# Fat Tailed Factors

Jan Rosenzweig111Pine Tree Funds, 107 Cheapside, London EC2V 6DN,
<EMAIL_ADDRESS>King’s College London, Strand, London WC2R
2LS<EMAIL_ADDRESS>
###### Abstract

Standard, PCA-based factor analysis suffers from a number of well known
problems due to the random nature of pairwise correlations of asset returns.
We analyse an alternative based on ICA, where factors are identified based on
their non-Gaussianity, instead of their variance.
Generalizations of portfolio construction to the ICA framework leads to two
semi-optimal portfolio construction methods: a fat-tailed portfolio, which
maximises return per unit of non-Gaussianity, and the hybrid portfolio, which
asymptotically reduces variance and non-Gaussianity in parallel.
For fat-tailed portfolios, the portfolio weights scale like performance to the
power of $1/3$, as opposed to linear scaling of Kelly portfolios; such
portfolio construction significantly reduces portfolio concentration, and the
winner-takes-all problem inherent in Kelly portfolios.
For hybrid portfolios, the variance is diversified at the same rate as Kelly
PCA-based portfolios, but excess kurtosis is diversified much faster than in
Kelly, at the rate of $n^{-2}$ compared to Kelly portfolios’ $n^{-1}$ for
increasing number of components $n$.

Key words: optimal portfolios, ICA, PCA, fat-tailed risk

Key messages:

  * •

tail-risk-based portfolios can address a number of know problems of
covariance-based portfolios

  * •

when portfolios are optimized for tail risk, weights of components scale sub-
linearly with their performance, reducing portfolio concentration and the
winner-takes-all problem

  * •

portfolios optimized for tail risk manage variance as efficiently as
covariance-based portfolios.

Word count: 3,996

Number of figures: 4

Number of tables: 0

##  1 Introduction

Analysis of portfolio returns is usually performed with reference to the
relationship between the portfolio’s returns vector and its covariance matrix,
a practice stemming back from Markowitz’s Modern Portfolio Theory in the 1950s
[13, 14].

This, in turn, is significantly aided by the use of the Principal Component
Analysis (PCA) [8]. Fast algorithms for extraction of principal components are
routinely used to partially invert covariance matrices and construct
approximate optimal portfolios.

PCA itself has an equally long history in finance, stemming back from the work
of Tobin [21], Lintner [9] and Sharpe [19] in the 1950s and 1960s in the
context of stock returns. It is also routiney used for simplification of the
dynamics of muti-asset complexes such as yield curves [10] and volatility
surfaces [2].

Covariance based methods, however, suffer from a number of well known
problems. The main problem is that correlations of asset returns are not
particularly robust, and they may even arise randomly [1]. This is in part due
to the relatively small number of data points used to calculate correlations,
leading to a wide confidence interval with the possibility of significant
errors. There have been a number of approaches aimed at alleviating that
problem (see [1, 20] and references therein).

We take a somewhat different approach, based on the Independent Component
Analysis (ICA) [23]. In signal processing, ICA is a computational method for
separating a multivariate signal into additive subcomponents. This is done by
assuming that the subcomponents are non-Gaussian signals and that they are
statistically independent from each other. ICA is a special case of blind
source separation. A common example application is the ”cocktail party
problem” of listening in on one person’s speech in a noisy room [7].

ICA is a robust method widely used in engineering signal processing
applications ranging from acoustic signal processing [5], visual processing
[15] and telecommunications [17], to electroencephalography [22] and many
others.

Practically, ICA extracts factors based on their non-Gaussianity, rather than
on their variance, as PCA. Traditionally, the chosen measure of non-
Gaussianity was excess kurtosis. Other measures of non-Gaussianity have also
been popular, such as various empirical approximations to negentropy [6].

ICA has been sporadically used in finance, starting with the pioneering work
of Back [3] in the 1990s, and proceeding to this date [16, 18, 12, 11, 4].

This paper is set out as follows. Section 2 outlines the the main differences
between PCA and ICA, and sets out the notation. Section 3 is dedicated to Fat-
Tailed Portfolios, a generalization of Kelly portfolios to the non-Gaussianity
framework. Section 4 analyses Hybrid Portfolios, a class of kurtosis- and
variance- minimizing portfolios. Section 5 looks at a practical analysis of
S&P500 constituents over a 12 year period, and Section 6 is the Discussion.

##  2 PCA vs ICA

Let $S^{(1)}_{t}..S^{(N)}_{t}$ denote prices of $N$ assets at time $t$,
forming a vector of asset prices ${\bf S}_{t}$. Let ${\bf w}^{(i)}$ be
portfolio weight $N$-vectors, and ${\Pi}^{(i)}$ be the resulting portfolios,

$\Pi_{t}^{(i)}={\bf w}^{(i)}.{\bf S}_{t}$ (1)

with portfolio variances $\sigma^{(i)}$ and kurtoses $\kappa^{(i)}$.

Both Principal Component Decomposition (PCA) and Independent Component
Decomposition (ICA) are decompositions of the form (1), and they can both be
obtained iteratively.

For the $i$th component of the PCA, the unit weight vector ${\bf w}^{(i)}$ is
selected so as to maximise the variance $\sigma^{(i)}$, the residues are
projected to the hyperplane orthogonal to the resulting component $\Pi^{(i)}$,
and the iteration then proceeds to the component $i+1$ for the residues.

Note that the resulting principal components are generally long-short, and
that we make no assumptions on positivity of weights, or them adding up to a
positive number. They are only normalized to unit length.

For the $i$th component of the ICA, the process is analogous, except that the
weight vector ${\bf w}^{(i)}$ is selected so as to maximise the kurtosis
$\kappa^{(i)}$ instead of the variance $\sigma^{(i)}$. Also, by convention,
the weight vectors are normalised differently in PCA and ICA. While PCA weight
vectors are normalised to the same portfolio weight (typically unity), ICA
weights are normalised to the same variance (typically the variance of the
first component, whose weight vector is normalised to unity).

The principal drawback of PCA is that it assumes that the underlying
distribution is multivariate Gaussian; specifically, that each marginal is
Gaussian, fully specified by its mean and variance, and that the marginals
interact through a Gaussian copula, with their codependence fully specified by
pairwise correlations. Problems arise when these assumptions are violated, as
illustrated by a simple example of a multivatriate stochastic volatility
process with local correlations; for such a process, any measured volatilities
and pair correlations are only random snapshots of a more complicated process,
and using them to extract PCs leads to components that are only valid
instantaneously [1, 20]. In contrast, ICA makes no such assumptions, and it is
therefore more robust when those assumptions are broken.

