla_{a}\,V^{b}\,=\,\partial_{a}\,V^{b}\,+\,\Gamma^{b}_{ac}\,V^{c}\,,$

where $\\{a,b,c,\ldots\\}$ denote coordinate indices and $\Gamma^{b}_{ac}$ is
the connection symbol. Since the symbol $\Gamma^{b}_{ac}$ does not transform
homogeneously under coordinate transformations, it turns out that it is not a
tensor. Nevertheless, its skew-symmetric part in the lower indices is a tensor
known as the Torsion tensor,

$T^{c}_{\phantom{c}ab}\,\equiv\,2\,\Gamma^{c}_{[ab]}\,=\,\Gamma^{c}_{ab}\,-\,\Gamma^{c}_{ba}\,.$

Where, as usual, indices enclosed by square brackets are anti-symmetrized
whereas indices enclosed by round brackets are symmetrized. Because of its
skew-symmetry in the last pair of indices it follows that the Torsion can be
written as

$T_{abc}\,=\,\frac{2}{3}\,T_{(ab)c}\,-\,\frac{2}{3}\,T_{(ac)b}\,+\,T_{[abc]}\,.$
(1)

Physically, it is reasonable to work with connections that preserve lengths
and angles under parallel transports over the manifold, namely connections
compatible with the metric. Therefore, we shall assume that the metric is
covariantly constant, $\nabla_{a}g_{bc}\,=\,0$. One can solve the latter
requirement for the connection and the solution is uniquely given by222A
manifold endowed with this connection is called a Riemann-Cartan space.

$\Gamma^{c}_{ab}\,=\,\frac{1}{2}\,g^{ce}\,\left(\partial_{a}\,g_{be}\,+\,\partial_{b}\,g_{ae}\,-\,\partial_{e}\,g_{ab}\,-\,T_{abe}\,-\,T_{bae}\,+\,T_{eab}\right)\,=\,\mathring{\Gamma}^{c}_{ab}\,+\,K^{c}_{\phantom{c}ab}\,,$
(2)

with $\mathring{\Gamma}^{c}_{ab}$ denoting the familiar Levi-Civita
connection, which is metric-compatible and torsion-free, whereas
$K^{c}_{\phantom{c}ab}$ is the so-called contortion tensor, whose definition
is

$K^{c}_{\phantom{c}ab}\,\equiv\,\,-\,\frac{1}{2}\,g^{ce}\,\left(T_{abe}\,+\,T_{bae}\,-\,T_{eab}\right)\,.$
(3)

It is interesting noting that, although the torsion arises from the anti-
symmetric part of the connection, in general, it also contributes to the
symmetric part of the connection. Indeed,

$\Gamma^{c}_{(ab)}\,=\,\mathring{\Gamma}^{c}_{ab}\,-\,T_{(ab)}^{\phantom{(ab)}c}\,.$
(4)

When we are dealing with a torsion-free connection, there are two equivalent
ways of defining a geodesic: (i) Geodesics are the curves that minimize the
distance between two points in the manifold; (ii) Geodesics are the integral
curves of the vector fields with vanishing acceleration. However, for
connections with non-zero torsion, these definitions cease to be equivalent.
Indeed, if $\\{x^{a}\\}$ is a coordinate system, these two definitions lead to
the following differential equations respectively:

$\displaystyle\delta\,s\,\,=\,\delta\int\sqrt{g_{ab}\,dx^{a}\,dx^{b}\,}\,=\,0\,\quad\Longleftrightarrow\quad$
$\displaystyle\frac{d^{2}x^{a}}{ds^{2}}\,+\,\mathring{\Gamma}^{a}_{bc}\,\frac{dx^{b}}{ds}\frac{dx^{c}}{ds}\,=\,0$
$\displaystyle V^{a}\nabla_{a}\,V^{b}\,=\,0\quad\Longleftrightarrow\quad$
$\displaystyle\frac{d^{2}x^{a}}{ds^{2}}\,+\,\Gamma^{a}_{bc}\,\frac{dx^{b}}{ds}\frac{dx^{c}}{ds}\,=\,0\,,$

where $s$ represents the arc-length functional and $V^{a}=\frac{dx^{a}}{ds}$.
Note that the first of the these differential equations can be written as
$V^{a}\mathring{\nabla}_{a}\,V^{b}=0$, with $\mathring{\nabla}$ standing for
the Levi-Civita covariant derivative. Due to Eq. (4), it follows that these
differential equations are equivalent if, and only if, $T_{(ab)c}=0$.
Therefore, because of (1), we conclude these two definitions of a geodesic
curve are equivalent if, and only if, the torsion is totally anti-symmetric,
$T_{abc}=T_{[abc]}$. The case of totally skew-symmetric torsion is also of
great relevance for string theory, in which the torsion is generally given by
the field strength of a 2-form field. In spite of the physical appeal of the
totally anti-symmetric torsions, in what follows most of the results are
worked out without assuming this constraint.

It is worth remarking that none of these geodesics represent the orbits of
general free falling particles in the presence of torsion. Indeed, if
$V^{a}\mathring{\nabla}_{a}V^{b}=0$ had been the equation of motion of a
general particle interacting just with the gravitational field then the
particles would not have been affected by the torsion at all, so that it would
have been physically unnecessary to introduce the concept of torsion. On the
other hand, had $V^{a}\nabla_{a}V^{b}=0$ been the correct equation of motion
of a general free falling particle then all particles would have been affected
by the torsion. However, this cannot be true, since it is well established
that only particles with intrinsic spin feel the torsion [30, 31]. The
requirement of diffeomorphism invariance of the matter action implies a
conservation law involving the energy momentum tensor. It turns out that by
means of this conservation law it is possible to deduce the equations of
motion of a test particle in a gravitational field, for an alternative
derivation see [32]. For instance, in the torsion-less case the conservation
law states that the divergence of the energy-momentum tensor vanishes and,
using this fact, it is possible to obtain differential equations involving the
momentum and the spin of the test particle [33, 34], the so-called Mathisson-
Papapetrou-Dixon equations.333There are two inequivalent versions for these
equations of motion, it turns out that the version deduced by Dixon implies
the one obtained by Papapetrou but not the converse [35]. The analogous of
these equations in the presence of torsion have been obtained in [9, 30].
Nevertheless, it is worth pointing out that such equations of motion do not
predict the whole motion of the test particle, since the dynamic of the center
of mass of the particle is not fixed by these conservation laws. In spite of
this, in the torsion-less case one can use the requirement of energy
positiveness in order to argue that the momentum is proportional to the
velocity of the center of mass [33, 35] and then deduce that a point particle
without intrinsic spin follow the geodesic path, but some ambiguities remain
in the case of non-vanishing torsion [30]. However, it is worth stressing
that, due to spin-orbit couplings, a point particle with non-zero spin
generally will not follow the geodesic path even in the absence of torsion, an
exception being the massless particles since in this case, semi-classically
speaking, the spin must be aligned with the momentum [36]. Irrespective of the
correct path followed by a point test particle in the presence of torsion, the
two concepts of geodesic presented here are of great geometrical significance
and are worth studying in their own rights.

Whenever a fiber bundle is endowed with a connection, its curvature operator
is defined by

$\nabla_{\boldsymbol{X}}\nabla_{\boldsymbol{Z}}-\nabla_{\boldsymbol{Z}}\nabla_{\boldsymbol{X}}-\nabla_{[\boldsymbol{X},\boldsymbol{Z}]}\,,$

where $\boldsymbol{X}$ and $\boldsymbol{Z}$ are tangent vector fields and
$[\boldsymbol{X},\boldsymbol{Z}]$ denotes their Lie Bracket. Particularly,
assuming $\boldsymbol{X}$ and $\boldsymbol{Z}$ to be the coordinate vectors
$\boldsymbol{\partial}_{a}$ and $\boldsymbol{\partial}_{b}$, we end up with
the following form for the curvature operator on the tangent bundle

$\left(\nabla_{a}\nabla_{b}\,-\,\nabla_{b}\nabla_{a}\,+\,T^{e}_{\phantom{e}ab}\nabla_{e}\right)\,.$

Since the action of the above operator in any scalar function gives zero, it
follows that

$\left(\nabla_{a}\nabla_{b}\,-\,\nabla_{b}\nabla_{a}\,+\,T^{e}_{\phantom{e}ab}\nabla_{e}\right)(f\boldsymbol{L})\,=\,f\left(\nabla_{a}\nabla_{b}\,-\,\nabla_{b}\nabla_{a}\,+\,T^{e}_{\phantom{e}ab}\nabla_{e}\right)\boldsymbol{L}$

for any tensorial field $\boldsymbol{L}$. Therefore, such operator defines a
tensor called the curvature tensor and denoted by $R_{abc}^{\phantom{abc}e}$.
More precisely, the action of such operator in a tensor of rank $p$ is given
by:

$\left(\nabla_{a}\nabla_{b}\,-\,\nabla_{b}\nabla_{a}\,+\,T^{e}_{\phantom{e}ab}\nabla_{e}\right)\,L_{c_{1}c_{2}\cdots
c_{p}}\,=\,\sum_{i=1}^{p}R_{abc_{i}}^{\phantom{abc_{i}}e}\,L_{c_{1}\cdots\check{c}_{i}\,e\,c_{i+1}\cdots
c_{p}}\,,$ (5)

where $\check{c}_{i}$ means that the index $c_{i}$ has been withdrawn. Writing
the covariant derivatives in the left hand side of (5) in terms of partial
derivatives and the connection symbol, we arrive at the following expression
for the curvature:

$R_{abc}^{\phantom{abc}e}\,=\,\partial_{b}\Gamma^{e}_{ac}\,-\,\partial_{a}\Gamma^{e}_{bc}\,+\,\Gamma^{d}_{ac}\Gamma^{e}_{bd}\,-\,\Gamma^{d}_{bc}\Gamma^{e}_{ad}\,=\,\,-\,2\,\partial_{[a}\Gamma^{e}_{b]c}\,+\,2\,\Gamma^{d}_{[a|c|}\Gamma^{e}_{b]d}\,.$
(6)

Now, inserting Eq. (2) into the above relation, i.e., assuming the connection
to be metric-compatible, one arrive at the following relation

$R_{abc}^{\phantom{abc}e}\,=\,\mathring{R}_{abc}^{\phantom{abc}e}\,-\,2\,\nabla_{[a}K^{e}_{\phantom{e}b]c}\,+\,2\,K^{e}_{\phantom{e}[a|d|}K^{d}_{\phantom{d}b]c}\,-\,\,T^{d}_{\phantom{d}ab}K^{e}_{\phantom{e}dc}\,,$
(7)

with $\mathring{R}_{abc}^{\phantom{abc}e}$ standing for the curvature of the
Levi-Civita connection. By means of the latter identity, one can prove that
the curvature of the general metric-compatible connection obeys the following
identities:

$\displaystyle R_{abcd}\,=\,$ $\displaystyle R_{[ab][cd]}$ $\displaystyle
R_{[abc]}^{\phantom{[abc]}e}\,=\,$
$\displaystyle-\,\nabla_{[a}T^{e}_{\phantom{e}bc]}\,+\,T^{d}_{\phantom{d}[ab}T^{e}_{\phantom{e}c]d}$
$\displaystyle\nabla_{[a}R_{bc]de}\,=\,$ $\displaystyle
T^{f}_{\phantom{e}[ab}R_{c]fde}$ (8) $\displaystyle R_{ab}\,=\,$
$\displaystyle
R_{ba}\,+\,\nabla_{c}T^{c}_{\phantom{c}ab}\,+\,2\,\nabla_{[a}T^{c}_{\phantom{c}b]c}\,+\,T^{c}_{\phantom{c}ab}T^{d}_{\phantom{d}cd}$

Where $R_{ab}\equiv R^{c}_{\phantom{c}acb}$ denotes the Ricci tensor. Note, in
particular, that in the presence of torsion the Ricci tensor generally is not
symmetric. In the special case in which the torsion is totally skew-symmetric,
we further have the following relations:

$T_{abc}=T_{[abc]}\;\Longrightarrow\;\left\\{\begin{array}[]{cl}R_{abcd}\,=&R_{cdab}\,+\,\nabla_{[c}T_{d]ab}\,-\,\nabla_{[a}T_{b]cd}\\\
R_{ab}\,=&R_{ba}\,+\,\nabla_{c}T^{c}_{\phantom{c}ab}\,.\\\ \end{array}\right.$
(9)

The exterior derivative of a differential form can be defined irrespective of
the existence of a connection on the bundle of differential forms. Indeed, if
$\boldsymbol{F}$ is a $p$-form then its exterior derivative is a $(p+1)$-form
whose components are:

$(dF)_{ab_{1}b_{2}\cdots b_{p}}\,=\,(p+1)\,\partial_{[a}F_{b_{1}b_{2}\cdots
b_{p}]}\,.$

It is well-known that in the case of a torsion-free connection it is harmless
to replace the partial derivative in the latter expression by a covariant
derivative. Nevertheless, for connections with torsion this replacement is not
allowed anymore. Instead, the following relation holds:

$(dF)_{ab_{1}b_{2}\cdots b_{p}}\,=\,(p+1)\,\nabla_{[a}F_{b_{1}b_{2}\cdots
b_{p}]}\,+\,\frac{p(p+1)}{2}\,F_{c[b_{2}\cdots
b_{p}}\,T^{c}_{\phantom{c}ab_{1}]}\,.$

As usual, if $d\boldsymbol{F}=0$ we shall say that the differential form
$\boldsymbol{F}$ is closed, whereas if $\nabla_{[a}F_{b_{1}b_{2}\cdots
b_{p}]}=0$ then $\boldsymbol{F}$ will be said to be covariantly closed. These
concepts coincide just for connections with vanishing torsion.

###  2.1 Killing Vectors and Torsion

In the absence of torsion there are two equivalent ways of saying that a
vector field $\boldsymbol{\eta}$ is a Killing vector: (i) The metric is
invariant under the Lie dragging along the orbits of $\boldsymbol{\eta}$,
namely $\mathcal{L}_{\boldsymbol{\eta}}g_{ab}=0$, with
$\mathcal{L}_{\boldsymbol{\eta}}$ standing for the Lie derivative along
$\boldsymbol{\eta}$; (ii) The vector field $\boldsymbol{\eta}$ obeys the
Killing equation $\nabla_{(a}\eta_{b)}=0$. However, if the connection has
torsion these two definitions generally are not equivalent anymore. Indeed,
one can check that the following relation holds:

$\nabla_{a}\eta_{b}\,+\,\nabla_{b}\eta_{a}\,=\,\mathcal{L}_{\boldsymbol{\eta}}g_{ab}\,+\,2\,T_{(ab)e}\,\eta^{e}\,.$
(10)

If we choose a coordinate frame in which the Killing vector is one of the
basis vectors, $\boldsymbol{\eta}=\boldsymbol{\partial}_{x^{1}}$, then, in
these coordinates, the operator $\mathcal{L}_{\boldsymbol{\eta}}$ is given by
the partial derivative $\partial_{x^{1}}$. Therefore, the definition (i) means
that the components of the metric do not depend on the coordinate $x^{1}$.
Differently, in the presence of torsion, the existence of a vector field
obeying $\nabla_{(a}\eta_{b)}=0$ does not guarantee that the metric is
independent some coordinate in a suitable coordinate frame.

These two notions of Killing vector are intimately related to the two distinct
definitions of geodesic. In fact, if $\boldsymbol{V}$ is a vector field
tangent to a geodesic and $\boldsymbol{\eta}$ is a Killing vector according to
the definition (i) then the scalar $(\eta_{a}V^{a})$ is conserved along the
curves of minimum length, while if $\boldsymbol{\eta}$ is a Killing vector
according to the definition (ii) then $(\eta_{a}V^{a})$ is conserved along the
curves of zero acceleration. More precisely,

$\displaystyle V^{a}\mathring{\nabla}_{a}\,V^{b}=0\quad\textrm{ and
}\quad\mathcal{L}_{\boldsymbol{\eta}}g_{ab}=0\quad$
$\displaystyle\Longrightarrow\quad V^{a}\nabla_{a}(\eta_{b}V^{b})\,=\,0\,,$
$\displaystyle V^{a}\nabla_{a}\,V^{b}=0\quad\textrm{ and
}\quad\nabla_{(a}\eta_{b)}=0\quad$ $\displaystyle\Longrightarrow\quad
V^{a}\nabla_{a}(\eta_{b}V^{b})\,=\,0\,.$

Looking at Eq. (10), one conclude that in order for both definitions of a
Killing vector to be equivalent we must have $T_{(ab)c}=0$, which means that
the torsion is totally skew-symmetric. Likewise, this is exactly the necessary
condition for the two concepts of geodesic to be equivalent. In what follows
we shall mainly stick to the definition (ii) of a Killing vector.

Before proceeding to the integrability conditions of the conformal Killing-
Yano tensors, let us write the second derivative of a Killing vector in a
convenient way for later purposes. Here, by a Killing vector it is meant a
vector field obeying the Killing equation with the general metric-compatible
connection. Thus, if $\boldsymbol{\eta}$ is a Killing vector then the
following equation hold:

$\nabla_{a}\nabla_{b}\eta_{c}\,+\,\nabla_{a}\nabla_{c}\eta_{b}\,=\,\nabla_{a}\left(\,\nabla_{b}\eta_{c}\,+\,\nabla_{c}\eta_{b}\,\right)\,=\,0\,.$

Using permutations of this equation along with (5) we find

$\displaystyle 2\,\nabla_{a}\nabla_{b}\eta_{c}$
$\displaystyle\,=\,(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})\eta_{c}\,+\,(\nabla_{c}\nabla_{a}-\nabla_{a}\nabla_{c})\eta_{b}\,+\,(\nabla_{c}\nabla_{b}-\nabla_{b}\nabla_{c})\eta_{a}$
$\displaystyle\,=\,(R_{abc}^{\phantom{abc}e}+R_{cab}^{\phantom{cab}e}+R_{cba}^{\phantom{abc}e})\eta_{e}\,-\,(T^{e}_{\phantom{e}ab}\nabla_{e}\eta_{c}+T^{e}_{\phantom{e}ca}\nabla_{e}\eta_{b}+T^{e}_{\phantom{e}cb}\nabla_{e}\eta_{a})$
$\displaystyle\,=\,\left(3\,R_{[abc]}^{\phantom{[abc]}e}\,+\,2\,R_{cba}^{\phantom{cba}e}\right)\eta_{e}\,-\,(T^{e}_{\phantom{e}ab}\nabla_{e}\eta_{c}+T^{e}_{\phantom{e}ca}\nabla_{e}\eta_{b}+T^{e}_{\phantom{e}cb}\nabla_{e}\eta_{a})\,.$
(11)

##  3 Killing-Yano and Covariantly Closed Conformal Killing-Yano Tensors

In this section we shall define Killing-Yano tensors and covariantly closed
conformal Killing-Yano tensors and argue that these objects are two sides of
the same coin. Then, the integrability conditions for the existence of these
objects will be obtained.

A non-zero $p$-form $\boldsymbol{Y}$ is called a conformal Killing-Yano (CKY)
tensor of order $p$ whenever it obeys the following differential equation

$\nabla_{a}\,Y_{b_{1}b_{2}\cdots b_{p}}\,+\,\nabla_{b_{1}}\,Y_{ab_{2}\cdots
b_{p}}\,=\,2\,g_{a[b_{1}}\,h_{b_{2}\cdots
b_{p}]}\,+\,2\,g_{b_{1}[a}\,h_{b_{2}\cdots b_{p}]}\,,$ (12)

with $h_{b_{2}\cdots b_{p}}$ being some totaly anti-symmetric tensor of rank
$p-1$. Actually, contracting the above equation with $g^{ab_{1}}$ we find that

$h_{b_{2}\cdots b_{p}}\,=\,\frac{p}{2(n+1-p)}\,\nabla^{a}\,Y_{ab_{2}\cdots
b_{p}}\,,$ (13)

with $n$ standing for the dimension of the manifold. By means of algebraic
manipulations, one can show that the conformal Killing-Yano equation (12) is
equivalent to the following condition:

$\nabla_{a}\,Y_{b_{1}b_{2}\cdots b_{p}}\,=\,\nabla_{[a}\,Y_{b_{1}b_{2}\cdots
b_{p}]}\,+\,2\,g_{a[b_{1}}\,h_{b_{2}\cdots b_{p}]}\,.$ (14)

There are two very special cases of CKY tensors. If $h_{b_{2}\cdots b_{p}}$
vanishes, i.e., if $\boldsymbol{Y}$ has vanishing divergence, then the CKY
tensor is called a Killing-Yano (KY) tensor. While if
$\nabla_{[a}\,Y_{b_{1}b_{2}\cdots b_{p}]}$ vanishes then we shall say that
$\boldsymbol{Y}$ is a covariantly closed conformal Killing-Yano (CCCKY)
tensor.

It turns out that every KY tensor is the Hodge dual of a CCCKY tensor and vice
versa. Indeed, if $\epsilon_{a_{1}a_{2}\cdots a_{n}}$ is the volume-form of
the manifold then it follows that

$\epsilon^{a_{1}\ldots a_{q}\,b_{q+1}\ldots b_{n}}\,\epsilon_{a_{1}\ldots
a_{q}\,c_{q+1}\ldots
c_{n}}\;=\;q!(n-q)!\,(-1)^{\frac{n-s}{2}}\delta_{c_{q+1}}^{\;[b_{q+1}}\ldots\delta_{c_{n}}^{\;b_{n}]}\,,$
(15)

with $s$ being the signature of the metric. Taking the covariant derivative of
the latter equation we conclude that $\boldsymbol{\epsilon}$ is covariantly
constant with respect to any metric-compatible connection. Now, let
$\boldsymbol{A}$ be a KY tensor of rank $n-p$, so that

$\nabla_{a}A_{b_{1}\cdots b_{n-p}}\,=\,\nabla_{[a}\,A_{b_{1}b_{2}\cdots
b_{n-p}]}\,.$ (16)

Then, let us define the tensors

$H^{a_{1}a_{2}\cdots a_{p}}\,\equiv\,\epsilon^{a_{1}a_{2}\cdots
a_{p}b_{1}\cdots b_{n-p}}\,A_{b_{1}\cdots b_{n-p}}\quad\textrm{and}$
$h^{a_{1}a_{2}\cdots
a_{p-1}}\,\equiv\,\frac{(-1)^{(p-1)}\,p!\,(n-p)!}{(p-1)!\,(n-p+1)!}\,\epsilon^{a_{1}a_{2}\cdots
a_{p-1}cb_{1}\cdots b_{n-p}}\,\nabla_{[c}\,A_{b_{1}b_{2}\cdots b_{n-p}]}\,.$

Now, contracting the above definition of $h^{a_{1}a_{2}\cdots a_{p-1}}$ with
$\epsilon_{a_{1}a_{2}\cdots a_{p-1}de_{1}\cdots e_{n-p}}$ we find that

$\nabla_{[a}\,A_{b_{1}b_{2}\cdots
b_{n-p}]}\,=\,\frac{(-1)^{(\frac{n-s}{2}+p-1)}}{p!\,(n-p)!}\,h^{c_{1}c_{2}\cdots
c_{p-1}}\epsilon_{c_{1}c_{2}\cdots c_{p-1}ab_{1}b_{2}\cdots b_{n-p}}\,.$

Inserting the latter identity into (16) then contracting the final equation
with $\epsilon^{d_{1}d_{2}\cdots d_{p}b_{1}\cdots b_{n-p}}$, and using the
fact that the volume form is covariantly constant, eventually lead us to the
following differential equation:

$\nabla_{a}H_{b_{1}\cdots b_{p}}\,=\,2\,g_{a[b_{1}}h_{b_{2}\cdots b_{p}]}\,.$

Thus, $\boldsymbol{H}$ is a CCCKY tensor. Since, apart from a non-important
multiplicative constant, $\boldsymbol{H}$ is the Hodge dual of
$\boldsymbol{A}$, we have proved that the Hodge dual of a KY tensor is a CCCKY
tensor. In a completely analogous fashion, one can prove that the Hodge dual
of every CCCKY tensor is a KY tensor. Hence, studying the CCCKY equation is
equivalent to analysing KY equation. Particularly, in this article we have
made the choice of working out the integrability conditions for CCCKY tensors
of arbitrary rank. The choice of dealing with CCCKY tensors instead of KY
tensors is mainly based on the fact that in Kerr-NUT-(A)dS spacetimes of
arbitrary dimension a CCCKY tensor of rank two is the origin of the
integrability of Klein-Gordon and Dirac equations in this background [20, 21].

The importance of Killing-Yano tensors relies on the fact that they generate
conserved quantities along the geodesic curves, where here a geodesic means a
curve of zero acceleration. Indeed, if $\boldsymbol{V}$ is a geodesic vector
field, $V^{a}\nabla_{a}V^{b}=0$, and $\boldsymbol{A}$ is a KY tensor of order
$p$ then it follows that the scalar

$C\,=\,V^{a}\,V^{b}\,A_{ac_{2}\cdots c_{p}}\,A_{b}^{\phantom{b}c_{2}\cdots
c_{p}}$

is conserved along the geodesic tangent to $\boldsymbol{V}$, i.e.,
$V^{a}\nabla_{a}C=0$. In particular, this implies that the symmetric tensor

$Q_{ab}\,=\,A_{ac_{2}\cdots c_{p}}\,A_{b}^{\phantom{b}c_{2}\cdots c_{p}}$

is a Killing tensor of rank two, namely $\nabla_{(a}Q_{bc)}\,=\,0$. More
generally, if $\boldsymbol{A}$ and $\widehat{\boldsymbol{A}}$ are both KY
tensors of order $p$ then

$\widetilde{Q}_{ab}\,=\,A_{(a}^{\phantom{a}c_{2}\cdots
c_{p}}\,\widehat{A}_{b)c_{2}\cdots c_{p}}$

is a Killing tensor of rank two. On the other hand, the utility of CCCKY
tensors relies on the fact that the exterior product of two CCCKY tensors is
another CCCKY tensor. Thus, if $\boldsymbol{H}$ and $\widehat{\boldsymbol{H}}$
are CCCKY tensors of order $p$ and $q$ respectively then

$\widetilde{H}_{a_{1}\cdots a_{p}b_{1}\cdots b_{q}}\,=\,H_{[a_{1}\cdots
a_{p}}\,\widehat{H}_{b_{1}\cdots b_{q}]}$

is a CCCKY tensor of order $(p+q)$. Therefore, once we a have a CCCKY tensor
we can, in principle, build a tower of CCCKY tensors by means of taking
exterior products of the CCCKY tensor with itself. Then, taking the Hodge dual
of these CCCKY tensors we end up with a tower of KY tensors and, hence, a
tower of conserved scalars. Note that, since Killing tensors lead to conserved
quantities along geodesics, it follows from Noether’s theorem that such
tensors might generate symmetry transformations that leave particle’s action
invariant, a fact that have been addressed in Refs. [29, 37]. In the case of
KY tensors they also generate symmetries in superspace [29]. For more on the
relation between KY tensors and conserved quantities as well as their
importance in general relativity, the reader is referred to Ref. [18] and
references therein.

###  3.1 Integrability Condition

Now let us obtain the integrability conditions for the existence of a
covariantly closed conformal Killing-Yano tensor of arbitrary order in the
presence of torsion. The torsion-free case have already been addressed in [24,
25]. A skew-symmetric tensor $\boldsymbol{H}$ of rank $p$ is said to be a
CCCKY tensor if it obeys the following equation:

$\nabla_{a}H_{b_{1}\cdots b_{p}}\,=\,2\,g_{a[b_{1}}h_{b_{2}\cdots b_{p}]}\,,$
(17)

where $\boldsymbol{h}$ is a skew-symmetric tensor of rank $(p-1)$ given by

$h_{b_{2}\cdots b_{p}}\,=\,\frac{p}{2(n+1-p)}\,\nabla^{a}\,H_{ab_{2}\cdots
b_{p}}\,.$ (18)

In order to simplify the notation, let us define the tensor
$\boldsymbol{h}^{\prime}$ as

$h^{\prime}_{ab_{2}\cdots b_{p}}\,=\,\nabla_{a}h_{b_{2}\cdots b_{p}}\,.$

Particularly, note that $h^{\prime}_{ab_{2}\cdots
b_{p}}=h^{\prime}_{a[b_{2}\cdots b_{p}]}$. Now, let us use Eq. (5) in order to
compute the trace of $\boldsymbol{h}^{\prime}$:

$\displaystyle h^{\prime a}_{\phantom{a}ab_{3}\cdots b_{p}}\,=\,$
$\displaystyle\nabla^{a}\,h_{ab_{3}\cdots
b_{p}}\,=\,\frac{p}{2(n+1-p)}\,\nabla^{a}\nabla^{c}\,H_{cab_{3}\cdots b_{p}}$
$\displaystyle\,=\,$
$\displaystyle\frac{-\,p}{4(n+1-p)}\,(\nabla^{a}\nabla^{c}-\nabla^{c}\nabla^{a})\,H_{acb_{3}\cdots
b_{p}}$ $\displaystyle\,=\,$
$\displaystyle\frac{p}{4(n+1-p)}\left[2\,R^{ac}\,H_{acb_{3}\cdots
b_{p}}+\sum_{i=3}^{p}R^{ace}_{\phantom{[ace]}b_{i}}\,H_{acb_{3}\cdots\check{b}_{i}\,e\,b_{i+1}\cdots
b_{p}}\,+\,T^{eac}\nabla_{e}H_{acb_{3}\cdots b_{p}}\right]$
$\displaystyle\,=\,$
$\displaystyle\frac{p}{4(n+1-p)}\left[2\,R^{[ac]}\,H_{acb_{3}\cdots
b_{p}}+\sum_{i=3}^{p}R^{[ace]}_{\phantom{[ace]}b_{i}}\,H_{acb_{3}\cdots\check{b}_{i}\,e\,b_{i+1}\cdots
b_{p}}\right.$
$\displaystyle\phantom{\frac{p}{4(n+1-p)}}\;\;\,\,+\,\,\left.\frac{4}{p}\,T_{a}^{\phantom{a}ac}h_{cb_{3}\cdots
b_{p}}\,+\,\frac{2(p-2)}{p}\,h_{ac[b_{4}\cdots
b_{p}}T_{b_{3}]}^{\phantom{b_{3}]}ac}\right]$ (19)

Where it is worth recalling that the notation $\check{b}_{i}$ means that the
index $b_{i}$ is absent. In order to attain (3.1) it has been used the
following useful algebraic identity valid for any totally skew-symmetric
tensor $\boldsymbol{A}$ of rank $(p-1)$:

$\displaystyle g_{e[a}A_{cb_{3}\cdots b_{p}]}\,\,=\,$
$\displaystyle\frac{1}{p}\,g_{ea}\,A_{cb_{3}\cdots
b_{p}}\,-\,\frac{p-1}{p}\,\,A_{a[b_{3}\cdots b_{p}}g_{c]e}$
$\displaystyle\,=\,$ $\displaystyle\frac{1}{p}\,g_{ea}\,A_{cb_{3}\cdots
b_{p}}\,-\,\frac{1}{p}\,g_{ec}\,A_{ab_{3}\cdots
b_{p}}\,+\,\frac{p-2}{p}\,\,A_{ac[b_{4}\cdots b_{p}}g_{b_{3}]e}\,.$ (20)

Note that the right hand side of (3.1) is zero for vanishing torsion. Now,
taking the covariant derivative of (17), we find that

$\nabla_{a}\nabla_{b}H_{c_{1}c_{2}\cdots
c_{p}}\,=\,2\,h^{\prime}_{a[c_{2}\cdots c_{p}}\,g_{c_{1}]b}$

Thus, using this equation along with (5) we arrive at

$\displaystyle 2\,h^{\prime}_{a[c_{2}\cdots
c_{p}}\,g_{c_{1}]b}\,-\,2\,h^{\prime}_{b[c_{2}\cdots c_{p}}\,g_{c_{1}]a}\,=\,$
$\displaystyle(\nabla_{a}\nabla_{b}\,-\,\nabla_{b}\nabla_{a})H_{c_{1}c_{2}\cdots
c_{p}}$ $\displaystyle\,=\,$
$\displaystyle\sum_{i=1}^{p}R_{abc_{i}}^{\phantom{abc_{i}}e}\,H_{c_{1}c_{2}\cdots\check{c}_{i}\,e\,c_{i+1}\cdots
c_{p}}\,-\,T^{e}_{\phantom{e}ab}\nabla_{e}H_{c_{1}c_{2}\cdots c_{p}}$
$\displaystyle\,=\,$
$\displaystyle\sum_{i=1}^{p}R_{abc_{i}}^{\phantom{abc_{i}}e}\,H_{c_{1}c_{2}\cdots\check{c}_{i}\,e\,c_{i+1}\cdots
c_{p}}\,-\,2\,h_{[c_{2}\cdots c_{p}}T_{c_{1}]ab}\,.$ (21)