In further text, we denote the $i$th principal component as $PC^{(i)}$, and
the $i$th independent component as $IC^{(i)}$. The respective means,
volatilities and kurtoses of the principal and independent components are
denoted
$\mu_{PCA}^{(i)},\mu_{ICA}^{(i)},\sigma_{PCA}^{(i)},\sigma_{ICA}^{(i)},\kappa_{PCA}^{(i)}$
and $\kappa_{ICA}^{(i)}$, respectively.

##  3 Fat Tailed Portfolios

We denote by

${\bf m}=E\left({\bf S}\right),$ ${\bf V}=E\left({\bf S}\otimes{\bf
S}\right)-{\bf m}^{2}$ ${\bf K}=E\left(({\bf S}-{\bf m})\otimes({\bf S}-{\bf
m})\otimes({\bf S}-{\bf m})\otimes({\bf S}-{\bf m})\right)-3{\bf V}\otimes{\bf
V}$

the return, covariance and excess cokurtosis of the joint distribution of the
asset process ${\bf S}$.

The Kelly criterion for portfolio construction is easily obtained by looking
for the weights that maximize the portfolio return, while penalising for
portfolio variance:

${\bf w}=\underset{\bf w}{\operatorname{arg}\,\operatorname{max}}\;\left({\bf
w}.{\bf m}-\lambda\ {\bf w}.{\bf V}.{\bf w}\right)$ (2)

for some risk aversion parameter $\lambda$.

Differentiating the right hand side of (2) wrt ${\bf w}$ and setting the
result to zero yieds the familiar Kelly criterion

${\bf w}\propto{\bf V}^{-1}.{\bf m}$ (3)

where the constant of proportionaity depends on the investor’s risk aversion.
Equation (3) is easily solved using PCA; noting that each eigenvector of ${\bf
V}$ corresponds to the weights vector of a principal component, and that the
corresponding eigenvalue corresponds to its variance, we can write the optimal
portfoio weights in terms of PCs as

$w_{PCA}^{(i)}\propto\frac{\mu_{PCA}^{(i)}}{{\sigma_{PCA}^{(i)}}^{2}}$ (4)

The optimal portfolio is given as

$\Pi\propto\sum_{i}w_{PCA}^{(i)}PC^{(i)}=\sum_{i}\frac{\mu_{PCA}^{(i)}}{{\sigma_{PCA}^{(i)}}^{2}}PC^{(i)}$
(5)

where the sum is typically truncated to a small number of principal
components.

The optimization (2) is easily generalised if we are penalising for kurtosis
instead of variance; the equivalent of (2) is

${\bf w}=\underset{\bf w}{\operatorname{arg}\,\operatorname{max}}\;\left({\bf
w}.{\bf m}-\nu\ {\bf w}.{\bf w}.{\bf K}.{\bf w}.{\bf w}\right),$ (6)

where $\nu$ is a different risk aversion parameter, aversion to kurtosis risk.
This leads to portfolio weights

$w_{ICA}^{(i)}\propto\left(\frac{\mu_{ICA}^{(i)}}{\kappa_{ICA}^{(i)}}\right)^{1/3}$
(7)

and the optimal portfolio is

$\Pi\propto\sum_{i}w_{ICA}^{(i)}IC^{(i)}=\sum_{i}\left(\frac{\mu_{ICA}^{(i)}}{\kappa_{ICA}^{(i)}}\right)^{1/3}IC^{(i)},$
(8)

which can again practically be truncated to a small number of independent
components. We call the portfolio (8) the Fat-Tailed Portfolio.

The Fat-tailed portfolio (8) has several interesting properties; in
particular, the portfolio weight no longer scales linearly with the
performance of a component, as it does in a Kelly portfolio, but as its cubed
root. Practically, this leads to more diversified portfolios with a less
pronounced winner-takes-all profile than that dictated by Kelly.

In particular, let us imagine two components with same variances and kurtoses,
but the first one has twice the return of the second one; Kelly dictates that
the first component would get twice the leverage of the second, while the fat
tailed portfolio would only allocate it $2^{1/3}\approx 125.9\%$ of the
leverage of the second component.

It is, however, important to note that the fat-tailed portfolio (8) only
maximizes the return per unit of kurtosis; it does not specifically target low
variance, and its variance is not guaranteed to be low. Even the independence
of the ICs does not necessarily lead to variance decay through
diversification. Diversification, in the sense of reducing riskiness of the
portfolio by including more components, would require summation of components
of equal variance; the ICs themselves have equal variances by construction,
but in the portfolio they are weigthed by the cubed root of the ratio of their
mean to kurtosis. Diversification is only likely if this ratio is relatively
stable across the ICs. On the other hand, the cubed root scaling serves to
flatten the portfolio weights, so the requirement that they are stable is
easier to achieve in practice, as will be shown in Sections 4 and 5.

In the absence of such diversification, the fat-tailed portfolio only moves
the risk from the fourth moment to the second; it pushes the risks from the
tails of the distribution towards its centre, but it does not eliminate them
from the centre.

##  4 Hybrid Portfolios

We now turn to the construction of portfolios which manage both variance and
kurtosis.

In analogy to (2) and (6), one could naively look for portfolio weights by
solving the combined optimization problem

${\bf w}=\underset{\bf w}{\operatorname{arg}\,\operatorname{max}}\;\left({\bf
w}.{\bf m}-\lambda\ {\bf w}.{\bf V}.{\bf w}-\nu\ {\bf w}.{\bf w}.{\bf K}.{\bf
w}.{\bf w}\right),$ (9)

where we now have two risk aversion parameters; aversion to variance,
$\lambda$, and aversion to kurtosis, $\nu$.

This is, however, not ideal. The Kelly portfolio (5) and the Fat-tailed
portfolio (8) have a certain measure of universality; the shape of the
portfolio is fixed, and the investor risk aversion only affects the leverage.
With (9), this would no longer be the case. The shape of the portfolio would
now depend on the ratio of variance aversion to kurtosis aversion, and it woud
be different from investor to investor.

We are, on the other hand, interested in finding universal portfolio shapes
that control both variance and kurtosis, while being independent of the
investor risk preferences. In other words, the investor risk preferences
should only affect the leverage, but not the shape of the portfolio.

The solution comes from the Central Limit Theorem, which loosely states that
the sum of independent random variables, de-meaned and normalized to unit
volatility, tends to a normal distribution. In particular, its excess kurtosis
vanishes due to the Central Limit Theorem, and its variance decays through
diversification.