As it is, this equation is an integrability condition involving the CCCKY
tensor $\boldsymbol{H}$ and its second derivative through the tensor
$\boldsymbol{h}^{\prime}$. However, in order to apply an integrability
condition it is more useful when it is purely algebraic. Therefore, let us try
to express the tensor $\boldsymbol{h}^{\prime}$ in terms of $\boldsymbol{H}$
and $\boldsymbol{h}$. Expanding Eq. (3.1) by means the algebraic identity
(3.1) lead us to

$\displaystyle\frac{2}{p}\,g_{c_{1}b}$
$\displaystyle\,h^{\prime}_{ac_{2}\cdots c_{p}}\,-\,\frac{2(p-1)}{p}\
h^{\prime}_{ac_{1}[c_{3}\cdots
c_{p}}g_{c_{2}]b}\,-\,\frac{2}{p}\,g_{c_{1}a}\,h^{\prime}_{bc_{2}\cdots
c_{p}}\,+\,\frac{2(p-1)}{p}\ h^{\prime}_{bc_{1}[c_{3}\cdots c_{p}}g_{c_{2}]a}$
$\displaystyle\,=\,$ $\displaystyle
R_{abc_{1}}^{\phantom{abc_{1}}e}\,H_{ec_{2}\cdots
c_{p}}\,+\,\sum_{i=2}^{p}R_{abc_{i}}^{\phantom{abc_{i}}e}\,H_{c_{1}c_{2}\cdots\check{c}_{i}\,e\,c_{i+1}\cdots
c_{p}}\,-\,\frac{2}{p}\,T_{c_{1}ab}\,h_{c_{2}\cdots
c_{p}}\,+\,\frac{2(p-1)}{p}h_{c_{1}[c_{3}\cdots c_{p}}T_{c_{2}]ab}\,.$

So, contracting the latter equation with $g^{bc_{1}}$, we end up with the
following expression for $\boldsymbol{h}^{\prime}$:

$\displaystyle h^{\prime}_{ac_{2}\cdots c_{p}}\,=\,$
$\displaystyle\frac{-\,p}{2(n-p)}\left[R_{a}^{\phantom{a}e}\,H_{ec_{2}\cdots
c_{p}}\,+\,\sum_{i=2}^{p}R_{\phantom{b}ac_{i}}^{b\phantom{ac_{i}}e}\,H_{bc_{2}\cdots\check{c}_{i}e\cdots
c_{p}}\,+\,\frac{2(p-1)}{p}\,h^{\prime e}_{\phantom{e}e[c_{3}\cdots
c_{p}}g_{c_{2}]a}\right.$
$\displaystyle\phantom{\frac{-\,p}{2(n-p)}}\quad\,+\,\left.\frac{2(p-1)}{p}\,h^{e}_{\phantom{e}[c_{3}\cdots
c_{p}}T_{c_{2}]ea}\,-\,\frac{2}{p}\,h_{c_{2}\cdots
c_{p}}\,T^{e}_{\phantom{e}ea}\right]$ (22)

Then, inserting (3.1) into (3.1) we can find the wanted expression for
$\boldsymbol{h}^{\prime}$. Finally, inserting such expression into (3.1) we
arrive at the integrability condition for the existence of a CCCKY tensor
involving just $\boldsymbol{H}$ and $\boldsymbol{h}$, without higher
derivatives of $\boldsymbol{H}$. Thus, the integrability condition amounts to
the Eqs. (3.1), (3.1) and (3.1).

As a first consequence of such integrability condition, let us obtain a
constraint on the torsion tensor. Taking the totally skew-symmetric part of
indices $abc_{1}\cdots c_{p}$ in Eq. (3.1) we find that

$2\,T_{[abc_{1}}\,h_{c_{2}\cdots
c_{p}]}\,=\,\sum_{i=1}^{p}R_{[abc_{i}}^{\phantom{[abc_{i}}e}\,H_{c_{1}c_{2}\cdots\check{c}_{i}|e|c_{i+1}\cdots
c_{p}]}\,=\,(-1)^{(p-1)}\,p\,R_{[abc_{1}}^{\phantom{[abc_{1}}e}\,H_{c_{2}\cdots
c_{p}]e}\,.$ (23)

Now, using the first Bianchi identity in (8) we can write the above equations
as

$2\,T_{[abc_{1}}\,h_{c_{2}\cdots
c_{p}]}\,=\,(-1)^{p}\,p\,\left(\,\nabla_{[a}T^{e}_{\phantom{e}bc_{1}}\,H_{c_{2}\cdots
c_{p}]e}\,+\,T^{e}_{\phantom{e}d[a}T^{d}_{\phantom{d}bc_{1}}H_{c_{2}\cdots
c_{p}]e}\,\right)\,.$ (24)

This is a constraint involving just the torsion. Therefore, besides the
metric, the torsion is also constrained by the existence of a covariantly
closed conformal Killing-Yano tensor.

As another consequence of these integrability conditions, let us consider the
case $p=2$, namely a bivector $\boldsymbol{H}$ obeying to
$\nabla_{a}H_{bc}=2g_{a[b}h_{c]}$. It is well known that in the absence of
torsion the vector field $h_{a}$ is a Killing vector whenever the Ricci tensor
is proportional to the metric [25]. Differently, since
$\nabla_{(a}h_{b)}=h^{\prime}_{(ab)}$ it follows from (3.1) that if torsion is
different from zero then, in general, $h_{a}$ will not be a Killing vector
even if the Ricci tensor is proportional to the metric. For instance, if the
torsion is totally skew-symmetric and $R_{ab}\propto g_{ab}$ then Eqs. (3.1),
(3.1), (8) and (9) imply that $\nabla_{(a}h_{b)}$ vanishes if, and only if,

$\left(\,\nabla_{[a}T_{e]cb}+\nabla_{[c}T_{e]ab}\,\right)\,H^{eb}$

also vanishes, which generally is not the case. Now, let us apply the results
obtained in the present section for the case $p=1$, i.e., the case of a
covariantly closed conformal Killing vector.

##  4 Covariantly Closed Conformal Killing Vectors

The aim of the present section is to find all metrics and torsions compatible
with the existence of a covariantly closed conformal Killing vector (CCCKV),
namely a vector field $\boldsymbol{H}$ obeying

$\nabla_{a}\,H_{b}\,=\,2\,h\,g_{ab}\quad\textrm{ with }\quad
h\,=\,\frac{1}{2\,n}\,\nabla_{a}H^{a}\,.$ (25)

Since every Killing-Yano tensor of order $n-1$ is the Hodge dual of a CCCKV,
we will, equivalently, find all metrics and torsions compatible with a KY
tensor of order $n-1$.

Assuming $p=1$ in Eqs. (3.1) and (24) we arrive at the following integrability
conditions:

$R_{abc}^{\phantom{abc}e}H_{e}\,+\,2\,g_{ca}\,h^{\prime}_{b}\,-\,2\,g_{cb}\,h^{\prime}_{a}\,=\,2\,h\,T_{cab}\,,$
(26)
$H_{e}\,\nabla_{[a}T^{e}_{\phantom{e}bc]}\,+\,H_{e}\,T^{e}_{\phantom{e}d[a}T^{d}_{\phantom{d}bc]}\,+\,2\,h\,T_{[abc]}\,=\,0\,.$
(27)

Where, using (3.1), we have that $h^{\prime}_{a}$ is given by

$h^{\prime}_{a}\,=\,\nabla_{a}h\,=\,\frac{-\,1}{2(n-1)}\,\left(\,R_{a}^{\phantom{a}e}\,H_{e}\,-\,2\,h\,T^{e}_{\phantom{e}ea}\,\right)\,.$
(28)

Therefore, inserting (28) into (26) we arrive at the following algebraic
integrability condition:

$\left[\,(n-1)R_{abc}^{\phantom{abc}e}\,-\,g_{ac}\,R_{b}^{\phantom{b}e}\,+\,g_{bc}\,R_{a}^{\phantom{a}e}\,\right]\,H_{e}\,=\,2\,h\,\left[\,(n-1)\,T_{cab}\,-\,g_{ac}\,T^{e}_{\phantom{e}eb}\,+\,g_{bc}\,T^{e}_{\phantom{e}ea}\,\right]\,.$
(29)

Now, let us denote the squared norm of $\boldsymbol{H}$ by $N$,

$N\,\equiv\,H_{a}\,H^{a}\,.$

Then, taking the derivatives of this equation and using (25) we find that

$\displaystyle\nabla_{b}N\,=\,$ $\displaystyle 4\,h\,H_{b}\,,$ (30)
$\displaystyle\nabla_{a}\nabla_{b}N\,=\,$ $\displaystyle
4\,h^{\prime}_{a}\,H_{b}\,+\,8\,h^{2}\,g_{ab}\,.$ (31)

The commutator of covariant derivatives acting in a scalar gives

$(\nabla_{a}\nabla_{b}\,-\,\nabla_{b}\nabla_{a})N\,=\,-\,T^{e}_{\phantom{e}ab}\nabla_{e}N\,.$

Thus, inserting (30) and (31) into this identity we find the following useful
relation

$H_{a}\,h^{\prime}_{b}\,-\,H_{b}\,h^{\prime}_{a}\,=\,h\,H_{e}\,T^{e}_{\phantom{e}ab}\,.$
(32)

In particular, such equation implies

$H_{e}\,T^{e}_{\phantom{e}[ab}\,H_{c]}\,=\,0\,\,,\quad\;\textrm{ if $h\neq
0$}\,.$ (33)

Note that $h\neq 0$ means that the vector $\boldsymbol{H}$ is not covariantly
constant. Now, since $\boldsymbol{H}$ is covariantly closed it follows that

$0\,=\,\nabla_{[a}H_{b]}\,=\,\partial_{[a}H_{b]}-\Gamma^{e}_{[ab]}H_{e}\quad\Longrightarrow\quad\quad\partial_{[a}H_{b]}\,=\,\frac{1}{2}\,H_{e}\,T^{e}_{\phantom{e}ab}\,.$
(34)

So, using (34) and (33) we conclude that

$H_{[c}\,\partial_{a}H_{b]}\,=\,0\,\,,\quad\;\textrm{ if $h\neq 0$}\,.$ (35)

In what follows we shall assume the condition $h\neq 0$ to hold, the case
$h=0$ will be considered separately later. According to the Frobenius theorem,
the relation (35) guarantees that the vector field $\boldsymbol{H}$ is
orthogonal to a family of hyper-surfaces. Thus, if $\lambda$ is a parameter
along the orbits of $\boldsymbol{H}$ we can locally introduce coordinates
$\\{x^{i},\lambda\\}$, with $i,j\in\\{1,2,\cdots,n-1\\}$, such that the vector
field $\boldsymbol{H}=\boldsymbol{\partial}_{\lambda}$ is orthogonal to the
$(n-1)$ basis vectors $\boldsymbol{\partial}_{i}$. Therefore, in these
coordinates the line element is given by

$ds^{2}\,=\,N\,d\lambda^{2}\,+\,g_{ij}\,dx^{i}dx^{j}\,.$ (36)

Where, in principle, $N$ and $g_{ij}$ are functions of $\lambda$ and
$\\{x^{i}\\}$. However, since in these coordinates the components of
$\boldsymbol{H}$ are

$H^{a}\,=\,\delta^{a}_{\lambda}\quad\textrm{ and }\quad
H_{a}\,=\,N\,\delta^{\lambda}_{a}$

it follows from (30) that

$\partial_{a}N=4\,h\,N\,\delta^{\lambda}_{a}\quad\Longrightarrow\quad\left\\{\begin{array}[]{ll}\partial_{\lambda}N\,=\,4\,h\,N\,,\\\
\partial_{i}N\,=\,0\;\;\Rightarrow\;\;N=N(\lambda)\,.\\\ \end{array}\right.$
(37)

Therefore, $N$ is just a function of $\lambda$. Consequently, we have that

$\partial_{[a}H_{b]}\,=\,\partial_{[a}N\delta^{\lambda}_{b]}\,=\,\partial_{\lambda}N\,\delta^{\lambda}_{[a}\delta^{\lambda}_{b]}\,=\,0\,.$
(38)

Thus, besides being covariantly closed, it turns out that $\boldsymbol{H}$ is
also closed. As a consequence of (38) and (34), we arrive at the following
constraint for the torsion:

$H^{e}\,T_{eab}\,=\,T_{\lambda ab}\,=\,0\,.$ (39)

This, along with (32), implies that

$h^{\prime}_{a}\,\propto\,H_{a}\quad\Longrightarrow\quad h=h(\lambda)\,,$ (40)

which could be anticipated from (37). In addition, taking the covariant
derivative of (39) and then using (25) we immediately find

$H_{e}\nabla_{c}T^{e}_{\phantom{e}ab}\,=\,-\,2\,h\,T_{cab}\,.$ (41)

The latter identity along with (39) implies that the torsion integrability
condition (27) is readily satisfied. Now, writing down the equation
$\nabla_{a}H_{b}=2hg_{ab}$ in these coordinates and using (2) we eventually
find

$2\,h\,g_{ab}\,=\,(\partial_{\lambda}N)\,\delta^{\lambda}_{a}\delta^{\lambda}_{b}\,+\,\frac{1}{2}\left[\,\partial_{\lambda}g_{ab}\,-\,\partial_{a}g_{b\lambda}\,-\,\partial_{b}g_{a\lambda}\,+\,2\,T_{(ab)\lambda}\,\right]\,.$

Apart from the relations already obtained, the above equation is equivalent to

$\partial_{\lambda}\,g_{ij}\,=\,4\,h\,g_{ij}\,-\,2\,T_{(ij)\lambda}\,.$ (42)

Therefore, we have proved that a manifold endowed with a metric-compatible
connection admits a vector field $\boldsymbol{H}$ obeying the CCCKV equation,
$\nabla_{a}H_{e}=2hg_{ab}$, with $h\neq 0$ if, and only if, its line element
can be written as (36) with $N$ being a function of $\lambda$, $g_{ij}$
obeying the differential equation (42) and the torsion obeying the constraint
(39). In the previous coordinates the CCCKV is given by
$\boldsymbol{H}=\boldsymbol{\partial}_{\lambda}$.

###  4.1 The Case of a Totally Skew-Symmetric Torsion

Let us now consider the important special case of a totally skew-symmetric
torsion, $T_{abc}=T_{[abc]}$. In such a case the Eq. (42) can be nicely
integrated for the metric $g_{ij}$. Indeed, since in this case $T_{(ab)c}=0$,
equations (42) and (37) yield

$N\,\partial_{\lambda}\,g_{ij}\,=\,g_{ij}\,\partial_{\lambda}N\quad\Longrightarrow\quad
g_{ij}(\lambda,x)=N(\lambda)\,\tilde{g}_{ij}(x)\,,$

where the functions $\tilde{g}_{ij}$ are functions just of the coordinates
$\\{x^{i}\\}$. Therefore, the metric (36) can be written as

$ds^{2}\,=\,N(\lambda)\,\left[\,d\lambda^{2}\,+\,\tilde{g}_{ij}(x)\,dx^{i}dx^{j}\,\right]\,.$
(43)

Thus, a manifold endowed with a metric-compatible connection with totally
skew-symmetric torsion admits a vector field $\boldsymbol{H}$ obeying the
CCCKV equation if, and only if, its metric can be written as (43) and the
torsion is such that $T_{\lambda ab}=0$. In the latter case the CCCKV is given
by $\boldsymbol{H}=\boldsymbol{\partial}_{\lambda}$. It is interesting noting
that, in the case of a totally skew-symmetric torsion, Eqs. (40) and (28)
imply that $\boldsymbol{H}$ is an eigenvector of the Ricci tensor,
$R_{a}^{\phantom{a}e}H_{e}\propto H_{a}$.

Now, suppose that a manifold admits a CCCKV $\boldsymbol{H}$ and a Killing
vector $\boldsymbol{\eta}$. Then, let us prove that in this case the vector
field $\chi_{b}=\nabla_{b}(H^{c}\eta_{c})$ is a conformal Killing vector.
Indeed, using (25) we have that

$\chi_{b}\,=\,\nabla_{b}(H^{c}\eta_{c})\,=\,2\,h\,\eta_{b}\,+\,H^{c}\,\nabla_{b}\eta_{c}\quad\Rightarrow\quad$

$\nabla_{a}\chi_{b}\,=\,2\,h^{\prime}_{a}\,\eta_{b}\,+\,H^{c}\nabla_{a}\nabla_{b}\eta_{c}\,.$
(44)

Where it has been used the fact that $\boldsymbol{\eta}$ obeys the Killing
equation.444Recall that in the case of a totally skew-symmetric torsion the
two traditional definitions of Killing vector coincide, so that it is not
necessary to specify what it is meant by a Killing vector (see section 2.1).
Now, inserting Eqs. (28) and (11) into (44) and using the condition
$H^{e}T_{eab}=0$ we find that:

$\nabla_{a}\chi_{b}\,=\,\frac{-\,1}{n-1}\,R_{a}^{\phantom{a}e}\,H_{e}\,\eta_{b}\,+\,H^{c}\,\eta^{e}\left[\frac{3}{2}\,R_{[abc]e}\,+\,R_{bcea}\right]\,-\,\frac{1}{2}\,T^{e}_{\phantom{e}ab}\,H^{c}\,\nabla_{e}\eta_{c}\,.$

Then, using (8) and (9) to rewrite the terms $R_{[abc]e}$ and $R_{bcea}$ in
the latter equation we have

$\nabla_{a}\chi_{b}\,=\,\frac{-\,1}{n-1}\,R_{a}^{\phantom{a}e}\,H_{e}\,\eta_{b}\,+\,H^{c}\,\eta^{e}\left[-\,\frac{3}{2}\,\nabla_{[a}T_{bc]e}\,+\,\left(\,R_{eabc}\,+\,\nabla_{[e}T_{a]bc}\,-\,\nabla_{[b}T_{c]ea}\,\right)\right]\,-\,\frac{1}{2}\,T^{e}_{\phantom{e}ab}\,H^{c}\,\nabla_{e}\eta_{c}\,.$

Now, making use of (41) and manipulating the derivative on the last term of
the above relation, we end up with

$\nabla_{a}\chi_{b}\,=\,\frac{-\,1}{n-1}\,R_{a}^{\phantom{a}e}\,H_{e}\,\eta_{b}\,+\,\eta^{e}\,R_{eab}^{\phantom{eab}c}\,H_{c}\,-\,3\,h\,\eta^{e}\,T_{eab}\,-\,\frac{1}{2}\,T^{e}_{\phantom{e}ab}\,\left(\,\chi_{e}\,-\,\eta_{c}\,\nabla_{e}H^{c}\,\right)\,.$

Finally, using the integrability condition (29) to rewrite the term
$\eta^{e}\,R_{eab}^{\phantom{eab}c}\,H_{c}$ lead us to the following result:

$\nabla_{a}\chi_{b}\,=\,\frac{-\,1}{n-1}\,g_{ab}\,\left(\eta^{e}\,R_{ec}\,H^{c}\right)\,-\,\frac{1}{2}\,\chi_{e}\,T^{e}_{\phantom{e}ab}\,.$
(45)

In particular, taking the symmetric part of (45) we see that the right hand
side is proportional to the metric. Therefore, the following theorem can be
stated.

Theorem - _Let $(M,\boldsymbol{g})$ be a manifold endowed with a metric-
compatible connection whose torsion is totally skew-symmetric. Then, if
$\boldsymbol{\eta}$ is a Killing vector field in $M$ and $\boldsymbol{H}$
obeys $\nabla_{a}H_{b}=2\,h\,g_{ab}$ with $h\neq 0$ then the vector field
$\chi_{a}=\nabla_{a}(H^{c}\eta_{c})$ is a conformal Killing vector. Moreover,
if $\chi^{e}T_{eab}=0$ then $\boldsymbol{\chi}$ is covariantly closed,
$\nabla_{[a}\chi_{b]}=0$._

The torsion-free version of this theorem was proved in [38].

###  4.2 Covariantly Constant Vector Fields, the Case $h=0$

Previously, we have considered a vector field $\boldsymbol{H}$ obeying
$\nabla_{a}H_{b}=2\,h\,g_{ab}$ with $h$ being non-vanishing. Now, let us
consider the special case $h=0$, namely when the vector field $\boldsymbol{H}$
is covariantly constant,

$\nabla_{a}H_{b}\,=\,0\,.$ (46)

The integrability condition is this case is

$R_{abc}^{\phantom{abc}e}\,H_{e}\,=\,0\,.$

In the general case $h\neq 0$ we have made use of (32) to prove that
$\boldsymbol{H}$ is orthogonal to a family of hyper-surfaces. However, Eq.
(32) is trivial for the case $h=0$, so that we cannot arrive at the same
conclusion in such a case. Instead, using (34) we have that

$\partial_{[a}H_{b]}\,=\,\frac{1}{2}\,H_{e}\,T^{e}_{\phantom{e}ab}\quad\;\textrm{
and }\;\quad
H_{[c}\,\partial_{a}H_{b]}\,=\,\frac{1}{2}\,H_{e}\,T^{e}_{\phantom{e}[ab}H_{c]}\,.$
(47)

Note that for vanishing torsion Eq. (47) not only implies that
$\boldsymbol{H}$ is hyper-surface-orthogonal but also guarantees that it
$\boldsymbol{H}$ is closed. Nevertheless, for a general torsion neither
conclusions need to hold.