We therefore look at portfolios of ICs that satisfy the conditions of the
Central Limit Theorem. For this, we have the following result:

######  Theorem (Central Limit Theorem for ICA).

Consider two portfolios,

$\Pi_{PC}=\frac{1}{n}\sum_{i}^{n}\frac{PC^{(i)}}{\sigma_{PCA}^{(i)}},$
$\Pi_{IC}=\frac{1}{n}\sum_{i}^{n}\frac{IC^{(i)}}{\sigma_{ICA}^{(i)}}.$

Then, as $n\rightarrow\infty$, the variance of the returns of both portfolios
is

$O\left(\frac{1}{n}\right);$

the excess kurtosis of the returns of $\Pi_{PC}$ is

$o\left(\frac{1}{n}\right),$

and the excess kurtosis of the returns pf $\Pi_{IC}$ is

$o\left(\frac{1}{n^{2}}\right).$

The proof of the CLT for ICA is given in the Appendix. Note that there is
formally no need to normalise the components of $\Pi_{IC}$ by
$\sigma_{ICA}^{(i)}$ in constructing the IC portfolio, as we do in the
statement of the CLT, since volatilities of ICs are already all equal by
construction. We are only normalising them so that we would be able to compare
them directly to PCA portfolios in the numerical example in the next Section.

This form of the CLT tells us that portfolios of ICs can be equally good as
portfolios of PCs in suppressing the variance, but they have the added bonus
of suppressing the non-Gaussianity of the returns much faster.

##  5 S&P500 stocks

We looked at S&P500 stocks over a period of 12 years, from the 1st January
2007 until the 31st December 2018. To counteract the effects of stocks
drifting in and out of the index over such a long time frame, we have divided
the time frame into four buckets, each lasting three calendar years; from 1st
January 2007 until 31st December 2009, from 1st January 2010 until 31st
December 2012, from 1st January 2013 until 31st December 2015 and from 1st
January 2016 until 31st December 2018. The basket for each bucket was selected
as consisting of the index constituents on the last business day prior to the
start of the bucket, and these stocks were followed until the end of the
bucket. Any stock that was de-listed before the end of a bucket in which it
appeared was deemed to have returned $0\%$ from its last trading day until the
end of the bucket. There were no adjustments for stocks entering or leaving
the index over the duration of any of the buckets.

We have performed PCA and ICA on each of the buckets, extracting 10 principal
components and 10 independent components for each bucket. Decompositions used
the Python package scikit-learn 0.23.2, and in particuar the classes
sklearn.decomposition.PCA for PCA, and sklearn.decomposition.fastICA for ICA.

Resulting PCs and ICs are plotted in Figure 1. Qualitatively, the PCs appear
to be significantly more dispersed than the ICs; PC1 is immediately visually
identifiable as an outlier on each PCA graph, while IC1 is not an obvious
outlier in any of the buckets.

Additionally, PCs vary on the order $\pm 10^{3}$ in all buckets, and $10^{2}$
in the 2010-2012 bucket, while the ICs vary on the order 1 in every bucket.
This is not an issue of scaling; first ten PCs have higher variance than the
first ten ICs, but first ten ICs have higher kurtosis than the first ten PCs.
This counters any argument that the PCs and ICs are differently scaled; it is
impossible to scale them so that their variances and kurtoses are of the same
order of magnitude simultaneously, due to the fact that variance scales
$\propto w^{2}$ and kurtosis scales $\propto w^{4}$.

The correlation matrices between the PCs and the ICs in each bucket are shown
in Figure 2. As expected, the PCs are all orthogonal to each other, as are the
ICs. But the PCs are generally not orthogonal to the ICs, and the correlation
matrix has a block structure, with the PC-IC block generally non-zero. The
block structure of the PC-IC correlation matrix is significant insofar as it
illustrates that ICs are not just PCs by another name. Even though the top PCs
and the top ICs span roughly the same space, they are not the same and the
transformation from PCs to ICs is non-trivial. By extension, this is also a
test of non-Gaussianity; if the underlying processes were generated by a
multivariate Gaussian variable, the PCs and the ICs would coincide exactly;
kurtosis of a Gaussian variable is $\propto\sigma^{4}$, so a component
selected for the highest variance (PC) would also be selected for the highest
kurtosis (IC). This is clearly not the case for S&P500 stocks.

We have constructed the portfolios $\Pi_{PC}$ and $\Pi_{IC}$ from the Central
Limit Theorem from the increasing number of components $n$, starting from a
single component and ending with all $n=10$ components. The variances and
kurtoses of the resulting portfolios are shown in Figure 3. Each component was
scaled to unit variance, so the variance of each portfolo is exactly $1/n$, to
machine prexision. The kurtosis illustrates the difference between the
portfolios; kurtosis of $\Pi_{IC}$ decays faster than the variance with
increasing $n$, while the kurtosis of $\Pi_{PC}$ generally does not. This is
consistent with the respective $n^{-1}$ and $n^{-2}$ scaling for the kurtoses
of $\Pi_{PC}$ and $\Pi_{IC}$ as predicted by the Central Limit Theorem.

Finally, we have constructed the optimal Kelly portfolio from the ten PCs, and
the optimal Fat-tailed portfolio from the ten ICs, scaled to the same
portfolio volatility of $10\%$ in all cases. The resulting portfolios are
shown in Figure 4. The correlation between the Kelly portfolio and the Fat-
tailed portfolio is greater than $90\%$ for each bucket, and it is as high as
$97.247\%$ in the 2013-2015 bucket. Given that the total number of components
is 10, correlation over $90\%$ implies that both Kelly and Fat-tailed
portfolios capture the same factors and the same performance over each bucket.
This is indeed the case in our portfolios.

Unsurprisingly, due to their construction, the Kelly portfolio always has
lower volatility and higher Sharpe ratio, while the Fat-tailed portfolio
always has lower kurtosis and higher Fat-tailed ratio
$\left(\mu/\kappa\right)^{1/3}$. The differences between the Sharpe ratios of
the Kelly and Fat-tailed portfolios are always below $10\%$. Given such small
differences for a posteriori portfolio construction with perfect hindsight, in
real-world, a priori portfolio construction without perfect hindsight, the
differences in Sharpe ratios are unlikely to be observable to some reasonable
level of confidence.

The differences between the kurtoses and Fat-tailed ratios between the two
portfolio classes vary more widely, ranging from $10\%$ difference in
2013-2015 (0.49 vs 0.54) to $60\%$ difference in 2010-2012 (0.766 vs 1.226).
Some of these differences would arguably survive into practical a priori
portfolio construction, where they would be felt.

We can see an indirect confirmation of this by looking at the maximum drawdown
of the selected portfolios, as an independent benchmark of the success of the
two portfolio construction methods. The Kelly portfolio has smaller maximum
drawdown than the Fat-tailed portfoilio in the 2007-2009 bucket (6.5% vs
6.7%), larger in the 2010-2012 and 2013-2015 buckets (7.7% vs 7.3% and 9.2% vs
8.4%), and the drawdowns are indistinguishable in the 2016-2018 bucket (12.8%
vs 12.8%). While far from conclusive, this supports the notion that fat-tailed
portfolio construction keeps a better handle on the overall portfolio risks
per unit of volatility.

##  6 Discussion

Non-Gaussianity-based factors such as those described in this paper are an
interesting alternative to standard PCA-based factors. In enginering
applications ranging from acoustic signal processing [5], visual processing
[15], telecomunications [17], electroencephalography [22] and many others, ICA
is generally considered to be a considerably more powerful tool than PCA, and
it is a de facto standard in a wide range of applications.

This has historically not been the case in finance, however, where PCA based
methods have dominated since Markowitz’ modern portfolio theory [13] [14], and
only a handful of studies have even seriously looked at ICA [3, 16, 18, 12,
11, 4].

While PCA and ICA are conceptually similar, their stated purpose is different.
The purpose of PCA is to isolate the strongest signals (those with highest
variance), while the purpose of ICA is to isolate the noisiest signals (those
with highest deviation from Gaussianity).

This leads to a different distribution of factors, as illustrated in our
S&P500 example. The first IC is nowhere near as dominant as the first PC,
which directly leads to the slower, $1/3$ scaling in the fat-tailed portfolio
construction. This, in turn, reduces portfolio concentration and generates
smoother, more diversified portfolios. As illustrated in our example, fat-
tailed portfolios end up capturing the same factors as Kelly portfolios, but
they manage the fat-tailed risks better due to their $1/3$ scaling, as opposed
to linear scaling of the Kelly portfolios.

Despite this lesser dispersion of ICs when compared to PCs, the Central Limit
Theorem for ICs shows that they are nonetheless good diversifiers; in many
respects, they are better diversifiers than PCs, in the sense that they
diversify variance equally well, but they also diversify kurtosis much faster.
The numerical results in Figure 3 confirm this in our specific example.

There are two main advantages of our Fat-taied portfolio construction over
Kelly portfolios; one based on the properties of the ICA as opposed to the
PCA, and the other based on the $1/3$ scaling of the weights.

The key advantage of the ICA is that makes no assumptions about Gaussianity of
the underlying porcesses, and hence it does not automatically break down when
such assumptions are not met. We can therefore expect ICA-based components to
be more robust than PCA-based components.

The key advantage of the $1/3$ scaling of the portfolio weight is that it is
sub-linear, and that the portfolio weight changes more slowly with increasing
$\mu$ and decreasing $\kappa$. Therefore, errors in the estimation of these
quantities will have a smaller effect on portfolio weights, and tus on the
overall shape of the portfolio.

ICA does not solve all known problems of PCA. The problem of low confidence in
asset correlations is not fixed by moving to higher order codependence; it
arguably becomes worse, due to the same number of data points being used to
infer a higher-order statistic. On the other hand, as addresed in the previous
paragraph, the $1/3$ scaling of portfolio weights means that the portfolio
construction is less sensitive to the exact estimation of codependence
numbers.

##  7 Conclusions

We have proposed a new, ICA-based Fat-tailed portfolio construction method.
The method rests on identifying the statistically independent components
through ICA, and then weighting such components in proportion to the ratio of
their return to kurtosis, raised to the power of $1/3$.

This method is shown to have the following advantages over covariance-based
Kelly portfolios:

  * •

ICA is more robust than PCA in the face of non-Gaussianity of the underlying
processes

  * •

the $1/3$ portfolio scaling is more robust than the linear scaling in the face
of uncertainties in parameter estimation

  * •

the resulting portfolios maintain better control of tail risks

These advantages do not come at a cost in terms of increased computational or
operational complexity; ICA calculation is available as part of numerous
publicly available numerical packages, and the only additional calculation is
the calculation of the kurtosis of each component, which is equally
straightforward.

In conclusion, ICA based factor analysis and portfolio construction methods
are a powerful alternative to the existing array of methods, and they address
a number of known concerns with covariance-based methods.

This is only the first step in the analysis of such portfolio construction
methods. Anticipated further work includes looking at penalty functions other
than kurtosis, and looking at different versions of ICA which generate
statistical independence to orders higher than 4.

The ultimate goal is to develop a robust portfolio cosntruction and risk
framework which controls for all portfolio risks, as opposed to only those
that are measured by covariance matrices.

## Appendix: Proof of the Central Limit Theorem for ICA

The CLT for $\Pi_{PC}$ is the standard CLT for any sum of de-meaned,
normalized orthogonal random variables. We denote

$Y_{i}=\frac{dPC^{(i)}-\mu_{PCA}^{(i)}}{\sigma_{PCA}^{(i)}}$ (10)

where $dPC^{(i)}$ denotes the returns process of $PC^{(i)}$, so that all
$Y_{i}$ all have zero mean and unit variance, and their normalised sums as

$Z_{n}=\frac{1}{\sqrt{n}}\sum_{i}^{n}Y_{i}$ (11)

Given that $Y_{i}$ are all orthogonal, the characteristic function of $Z_{n}$
is

$\displaystyle\varphi_{Z_{n}}(t)$
$\displaystyle=\varphi_{\frac{1}{\sqrt{n}}\sum_{i}^{n}Y_{i}}(t)$ (12)
$\displaystyle=\varphi_{Y_{1}}\left(\frac{t}{\sqrt{n}}\right)\varphi_{Y_{1}}\left(\frac{t}{\sqrt{n}}\right)...\
\varphi_{Y_{n}}\left(\frac{t}{\sqrt{n}}\right)+o\left(\frac{t^{2}}{n}\right)$
(13)
$\displaystyle=\left[\varphi_{Y_{1}}\left(\frac{t}{\sqrt{n}}\right)\right]^{n}+o\left(\frac{t^{2}}{n}\right)$
(14)

as $t^{2}/n\rightarrow 0$.

The error term $o(t^{2}/n)$ arises due to the fact that we can not guarantee
that joint moments beyond order 2 are zero. The construction of PCs only
guarantees their orthogonality, but it does not guarantee either their
Gaussianity, nor their independence at orders higher than 2.

The second order Taylor expansion of $\varphi_{Y_{1}}$ around zero gives

$\varphi_{Y_{1}}=\left(1-\frac{t^{2}}{2n}+o\left(\frac{t^{2}}{n}\right)\right),$
(15)

so

$\varphi_{Z_{n}}=\left(1-\frac{t^{2}}{2n}+o\left(\frac{t^{2}}{n}\right)\right)^{n}+o\left(\frac{t^{2}}{n}\right).$
(16)

Expanding the known terms in (16) to fourth order, we get

$\varphi_{Z_{n}}=1-n\frac{t^{2}}{2n}+\frac{n(n-1)}{2}\frac{t^{4}}{4n^{2}}+o\left(\frac{t^{2}}{n}\right)$
(17)

where the term $o(t^{2}/n)$ may contain additional contributions at the order
$t^{4}$, so the best we can say about the error at $t^{4}$ is that it is
$o(1/n)$.