In order to attain these conclusions more explicitly, let us introduce
coordinates $\\{x^{i},\lambda\\}$ with $i,j\in\\{1,2,\cdots,n-1\\}$ and such
that $\boldsymbol{H}=\boldsymbol{\partial}_{\lambda}$. In these coordinates
the metric is generally written as

$ds^{2}\,=\,N\,d\lambda^{2}\,+\,2\,\xi_{i}\,dx^{i}\,d\lambda\,+\,g_{ij}\,dx^{i}dx^{j}\,.$
(48)

Where $N$ is a constant while $\xi_{i}$ and $g_{ij}$ can, in principle, be
general functions of $\lambda$ and $\\{x^{i}\\}$. Now, imposing the equation
$\nabla_{a}H^{b}=0$ in this coordinate frame we find that the connection
symbol $\Gamma^{b}_{a\lambda}$ must vanish, which is tantamount to

$\partial_{a}g_{\lambda
c}\,+\,\partial_{\lambda}g_{ac}\,-\,\partial_{c}g_{a\lambda}\,=\,T_{a\lambda
c}\,+\,T_{\lambda ac}\,-\,T_{ca\lambda}\,.$ (49)

This equality, in turn, is equivalent to the following three constraints:

$\partial_{\lambda}\xi_{i}\,=\,T_{\lambda\lambda
i}\quad,\quad\partial_{\lambda}g_{ij}\,=\,-\,2\,T_{(ij)\lambda}\quad,\quad\partial_{i}\xi_{j}\,-\,\partial_{j}\xi_{i}\,=\,T_{\lambda
ij}\,.$ (50)

In particular, the third condition in (50) guarantees that if $T_{\lambda ij}$
is non-vanishing then the functions $\xi_{i}$ cannot all vanish, so that
generally $\boldsymbol{H}$ is not orthogonal to a family of hyper-surfaces.
This is intriguing. Since the equation $\nabla_{a}H_{b}=0$ is more restrictive
than the CCCKV equation, $\nabla_{a}H_{b}=2\,h\,g_{ab}$, it is natural to
expect that the results valid for $h\neq 0$ would also be valid in the more
special case $h=0$. However, we have proved that there are constraints valid
in the case $h\neq 0$ that do not carry to the special case $h=0$, such as the
hyper-surface-orthogonal condition and the restriction $T_{\lambda ab}=0$.
Note, however, that such unexpected fact happens only when the torsion is non-
zero.

##  5 Maximally Symmetric Spaces

A Riemannian manifold endowed with the Levi-Civita connection is called
maximally symmetric when it admits the maximum number independent Killing
vector fields. In $n$ dimensions this maximal number is $\frac{1}{2}n(n+1)$,
which physically arises from $n$ translations an $\frac{1}{2}n(n-1)$
rotations. In this section, we shall consider the issue of defining a
maximally symmetric space in the presence of torsion. Here, we shall say that
$\boldsymbol{\eta}$ is a Killing vector field if it obeys the Killing equation

$\nabla_{a}\,\eta_{b}\,+\,\nabla_{b}\,\eta_{a}\,=\,0\,.$

Recall that we could have defined a Killing vector as a vector field such that
$\mathcal{L}_{\boldsymbol{\eta}}g_{ab}=0$. As pointed out in Sec. 2.1, these
two definitions are not equivalent in the presence of general torsion.
However, since

$\mathcal{L}_{\boldsymbol{\eta}}g_{ab}=\mathring{\nabla}_{a}\,\eta_{b}\,+\,\mathring{\nabla}_{b}\,\eta_{a}\,,$

it turns out that the latter definition is just a particular case of the
former. Namely, the equation $\mathcal{L}_{\boldsymbol{\eta}}g_{ab}=0$ can be
retrieved from the equation $\nabla_{(a}\,\eta_{b)}=0$ by choosing the
connection to be torsion-free. Maximally symmetric spaces in the presence of
torsion have been investigated before. In [39], a couple of integrability
conditions for the Killing equation are worked out, but along the calculations
some unjustified constraints are imposed over the torsion as well as over the
curvature. In Refs. [40, 41], it is imposed that the torsion tensor should
also be “maximally symmetric”, namely it is assumed that the torsion is
invariant under the Lie dragging along the Killing vectors of a maximally
symmetric manifold or submanifold, see also [42]. Here we go further and
deduce the integrability conditions of a maximally symmetric space without
imposing any constraint over the torsion. Moreover, we also investigate spaces
with the maximal number of KY tensors of order $n-1$. For a nice review of
maximally symmetric spaces in the torsion-free case, the reader is referred to
Ref. [43].

The first natural question is: what is the maximum number of Killing vectors
when the connection has non-zero torsion? The answer is held in Eq. (11),
according to which the second derivative of a Killing vector can be written in
terms of the vector itself and its first derivative. Thus, successively
differentiating this equation one can write all derivatives of the Killing
vector in terms of the Killing vector itself and its first derivative. So, if
the components of $\eta_{a}$ and $\nabla_{a}\eta_{b}$ are given in one point
of the manifold it is possible to reconstruct the vector field
$\boldsymbol{\eta}$ in the whole manifold. Therefore, since $\eta_{a}$ has $n$
components and $\nabla_{a}\eta_{b}$ is an antisymmetric rank 2 tensor with
$\frac{1}{2}n(n-1)$ components, it follows that maximum number of Killing
vector fields remains being $\frac{1}{2}n(n+1)$. Therefore, even in the
presence of torsion, we shall say that an $n$-dimensional manifold is
maximally symmetric if it admits $\frac{1}{2}n(n+1)$ independent Killing
vector fields.

Now, let us investigate how the existence of $\frac{1}{2}n(n+1)$ independent
Killing vector fields constrains the metric and the torsion of a manifold. In
order to accomplish this, we should find the integrability condition for the
Killing equation. Using the identity (5) we have that

$\left(\,\nabla_{d}\nabla_{a}\,-\,\nabla_{a}\nabla_{d}\,\right)\nabla_{b}\eta_{c}\,=\,R_{dab}^{\phantom{dab}e}\,\nabla_{e}\eta_{c}\,+\,R_{dac}^{\phantom{dac}e}\,\nabla_{b}\eta_{e}\,-\,T^{e}_{\phantom{e}da}\,\nabla_{e}\nabla_{b}\eta_{c}\
$

Now, rewriting the second derivative of $\boldsymbol{\eta}$ in the right hand
side of the above equation by means of (11) we obtain

$\displaystyle
2\left(\,\nabla_{d}\nabla_{a}\nabla_{b}\eta_{c}-\nabla_{a}\nabla_{d}\nabla_{b}\eta_{c}\,\right)\,=\,$
$\displaystyle
2R_{dabe}\nabla^{e}\eta_{c}\,+\,2\,R_{dace}\nabla_{b}\eta^{e}\,+\,R_{bcef}\,T^{f}{}_{ad}\eta^{e}\,-\,R_{bfce}\,T^{f}{}_{ad}\eta^{e}$
$\displaystyle\,+\,R_{cfbe}\,T^{f}{}_{ad}\eta^{e}$
$\displaystyle\,+\,T_{ebf}\,T^{f}{}_{ad}\nabla^{e}\eta_{c}\,+\,T_{ead}\,T_{fbc}\nabla^{f}\eta^{e}\,-\,T_{ecf}\,T^{f}{}_{ad}\nabla^{e}\eta_{b}$
(51)

Then, using (11) to write the terms $\nabla_{a}\nabla_{b}\eta_{c}$ and
$\nabla_{d}\nabla_{b}\eta_{c}$ on the left hand side of (51) and then using
(11) again to rewrite the second derivatives of $\boldsymbol{\eta}$ that
eventually appear, lead us to the following integrability condition:

$\displaystyle\left(\,R_{cdef}\,T^{f}{}_{ab}-R_{cfde}\,T^{f}{}_{ab}-R_{dfce}\,T^{f}{}_{ab}-R_{bdef}\,T^{f}{}_{ac}+R_{bfde}\,T^{f}{}_{ac}+R_{dfbe}\,T^{f}{}_{ac}-2R_{bcef}\,T^{f}{}_{ad}\right.$
$\displaystyle+2R_{bfce}\,T^{f}{}_{ad}-2R_{cfbe}\,T^{f}{}_{ad}-2\,R_{adef}\,T^{f}{}_{bc}-R_{acef}\,T^{f}{}_{bd}-R_{afce}\,T^{f}{}_{bd}-R_{cfae}\,T^{f}{}_{bd}+R_{abef}\,T^{f}{}_{cd}$
$\displaystyle+R_{afbe}\,T^{f}{}_{cd}+\left.R_{bfae}\,T^{f}{}_{cd}+2\nabla_{a}R_{bcde}+2\nabla_{a}R_{bdce}-2\nabla_{a}R_{cdbe}+2\nabla_{d}R_{abce}-2\nabla_{d}R_{acbe}-2\nabla_{d}R_{bcae}\,\right)\eta^{e}$
$\displaystyle\,=\,-2\,R_{bcde}\nabla_{a}\eta^{e}-2\,R_{bdce}\nabla_{a}\eta^{e}+2\,R_{cdbe}\nabla_{a}\eta^{e}-4\,R_{adce}\nabla_{b}\eta^{e}-2\,R_{abce}\nabla_{d}\eta^{e}+2R_{acbe}\nabla_{d}\eta^{e}$
(52)
$\displaystyle+2R_{bcae}\nabla_{d}\eta^{e}-T_{ecf}T^{f}{}_{bd}\nabla^{e}\eta_{a}+T_{ebf}T^{f}{}_{cd}\nabla^{e}\eta_{a}-2\nabla_{d}T_{ebc}\nabla^{e}\eta_{a}+T_{edf}T^{f}{}_{ac}\nabla^{e}\eta_{b}-2\,T_{ecf}T^{f}{}_{ad}\nabla^{e}\eta_{b}$
$\displaystyle+T_{eaf}T^{f}{}_{cd}\nabla^{e}\eta_{b}-2\nabla_{a}T_{ecd}\nabla^{e}\eta_{b}-2\nabla_{d}T_{eac}\nabla^{e}\eta_{b}-4R_{adbe}\nabla^{e}\eta_{c}-T_{edf}T^{f}{}_{ab}\nabla^{e}\eta_{c}+2T_{ebf}T^{f}{}_{ad}\nabla^{e}\eta_{c}$
$\displaystyle-
T_{eaf}T^{f}{}_{bd}\nabla^{e}\eta_{c}+2\nabla_{a}T_{ebd}\nabla^{e}\eta_{c}+2\nabla_{d}T_{eab}\nabla^{e}\eta_{c}-T_{ecf}T^{f}{}_{ab}\nabla^{e}\eta_{d}+T_{ebf}T^{f}{}_{ac}\nabla^{e}\eta_{d}+2\nabla_{a}T_{ebc}\nabla^{e}\eta_{d}$
$\displaystyle-
T_{ecd}T_{fab}\nabla^{f}\eta^{e}+T_{ebd}T_{fac}\nabla^{f}\eta^{e}+2T_{ebc}T_{fad}\nabla^{f}\eta^{e}+2\,T_{ead}T_{fbc}\nabla^{f}\eta^{e}+T_{eac}T_{fbd}\nabla^{f}\eta^{e}-T_{eab}T_{fcd}\nabla^{f}\eta^{e}\,.$