In particular, the second and fourth derivatives of $\varphi_{Z_{n}}$ at zero
are

$\displaystyle\varphi_{Z_{n}}^{\prime\prime}$ $\displaystyle=1$ (18)
$\displaystyle\varphi_{Z_{n}}^{iv}$
$\displaystyle=\frac{n(n-1)}{n^{2}}3+o\left(\frac{1}{n}\right),$ (19)

hence the excess kurtosis of $Z_{n}$ is $o(1/n)$. Given that the returns of
$\Pi_{PC}$ differ from $Z_{n}/\sqrt{n}$ by a deterministic mean process, the
excess kurtosis of the returns of $\Pi_{PC}$ is also $o(1/n)$.

Moving to $\Pi_{IC}$, the difference is that the components are now, by
construction, independent to the fourth order, instead of to second order for
$\Pi_{PC}$. Therefore, denoting

$y_{i}=\frac{dIC^{(i)}-\mu_{ICA}^{(i)}}{\sigma_{ICA}^{(i)}},$ (20)

where $dIC^{(i)}$ denotes the returns process of $IC^{(i)}$

$z_{n}=\frac{1}{\sqrt{n}}\sum_{i}^{n}y_{i},$ (21)

equations (15) and (16) read

$\varphi_{y_{1}}=\left(1-\frac{t^{2}}{2n}+o\left(\frac{t^{4}}{n^{2}}\right)\right),$
(22)
$\varphi_{z_{n}}=\left(1-\frac{t^{2}}{2n}+o\left(\frac{t^{4}}{n^{2}}\right)\right)^{n}+o\left(\frac{t^{4}}{n^{2}}\right),$
(23)

with the error at $t^{4}$ now of the order $o(1/n^{2})$.

The second and fourth derivatives become

$\displaystyle\varphi_{Z_{n}}^{\prime\prime}$ $\displaystyle=1$ (24)
$\displaystyle\varphi_{Z_{n}}^{iv}$
$\displaystyle=\frac{n(n-1)}{n^{2}}3+o\left(\frac{1}{n^{2}}\right),$ (25)

and the excess kurtoses of $z_{n}$ and the returns of $\Pi_{IC}$ are
$o(1/n^{2})$.

## Acknowledgments

The author reports no conflicts of interest. The author alone is responsible
for the content and writing of the paper.

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Figure 1: First ten PCs and first ten ICs, for each of the 2007-2009,
2010-2012, 2013-2015 and 2016-2018 buckets.

Figure 2: Correlations between the PCs and ICs for the 2007-2009 bucket (top),
2010-2012 (second from the top), 2013-2015 (second from the bottom) and
2016-2019 (bottom) buckets.

Figure 3: Kurtosis and kurtosis/variance ratio of equal volatility baskets
with increasing portfolio size $n$, PC portfolios vs IC portfolios.

Figure 4: Kelly portfolio vs Fat-tailed portfolio for each bucket; Fat-tailed
Ratio refers to the quantity $\left(\mu/\kappa\right)^{1/3}$, and MaxDD is the
maximum drawdown

.


# Killing-Yano Tensors of Order $n-1$

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

The properties of a Killing-Yano tensor of order $n-1$ in an $n$-dimensional
manifold are investigated. The integrability conditions are worked out and all
metrics admitting a Killing-Yano tensor of order $n-1$ are found. It is
pointed out a connection between such tensors and a generalization of the
concept of angular momentum. A theorem on how to generate closed conformal
Killing vectors using the symmetries of a manifold is proved and used to find
all Killing-Yano tensors of order $n-1$ of a maximally symmetric space.
(Keywords: Killing-Yano tensors, Killing tensors, Conformal Killing vectors,
General relativity, Angular momentum, Maximally symmetric spaces)

##  1 Introduction

It is hard to overemphasize the relevance of conservation laws to the
development of Physics and Mathematics. Indeed, the existence conserved
quantities are of fundamental importance in the integration of equations of
motion and differential equations in general, specially when these equations
are non-linear. Particularly, in General Relativity it is well-known that the
Killing vectors can be used to construct scalars that are conserved along
geodesic curves. For instance, the $4$-dimensional Schwarzschild spacetime
admits four independent Killing vectors, so that the geodesics in this
manifold can be found without solving the non-linear geodesic equation.
Nevertheless, in the case of the $4$-dimensional Kerr spacetime the issue of
finding the geodesic trajectories is much more complicated, since in this case
only two Killing vectors are available. However, in 1968 Carter was able to
find a new conserved scalar along Kerr geodesics that enabled him to fully
integrate the geodesic equation in such spacetime [1]. Differently from the
conserved scalars associated to Killing vectors, which are linear on the
tangent vector of the geodesic, the constant found by Carter is quadratic.
Later, Walker and Penrose proved that Carter’s constant is a consequence of
the fact that Kerr metric admits a Killing tensor of order two [2]. Killing
tensors are symmetric generalizations of the concept of Killing vectors that
also lead to conservation laws along geodesic curves, so that they are
valuable tools in General Relativity. Indeed, the Killing tensor of Kerr
spacetime was the key ingredient to allow the integration of Klein-Gordon and
Dirac equations [3, 4], as well as the separability of electromagnetic and
gravitational perturbations in Kerr background [5].

Besides the symmetric Killing tensors, there are also anti-symmetric
generalizations of the Killing vectors called Killing-Yano tensors [6, 7].
These tensors do also lead to conserved quantities along geodesics. In
addition, Killing-Yano tensors can be used to construct Killing tensors, since
the square of a Killing-Yano is a Killing tensor of order two. In this sense,
Killing-Yano tensors can be seen as more fundamental objects than the Killing
tensors. In particular, it turns out that the Killing tensor of the Kerr
spacetime is just the square of a Killing-Yano tensor [8, 9], so that instead
of working with the Killing tensor, in this case we can deal with the Killing-
Yano tensor with advantages. Recently, Killing-Yano tensors proved to be
valuable tools in higher-dimensional spacetimes as well. It was proved by
Frolov and Kubizňák that the whole family of Kerr-NUT-(A)dS spacetimes, in
arbitrary dimension, admits a collection of Killing-Yano tensors that generate
conserved quantities just enough to enable the explicit integration of the
geodesic equation [10, 11]. These Killing-Yano tensors were also used to
separate the Klein-Gordon and Dirac equations in Kerr-NUT-(A)dS background
[12, 13].