This integrability condition must hold for any vector field obeying the
Killing equation. In the particular case of a maximally symmetric space, we
can make linear combinations of the $\frac{1}{2}n(n+1)$ Killing vectors in
order to construct a Killing vector field such that $\eta_{a}$ has arbitrary
components at some point of the manifold and such that $\nabla_{a}\eta_{b}$ is
an arbitrary skew-symmetric matrix at the same point. In particular, we can
take $\nabla_{a}\eta_{b}=0$ and $\eta_{a}=\delta_{a}^{e}$, with $e$ being an
arbitrary fixed coordinate index. In this case we are led to the following
integrability condition:

$\displaystyle
R_{cdef}\,T^{f}{}_{ab}-R_{cfde}\,T^{f}{}_{ab}-R_{dfce}\,T^{f}{}_{ab}-R_{bdef}\,T^{f}{}_{ac}+R_{bfde}\,T^{f}{}_{ac}+R_{dfbe}\,T^{f}{}_{ac}-2R_{bcef}\,T^{f}{}_{ad}$
$\displaystyle+2R_{bfce}\,T^{f}{}_{ad}-2R_{cfbe}\,T^{f}{}_{ad}-2\,R_{adef}\,T^{f}{}_{bc}-R_{acef}\,T^{f}{}_{bd}-R_{afce}\,T^{f}{}_{bd}-R_{cfae}\,T^{f}{}_{bd}+R_{abef}\,T^{f}{}_{cd}$
(53)
$\displaystyle+R_{afbe}\,T^{f}{}_{cd}+R_{bfae}\,T^{f}{}_{cd}+2\nabla_{a}R_{bcde}+2\nabla_{a}R_{bdce}-2\nabla_{a}R_{cdbe}+2\nabla_{d}R_{abce}-2\nabla_{d}R_{acbe}-2\nabla_{d}R_{bcae}\,=\,0\,.$

Differently, we can take $\eta_{a}=0$ and
$\nabla_{a}\eta_{b}=2\,\delta_{[a}^{h}\delta_{b]}^{k}$ in some arbitrary point
of the manifold with $h$ and $k$ being arbitrary fixed indices. In the latter
case, Eq. (52) lead us to the following constraint that must be satisfied by a
maximally symmetric space:

$\displaystyle
4g_{ck}R_{dabh}+4g_{ch}R_{adbk}+4g_{bk}R_{adch}-4g_{bh}R_{adck}+2g_{ak}R_{bcdh}-2g_{ah}R_{bcdk}+2g_{ak}R_{bdch}-2g_{ah}R_{bdck}$
$\displaystyle-2g_{ak}R_{cdbh}+2g_{ah}R_{cdbk}-g_{ck}T_{fbd}T_{ha}{}^{f}+g_{bk}T_{fcd}T_{ha}{}^{f}+2g_{ck}T_{fad}T_{hb}{}^{f}+g_{ak}T_{fcd}T_{hb}{}^{f}-2g_{bk}T_{fad}T_{hc}{}^{f}$
$\displaystyle-
g_{ak}T_{fbd}T_{hc}{}^{f}-g_{ck}T_{fab}T_{hd}{}^{f}+g_{bk}T_{fac}T_{hd}{}^{f}+g_{ch}T_{fbd}T_{ka}{}^{f}-g_{bh}T_{fcd}T_{ka}{}^{f}-2g_{ch}T_{fad}T_{kb}{}^{f}-g_{ah}T_{fcd}T_{kb}{}^{f}$
(54)
$\displaystyle+2g_{bh}T_{fad}T_{kc}{}^{f}+g_{ah}T_{fbd}T_{kc}{}^{f}+g_{ch}T_{fab}T_{kd}{}^{f}-g_{bh}T_{fac}T_{kd}{}^{f}+2g_{ck}\nabla_{a}T_{hbd}-2g_{bk}\nabla_{a}T_{hcd}-2g_{ch}\nabla_{a}T_{kbd}$
$\displaystyle+2g_{bh}\nabla_{a}T_{kcd}+2g_{ck}\nabla_{d}T_{hab}-2g_{bk}\nabla_{d}T_{hac}-2g_{ak}\nabla_{d}T_{hbc}-2g_{ch}\nabla_{d}T_{kab}+2g_{bh}\nabla_{d}T_{kac}+2g_{ah}\nabla_{d}T_{kbc}$
$\displaystyle+g_{dk}(2R_{abch}-2R_{acbh}-2R_{bcah}+T_{fac}T_{hb}{}^{f}-T_{fab}T_{hc}{}^{f}+2\nabla_{a}T_{hbc})\;\quad$
$\displaystyle+g_{dh}(-2R_{abck}+2R_{acbk}+2R_{bcak}-T_{fac}T_{kb}{}^{f}+T_{fab}T_{kc}{}^{f}-2\nabla_{a}T_{kbc})\,=\,0\,.$

In particular, contracting the latter equation with $g^{bk}$ we find the
following simpler condition which can be proved to be equivalent to (54):

$\displaystyle(n-1)\left(\,4\,R_{adch}\,+\,T_{fcd}\,T_{ha}{}^{f}\,-\,T_{fca}T_{hd}{}^{f}\,-\,2\,T_{fad}\,T_{hc}{}^{f}\,+\,2\,\nabla_{a}T_{hdc}\,-\,2\nabla_{d}T_{hac}\,\right)$
$\displaystyle\,-\,g_{dh}(2\,R_{ac}\,+\,2\,R_{ca}\,-\,T^{f}{}_{a}{}^{k}T_{kcf}\,-\,T^{f}{}_{ac}T^{k}{}_{fk}\,-\,2\nabla_{a}T^{f}{}_{cf})$
$\displaystyle\,+\,g_{ah}(2\,R_{cd}\,+\,2\,R_{dc}\,-\,T^{f}{}_{c}{}^{k}T_{kdf}\,+\,T^{f}{}_{cd}T^{k}{}_{fk}\,-\,2\nabla_{d}T^{f}{}_{cf})$
(55)
$\displaystyle+2\,g_{ch}\left(\,T^{f}{}_{ad}\,T^{k}{}_{fk}+\nabla_{a}T^{f}{}_{df}-\nabla_{d}T^{f}{}_{af}\,\right)\,=\,0\,.$

Therefore, the integrability conditions satisfied by a maximally symmetric
space are (53) and (55). Now, contracting (55) with $g^{ac}$ and using the
curvature properties (8) we conclude that in a maximally symmetric space the
Ricci tensor might be given by:

$\displaystyle 4n\,R_{dh}\,=\,$ $\displaystyle
g_{dh}\left(\,4\,R-T^{abf}T_{baf}+2\,\nabla_{b}T^{a}{}_{a}{}^{b}\,\right)+(n-1)\left(\,T_{adb}T_{h}{}^{ab}-2\,\nabla_{a}T_{hd}{}^{a}\,\right)+2\,\nabla_{a}T^{a}{}_{dh}$
$\displaystyle\,+\,T^{a}{}_{d}{}^{b}T_{bha}+5\,T^{a}{}_{dh}T^{b}{}_{ab}+6\,\nabla_{d}T^{a}{}_{ha}-4\,\nabla_{h}T^{a}{}_{da}\,$
(56)

with $R\equiv g^{ab}R_{ab}$ denoting the Ricci scalar. Then, inserting the
above expression for the Ricci tensor into the integrability condition (55)
lead us to the following expression for the curvature tensor:

$\displaystyle
R_{adch}\,=\,\frac{1}{8n(n-1)}\left\\{(g_{ac}g_{dh}-g_{ah}g_{cd})(8R-2\,T^{bfk}T_{fbk}+4\nabla_{f}T^{b}{}_{b}{}^{f})+4ng_{ch}(2\,\nabla_{[d}T^{b}{}_{a]b}-T^{b}{}_{ad}T^{f}{}_{bf})\right.$
$\displaystyle+g_{dh}\left[(n-1)\left(T_{a}{}^{bf}T_{bcf}\,+\,T_{baf}T_{c}{}^{bf}-2\,T^{b}{}_{a}{}^{f}T_{fcb}-4\nabla_{b}T_{(ca)}{}^{b}\right)\right.$
$\displaystyle\left.\quad\;+2nT^{b}{}_{ca}T^{f}{}_{bf}+2\nabla_{c}T^{b}{}_{ab}+2\nabla_{a}T^{b}{}_{cb}-4n\nabla_{a}T^{b}{}_{cb}\,\right]$
$\displaystyle-
g_{ah}\left[(n-1)\left(T_{bdf}T_{c}{}^{bf}+T_{bcf}T_{d}{}^{bf}-2T^{b}{}_{c}{}^{f}T_{fdb}-4\nabla_{b}T_{(cd)}{}^{b}\right)\right.$
(57) $\displaystyle\left.\quad\;+2n\
T^{b}{}_{cd}T^{f}{}_{bf}+2\nabla_{c}T^{b}{}_{db}+2\nabla_{d}T^{b}{}_{cb}-4n\nabla_{d}T^{b}{}_{cb}\,\right]$
$\displaystyle\left.+2\,n(n-1)\left(2\nabla_{a}T_{hcd}+2\nabla_{d}T_{hac}+2\,T_{bad}T_{hc}{}^{b}-T_{bcd}T_{ha}{}^{b}-T_{bac}T_{hd}{}^{b}\right)\right\\}\,.$

Note that while the left hand side of (57) is skew-symmetric in the indices
$ch$, the right hand side is not automatically skew-symmetric. Therefore,
taking the symmetric part of the equation (57) in the indices $ch$ we are led
to a first order differential equation for the torsion. Particularly,
contracting (57) with $g^{ch}$ we find that

$\nabla_{a}\,T^{e}_{\phantom{e}de}\,-\,\nabla_{d}\,T^{e}_{\phantom{e}ae}\,=\,\,-\,T^{e}_{\phantom{e}ke}\,T^{k}_{\phantom{k}ad}\,.$

It is worth mentioning that the latter constraint is necessary in order to
prove the equivalence between (54) and (55).

Using (7) to write the curvature tensor on the left hand side of (57) in terms
of the Levi-Civita Riemann tensor lead us to a second order differential
equation for the metric with the torsion being the source. Then, once we have
found the metric we can substitute the expression (57) for the curvature
tensor into the integrability condition (53), which yields a second order
differential equation for the torsion. Going through these steps in the case
of general torsion is certainly rather involved. So, probably, the best way to
address these constraints is to make some simplifying assumptions on the
torsion. Particularly, in the special case of a totally skew-symmetric
torsion, it turns out that all these integrability conditions can be solved.
Indeed, in this case using the integrability condition (57) along with (7)
lead us to the following equation:

$T_{abc}\,=\,T_{[abc]}\quad\quad\Longrightarrow\quad\quad\mathring{R}_{adch}\,=\,\frac{\mathring{R}}{n\,(n-1)}\,\left(\,g_{ac}g_{dh}\,-\,g_{ah}g_{dc}\,\right)\,,$
(58)

with $\mathring{R}\equiv g^{ac}g^{bd}\mathring{R}_{abcd}$ denoting the Ricci
scalar of the Levi-Civita curvature. In particular, the second Bianchi
identity implies that $\mathring{R}$ is constant. Moreover, if (58) holds it
turns out that the integrability condition (53) is automatically satisfied.
Therefore, we conclude that the integrability conditions of a maximally
symmetric space, namely Eqs. (53) and (55), impose no constraint over the
torsion tensor if the torsion is totally skew-symmetric. The spaces obeying
Eq. (58) are the well-known maximally symmetric spaces according to the Levi-
Civita connection. For instance, in the case of Euclidean signature the
metrics compatible with (58) can always be written in suitable coordinates
$\\{x^{a}\\}$ as

$g_{ab}\,=\,\frac{1}{(1\,+\,\kappa\,r^{2})^{2}}\,\delta_{ab}\;,\;\textrm{
where }\quad
r^{2}\,=\,\left[\,(x^{1})^{2}\,+\,(x^{2})^{2}\,+\,\cdots\,+\,(x^{n})^{2}\,\right]\,,\;\;\textrm{
and }\;\;\kappa\,=\,\textrm{constant}\,.$ (59)

One can grasp this interesting result regarding maximally symmetric spaces
with respect to connections with totally skew-symmetric torsion by analysing
Eq. (10), according to which

$T_{abc}\,=\,T_{[abc]}\quad\quad\Longrightarrow\quad\quad\nabla_{a}\,\eta_{b}\,+\,\nabla_{b}\,\eta_{a}\,=\,\mathring{\nabla}_{a}\,\eta_{b}\,+\,\mathring{\nabla}_{b}\,\eta_{a}\,.$

Therefore, a vector field $\boldsymbol{\eta}$ obeys the Killing equation for a
connection with totally anti-symmetric torsion if, and only if, it obeys the
Killing equation for the Levi-Civita connection. Thus, if a manifold
$(M,\boldsymbol{g})$ is maximally symmetric according to the Levi-Civita
connection then this manifold is also maximally symmetric with respect to any
connection whose torsion is totally skew-symmetric. Moreover, the Killing
vectors are the same for both connections. For instance, using the coordinates
adopted in (59), the $\frac{1}{2}n(n+1)$ independent Killing vectors in the
Euclidean case are given by

$\boldsymbol{\eta}^{i}\,=\,(1-\kappa\,r^{2})\boldsymbol{\partial}_{i}\,+\,2\,\kappa\,x^{i}\,x^{a}\boldsymbol{\partial}_{a}\quad\textrm{
and
}\quad\boldsymbol{\eta}^{ij}\,=\,x^{i}\,\boldsymbol{\partial}_{j}\,-\,x^{j}\,\boldsymbol{\partial}_{i}\,,$

where $i,j$ are labels running from 1 to $n$ and $i<j$.

###  5.1 Maximal Number of Killing-Yano Tensors of Order $n-1$

One remarkable property of the maximally symmetric manifolds with respect to
the Levi-Civita connection is that, besides admitting the maximal number of
Killing vectors, they also admit the maximal number of hidden symmetries,
namely Killing tensors and Killing-Yano tensors. So, it is natural to wonder
whether this property remains valid for maximally symmetric manifolds with
respect to connections with non-zero torsion. In this section, we shall see
that the answer is negative. More precisely, by means of analysing the simpler
case of a Killing-Yano tensor of order $n-1$, we shall prove that in general a
maximally symmetric space will not admit the maximal number of Killing-Yano
tensors when the torsion is different from zero.

As mentioned before, every Killing-Yano tensor of rank $p$ is the Hodge dual
of a covariantly closed conformal Killing-Yano tensor of rank $(n-p)$ and
vice-versa. Therefore, instead of considering Killing-Yano tensors of order
$n-1$, we shall deal with covariantly closed conformal Killing vectors, i.e.,
vector fields $\boldsymbol{H}$ obeying (25). Taking the covariant derivative
of (25) and then using (3.1) lead us to:

$\nabla_{a}\nabla_{b}\,H_{c}\,=\,2\,g_{bc}\,\nabla_{a}h\,=\,\frac{-\,1}{n-1}\,g_{bc}\,\left(\,R_{a}^{\phantom{a}e}\,H_{e}\,-\,2\,h\,T^{e}_{\phantom{e}ea}\right)\,.$
(60)

Analogously, differentiating the latter equation and using (3.1) one find that
all derivatives of $\boldsymbol{H}$ can be written in terms of
$\boldsymbol{H}$ and $h=\frac{1}{2n}\nabla^{a}H_{a}$. Thus, if we know the
values of $H_{a}$ and $\nabla^{a}H_{a}$ at some point of the manifold then we
can obtain $H_{a}$ throughout the whole manifold. Therefore, the maximal
number of independent covariantly closed conformal Killing vectors in an
$n$-dimensional manifold is $(n+1)$. In addition, if a manifold admits the
maximal number of CCCKVs then it is always possible to find a CCCKV
$\boldsymbol{H}$ such that $H_{a}$ and $\nabla^{a}H_{a}$ have any desired
value at some arbitrary point of the manifold. Hence, integrability condition
(29) implies that if a manifold admits the maximal number of covariantly
closed conformal Killing vectors then the following constraints hold:

$\left\\{\begin{array}[]{cl}(n-1)R_{abc}^{\phantom{abc}e}\,-\,g_{ac}\,R_{b}^{\phantom{b}e}\,+\,g_{bc}\,R_{a}^{\phantom{a}e}\,=\,0\\\
(n-1)\,T_{cab}\,-\,g_{ac}\,T^{e}_{\phantom{e}eb}\,+\,g_{bc}\,T^{e}_{\phantom{e}ea}\,=\,0\,.\\\
\end{array}\right.$ (61)

Particularly, the second of these constraints implies that the torsion has the
following form

$T_{cab}\,=\,\frac{2}{n-1}\,T^{e}_{\phantom{e}e[b}g_{a]c}\,.$ (62)

Note, in particular, that the totally skew-symmetric part of the torsion must
be identically zero. This fact contrasts with what we have seen about spaces
admitting the maximum number of Killing vectors. While the existence of the
maximum number of CCCKVs implies that $T_{[abc]}$ must vanish, the existence
of the maximum number of Killing vectors imposes no constraint over the
torsion when it is totally antisymmetric.

Now, working out the first integrability condition in (61) lead us to the
following expression for the curvature tensor:

$R_{abcd}\,=\,\frac{R}{n(n-1)}\,\left(\,g_{ac}\,g_{bd}\,-\,g_{ad}\,g_{bc}\,\right)\,.$
(63)

Then, using (7) to rewrite the curvature in the above expression in terms of
the Levi-Civita curvature and then using (62), eventually lead us to the
following expression for the Levi-Civita curvature:

$\mathring{R}_{abcd}\,=\,\frac{\mathring{R}\,+\,2\,\nabla_{e}T^{ke}_{\phantom{ke}k}}{n(n-1)}\,\left(\,g_{ac}\,g_{bd}\,-\,g_{ad}\,g_{bc}\,\right)\,+\,\frac{2}{n-1}\left(\,g_{d[a}\nabla_{b]}T^{e}_{\phantom{e}ce}\,-\,g_{c[a}\nabla_{b]}T^{e}_{\phantom{e}de}\,\right)\,.$
(64)

The above condition provides a second order differential equation for the
metric, with the torsion being the source. Then, once we have found the
metric, Eq. (62) yields a first order differential equation for the torsion.
In addition, note that atisymmetrizing the indices $abc$ in (64) and using the
first Bianchi identity satisfied by the Levi-Civita Riemann tensor yields the
constraint $g_{d[a}\nabla_{b}T^{e}_{\phantom{e}c]e}=0$, which is equivalent to
$\nabla_{[a}T^{e}_{\phantom{e}b]e}=0$ if $n>2$.

##  6 Conclusions

In the present article the integrability conditions for the existence of
Killing-Yano tensors, of arbitrary order, with respect to general metric-
compatible connections have been obtained. Thus, extending the results of
Refs. [24, 25] to the case of non-zero torsion. As we have seen, the case of
totally skew-symmetric torsion is of special appeal, since in such a case the
concepts of geodesic and Killing vector are not ambiguous. Moreover, in
superstring theory the torsion is associated to the field strength of the
Kalb-Ramond field, so that it is automatically antisymmetric. In spite of such
motivations, for sake of completeness, here no restriction over the torsion
has been assumed. Once the curvature of the connection is known, the algebraic
integrability conditions found here can be used to eliminate several
components of the Killing-Yano tensor, facilitating the integration of the
Killing-Yano equation. Indeed, this kind of procedure have been successfully
applied in Ref. [23] for the torsion-free case. As an application of the
results obtained here, in Sec. 4 all metrics and torsions compatible with the
existence of a Killing-Yano tensor of order $n-1$ have been found.

It is worth pointing out the important role played by KY tensors in quantum
field theory in curved spacetimes. Although symmetric Killing tensors generate
conserved charges at the classical level, it turns out that at the quantum
level an anomaly term involving the curvature pops up. So, generally, Killing
tensors do not yield a conservation law in the quantum theory [29].
Differently, in the absence of torsion, KY tensors yield conserved charges for
the Klein-Gordon and Dirac equations both at classical and quantum levels [44,
29]. Nevertheless, in Ref. [27] it has been proved that a skew-symmetric
torsion generally spoils this property of KY tensors, namely in the quantum
theory an anomaly involving the torsion arises. Actually, an anomaly due to
the torsion shows up even at the semi-classical treatment of a spinning
particle [27].

Here, the issue of defining a maximally symmetric space in the presence of
torsion has also been addressed. Differently from the previous works on this
topic, in the present article we have not assumed that the torsion is
invariant by the isometries of the space, making the approach adopted here
more general. It has been obtained the restrictions that the curvature and the
torsion must obey in order for the space to admit the maximum number of vector
fields obeying the Killing equation with respect to a general metric-
compatible connection. Particularly, it has been proved that in the case of a
totally skew-symmetric torsion the metric of a maximally symmetric space must
be the same metric of the torsion-free case and that no restriction is imposed
over the torsion. Moreover, we have shown that, contrary to the torsion-less
case, a maximally symmetric space in the presence of torsion generally does
not admit the maximum number of Killing-Yano tensors. Hopefully, these results
regarding maximally symmetric spaces can be valuable for the study of
cosmological models in the presence of torsion.

## Acknowledgments

I really want to thank Tsuyoshi Houri for suggesting the theme of this article
as well as for valuable discussions. I also thank the Brazilian funding agency
CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) for part
of the financial support.

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