In addition to General Relativity, Killing-Yano tensors are also of relevance
in other branches of Physics. For instance, in conservative classical systems
the solutions of the equations of motion can be seen as geodesics of certain
curved spaces, so that Killing-Yano tensors can be used to find the
trajectories of these systems [14]. Such tensors have also been used to
construct symmetry operators commuting with the D’Alembertian and Dirac
operators [15], and to solve Maxwell’s equation in curved spaces [16].

In the present article it will be investigated the Killing-Yano tensors of
order $n-1$ in manifolds of dimension $n$. In Sec. 2, Killing-Yano tensors and
its associated conserved charges are reviewed. Then, the particular features
of the Killing-Yano tensors of order $n-1$ are investigated in Sec. 3. First,
it is shown that the existence of such a tensor is equivalent to the existence
of a closed conformal Killing vector and then the integrability conditions of
the latter object are analysed. After this, the metrics of manifolds admitting
a Killing-Yano tensor of order $n-1$ are explicitly found. Then, in Sec. 3.3
it is shown how such tensors can be associated to a generalized angular
momentum. In addition, in Sec. 3.5 it is proved a useful theorem stating that
one can use the Killing vectors of a space to generate Killing-Yano tensors of
order $n-1$. Finally, in Sec. 4, two examples are worked out. In particular,
all closed conformal Killing vectors of a maximally symmetric space are found.

##  2 Killing-Yano Tensors

In what follows we shall deal with a Riemannian manifold $(M,\boldsymbol{g})$
of arbitrary signature and arbitrary dimension $n$. The manifold is assumed to
be endowed with the Levi-Civita connection, $\nabla_{a}$, which is compatible
with the metric and torsion-free. The formal distinction between $1$-forms and
vector fields will be often ignored in what follows, as the metric provides a
one-to-one map between these objects. In this context, a vector field
$\boldsymbol{K}$ is called a Killing vector whenever it obeys to the so called
Killing equation:

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

The concept of Killing vectors can be easily generalized, giving rise to what
are called the Killing tensors. A tensor $\boldsymbol{K}$ is called a Killing
tensor of order $p$ if it is totally symmetric and obeys to the following
equation:

$\nabla_{(a}\,K_{b_{1}b_{2}\cdots b_{p})}\,=\,0\,,$ (1)

where the round brackets means that the enclosed indices are symmetrized. In
particular, the metric is a Killing tensor of order two. The Killing vectors
and tensors are important because they lead to conserved scalars along the
geodesic motion. Indeed, let $\boldsymbol{T}$ be an affinely parameterized
geodesic vector field, $T^{a}\nabla_{a}T^{b}=0$. Then, it easily follows from
(1) that the scalar

$C\,=\,K_{a_{1}a_{2}\cdots a_{p}}\,T^{a_{1}}T^{a_{2}}\cdots T^{a_{p}}$

is conserved along the geodesic curve tangent to $\boldsymbol{T}$, i.e.,
$T^{a}\nabla_{a}C=0$. Note that a Killing tensor of order $p$ provides a
conserved scalar that is of degree $p$ in the velocity $\boldsymbol{T}$. In
particular, a Killing vector lead to a conserved quantity that is linear in
the velocity, which can be interpreted as the momentum of the motion in the
direction of the Killing vector.

Instead of symmetric tensors, the idea of Killing vectors can also be
generalized as skew-symmetric tensors [6]. A totally skew-symmetric tensor
$\boldsymbol{Y}$ is called a Killing-Yano tensor of order $p$ if it obeys to
the following generalization of the Killing equation:

$\nabla_{a}\,Y_{b_{1}b_{2}\ldots b_{p}}\,+\,\nabla_{b_{1}}\,Y_{ab_{2}\ldots
b_{p}}\,=\,0\,.$ (2)

Note that the above equation is tantamount to the following one:

$\nabla_{a}\,Y_{b_{1}b_{2}\ldots b_{p}}\,=\,\nabla_{[a}\,Y_{b_{1}b_{2}\ldots
b_{p}]}\,,$ (3)

where the square brackets means that the indices on the right hand side are
completely anti-symmetrized. Killing-Yano tensors are also associated to
conservation laws. Indeed, let $\boldsymbol{T}$ be an affinely parameterized
geodesic vector field. Then, we can use the Killing-Yano tensor
$\boldsymbol{Y}$ to define the tensor

$P_{b_{2}\ldots b_{p}}\,=\,T^{b_{1}}\,Y_{b_{1}b_{2}\ldots b_{p}}$ (4)

that is conserved along the geodesic generated by $\boldsymbol{T}$. In order
to see that such tensor is constant along the geodesic, note that

$T^{a}\nabla_{a}\,P_{c_{2}\ldots
c_{p}}\,=\,T^{a}T^{b}\,\nabla_{a}Y_{bc_{2}\ldots
c_{p}}\,=\,T^{a}T^{b}\,\frac{1}{2}\left(\,\nabla_{a}Y_{bc_{2}\ldots
c_{p}}\,+\,\nabla_{b}Y_{ac_{2}\ldots c_{p}}\,\right)\,=\,0\,.$

In particular, since $\boldsymbol{P}$ is conserved it follows that the scalar
constructed using its square, $P^{2}\equiv P_{b_{2}\ldots b_{p}}P^{b_{2}\ldots
b_{p}}$, is also conserved. Note that such scalar can be written as

$P^{2}\,=\,K_{ab}\,T^{a}T^{b}\;,\;\textrm{ where
}\,K_{ab}=Y_{a}^{\phantom{a}c_{2}\cdots c_{p}}\,Y_{bc_{2}\cdots c_{p}}\,.$ (5)

Since the conserved scalar $P^{2}$ is quadratic in the velocity
$\boldsymbol{T}$, it follows that the tensor $K_{ab}$ defined in (5) must be a
Killing tensor. Therefore, we can take the square of a Killing-Yano tensor to
construct a Killing tensor of order two. Generally, if $\boldsymbol{Y}$ and
$\boldsymbol{Z}$ are both Killing-Yano tensors of order $p$, then the tensor

$Q_{ab}\,=\,Y_{(a}^{\phantom{a}c_{2}\cdots c_{p}}\,Z_{b)c_{2}\cdots c_{p}}$
(6)

is a Killing tensor of order two. However, there are Killing tensors that are
not the square of Killing-Yano tensors. The later issue was investigated by
Collinson [8] and Stephani [9], where it was proved that in vacuum
4-dimensional spacetimes of Petrov type D every Killing tensor turns out to be
the square of a Killing-Yano tensor of order two, see also [17]. More about
Killing tensors and conformal Killing tensors of order two is available in
[18]. Regarding Killing-Yano tensors of order two, some interesting results in
4-dimensional spacetimes can be found in Refs. [19] and [20].

One can say that any scalar function is a Killing-Yano tensor of order zero,
while a Killing-Yano tensor of order one is a Killing vector. As any skew-
symmetric tensor of rank $n$, a Killing-Yano tensor of order $n$ must be a
multiple of the volume-form of the manifold,

$Y_{a_{1}a_{2}\ldots a_{n}}\,=\,\lambda\,\epsilon_{a_{1}a_{2}\ldots a_{n}}\,.$
(7)

Then, using (3) and the fact that $\boldsymbol{\epsilon}$ is covariantly
constant, we conclude that $\lambda$ must be a constant. The Killing tensor
associated to (7) is just the metric, apart from a meaningless constant
multiplicative factor. Therefore, we can say that the Killing-Yano tensors of
order $p=0$, $p=1$ and $p=n$ are, in a sense, trivial. The simplest non-
trivial case is $p=n-1$. For instance, in [21] it was shown an example in
which Killing-Yano tensors of order $p=3$ in $n=4$ dimensions lead to non-
obvious conserved quantities. In spite of the simplicity, there are very few
comments about Killing-Yano tensors of order $n-1$ in the literature and the
intent of the present article is to fill this gap. However, it is worth noting
that some illuminating comments on the case of Killing-Yano tensors of order
$3$ in $4$-dimensional Lorentzian manifolds can be found in [19].

##  3 The Killing-Yano Tensors of Order $n-1$

Let $\boldsymbol{Y}$ be a Killing-Yano tensor of order $n-1$. Then, since it
is a totally anti-symmetric tensor, it can be seen as a differential form. In
particular, $\boldsymbol{Y}$ is Hodge dual of some 1-form $\boldsymbol{\xi}$:

$\boldsymbol{Y}\,=\,\star\,\boldsymbol{\xi}\quad\Rightarrow\quad
Y_{a_{2}a_{3}\cdots a_{n}}\,=\,\xi^{a_{1}}\,\epsilon_{a_{1}a_{2}a_{3}\cdots
a_{n}}\,.$ (8)

Inserting the latter relation into (3) and using the fact that every $n$-form
must be proportional to the volume form lead us to

$\epsilon_{b_{1}b_{2}b_{3}\cdots
b_{n}}\,\nabla_{a}\,\xi^{b_{1}}\,=\,\nabla_{[a}Y_{b_{2}b_{3}\cdots
b_{n}]}\,=\,S\,\epsilon_{ab_{2}\cdots b_{n}}\,,$

where $S$ is some scalar function. Then, contracting the above equation with
$\epsilon^{cb_{2}\cdots b_{n}}$ we find that $\nabla_{a}\xi_{c}=h\,g_{ac}$,
where $h$ is some function. Taking the trace of this equation, we conclude
that $h=\frac{1}{n}\nabla_{a}\xi^{a}$. Therefore, the Hodge dual of a Killing-
Yano tensor of order $n-1$ obeys to the following equation:

$\nabla_{a}\,\xi_{b}\,=\,h\,g_{ab}\;,\;\textrm{ where
}\;h\,=\,\frac{1}{n}\,\nabla_{a}\xi^{a}\,.$ (9)

The above equation is tantamount to say that $\boldsymbol{\xi}$ is a closed
conformal Killing vector, since $\nabla_{[a}\xi_{b]}=0$ and
$\nabla_{(a}\xi_{b)}\propto g_{ab}$ . Conversely, suppose that
$\boldsymbol{\xi}$ is a closed conformal Killing vector, meaning that Eq. (9)
is satisfied, then defining $\boldsymbol{Y}$ as in Eq. (8) we find

$\nabla_{a}Y_{b_{2}b_{3}\cdots b_{n}}\,=\,\epsilon_{b_{1}b_{2}b_{3}\cdots
b_{n}}\,\nabla_{a}\xi^{b_{1}}\,=\,h\,\epsilon_{ab_{2}b_{3}\cdots
b_{n}}\quad\Rightarrow\quad\nabla_{(a}Y_{b_{2})b_{3}\cdots b_{n}}\,=\,0\,,$

so that $\boldsymbol{Y}$ is a Killing-Yano tensor. Therefore, we conclude that
_$\boldsymbol{Y}$ is a Killing-Yano tensor of order $n-1$ if, and only if, its
Hodge dual is a closed conformal Killing vector_. Then, instead of working
with $\boldsymbol{Y}$ one can, equivalently, deal with $\boldsymbol{\xi}$.
Actually, this result is just a particular case of the general statement that
a differential form is a Killing-Yano tensor if, and only if, its Hodge dual
is a closed conformal Killing-Yano tensor [10].

Let us now display some geometric properties of a closed conformal Killing
vector. Since $\boldsymbol{\xi}$ is a closed 1-form, it follows that locally
we can find some function $\lambda$ such that $\xi_{a}=\nabla_{a}\lambda$. So,
as a vector field, $\boldsymbol{\xi}$ is orthogonal to the family of hyper-
surfaces of constant $\lambda$. Moreover, contracting Eq. (9) with $\xi^{a}$
we see that $\boldsymbol{\xi}$ is a geodesic vector field, although generally
not affinely parameterized. Now, define $f$ to be the squared norm of
$\boldsymbol{\xi}$,

$f\,=\,\xi^{a}\,\xi_{a}\,.$

Then, taking the derivative of $f$ lead us to

$\nabla_{a}\,f\,=\,2\,\xi^{b}\,\nabla_{a}\xi_{b}\,=\,2h\,\xi_{a}\,.$ (10)

Thus, applying $\nabla_{c}$ on the above identity and then taking the skew-
symmetric part of the resulting equation lead us to:

$\nabla_{c}\nabla_{a}\,f\,=\,2\,(\nabla_{c}h)\,\xi_{a}\,+\,2\,h^{2}\,g_{ca}\quad\Rightarrow\quad\xi_{[a}\nabla_{c]}h\,=\,0\quad\Rightarrow\quad\nabla_{c}h\,\propto\,\xi_{c}\,.$
(11)

As pointed out in the preceding section, one can use Killing-Yano tensors to
construct Killing tensors of order two, but the converse generally is not
true. In particular, if $\boldsymbol{Y}=\star\,\boldsymbol{\xi}$ is a Killing-
Yano tensor of order $n-1$ then using Eqs. (5) and (8) we find that the
associated Killing tensor, apart from an unimportant constant global factor,
is

$K_{ab}\,=\,\xi_{a}\,\xi_{b}\,-\,f\,g_{ab}\,.$ (12)

More generally, if $\boldsymbol{Y}=\star\,\boldsymbol{\xi}$ and
$\boldsymbol{Z}=\star\,\boldsymbol{\chi}$ are both Killing-Yano tensors of
order $n-1$ then, besides the Killing tensors of the form (12), one can
construct another Killing tensor of order two contracting $\boldsymbol{Y}$ and
$\boldsymbol{Z}$ as shown in (6), such Killing tensor is

$Q_{ab}\,=\,\xi_{(a}\,\chi_{b)}\,-\,(\xi^{c}\,\chi_{c})\,g_{ab}\,.$ (13)

###  3.1 Integrability Conditions

Now, let us work out the integrability conditions for a closed conformal
Killing vector, which is tantamount to studying the integrability conditions
of a Killing-Yano tensor of order $n-1$. Using (9) and the Ricci identity we
find that

$g_{bc}\,\nabla_{a}h\,-\,g_{ac}\,\nabla_{b}h\,=\,(\nabla_{a}\nabla_{b}\,-\,\nabla_{b}\nabla_{a})\,\xi_{c}\,=\,R_{abc}^{\phantom{abc}d}\,\xi_{d}\,.$
(14)

Then, contracting the above equation with $g^{ac}$ yields the following
relation:

$\nabla_{a}h\,=\,\frac{-1}{n-1}\,R_{a}^{\phantom{a}b}\,\xi_{b}\,.$ (15)

In particular, such relation along with (11) implies that $\boldsymbol{\xi}$
is an eigenvector of the Ricci tensor,

$R_{a}^{\phantom{a}b}\,\xi_{b}\,\propto\,\xi_{a}\,.$

Taking the covariant derivative of (9) and then using Eq. (15) lead to

$\nabla_{a}\nabla_{b}\,\xi_{c}\,=\,\frac{-1}{n-1}\,g_{bc}\,R_{a}^{\phantom{a}d}\,\xi_{d}\,.$
(16)

Such equation along with (9) implies that in order to know the vector field
$\boldsymbol{\xi}$ in the whole manifold we just need to know $\xi_{a}$ and
$\nabla_{a}\xi^{a}$ in one point of the manifold, the components of
$\boldsymbol{\xi}$ in the other points can then be obtained integrating
equations (9) and (16). In particular, this implies that the maximum number of
independent closed conformal Killing vectors in an $n$-dimensional manifold is
$n+1$. This maximum number is attained just in maximally symmetric spaces.

Then, inserting Eq. (15) into (14) yields the following integrability
condition:

$\left[\,(n-1)\,R_{abc}^{\phantom{abc}d}\,+\,g_{bc}\,R_{a}^{\phantom{a}d}\,-\,g_{ac}\,R_{b}^{\phantom{b}d}\right]\,\xi_{d}\,=\,0\,.$
(17)

Such identity states that in order for a manifold to admit a non-zero solution
to Eq. (9), the curvature must be constrained. In particular, in the case of
an Einstein space, a space in which $R_{ab}=\Lambda g_{ab}$, the above
integrability condition is equivalent to:

$C_{abcd}\,\xi^{d}\,=\,0\quad\textrm{ (Einstein spaces)}\,,$ (18)

where $C_{abcd}$ is the Weyl tensor. In 4-dimensional Lorentzian manifolds,
condition (18) implies that either the Weyl tensor vanishes or
$\boldsymbol{\xi}$ is a null vector field and the Petrov type of the Weyl
tensor is $N$. Moreover, in higher dimensions if $\boldsymbol{\xi}$ is null
and the signature is Lorentzian the constraint (18) implies that the algebraic
type of the Weyl tensor is special according to the CMPP classification [22],
although not necessarily type $N$ [23]. In the same vein, such condition
constrains the algebraic type of the Weyl tensor according to the
classification defined in [24]. Particularly, using the 6-dimensional
spinorial formalism presented in [25], it follows that if $\boldsymbol{\xi}$
is null then its spinorial representation can be chose to be
$\xi^{AB}=\chi_{1}^{[A}\chi_{2}^{B]}$, so that the condition (18) can be
proved to be equivalent to the following constrains on the spinorial version
of the Weyl tensor:

$\Psi^{A3}_{\phantom{A3}B1}\,=\,\Psi^{A3}_{\phantom{A3}B2}\,=\,\Psi^{A4}_{\phantom{A4}B1}\,=\,\Psi^{A4}_{\phantom{A4}B2}\,=\,0\quad;\quad\Psi^{A1}_{\phantom{A1}B1}\,+\,\Psi^{A2}_{\phantom{A3}B2}\,=\,0\,.$

However, it is worth noting that, differently from what happens in 4
dimensions, in higher-dimensional Einstein spacetimes with non-vanishing Weyl
tensor a vector field $\boldsymbol{\xi}$ obeying to (18) is not necessarily
null.

###  3.2 Metrics Admitting a Killing-Yano Tensor of Order $n-1$

In this section it will be presented the explicit form of the metrics allowing
the existence of a Killing-Yano tensor of order $n-1$, which is equivalent to
the existence of a closed conformal Killing vector $\boldsymbol{\xi}$. In
order to accomplish this, let us separate the analysis in two possibilities,
the cases $\boldsymbol{\xi}$ null and non-null. Before proceeding, it is worth
stressing that the results here are all local and that it is always being
assumed that the closed conformal Killing vector is non-trivial,
$\boldsymbol{\xi}\neq 0$.

####  3.2.1 The case $\boldsymbol{\xi}$ null

If the norm of $\boldsymbol{\xi}$ is identically zero, $f=0$, then Eq. (10)
implies that $h=0$. This, in turn, guarantees that $\boldsymbol{\xi}$ is a
covariantly constant null vector,

$\xi_{a}\,\xi^{a}\,=\,0\;\textrm{ and }\;\nabla_{a}\,\xi_{b}\,=\,0\,.$

In particular, since $\boldsymbol{\xi}$ is constant, the Killing-Yano tensor
defined in (8) is also constant. In 4 dimensions, spacetimes admitting a
covariantly constant null vector field are called $pp$-wave, they represent
gravitational waves with well-defined wave surfaces. In arbitrary dimension,
the general form of a Lorentzian metric with a covariantly constant null
vector is given by [26]:

$ds^{2}\,=\,2du\,\left[\,dv+H(u,x)\,du+F_{j}(u,x)\,dx^{j}\,\right]\,+\,\tilde{g}_{ij}(u,x)\,dx^{i}\,dx^{j}\,,$

with the indices $i,j$ running from $1$ to $n-2$. Where in the above
coordinates the covariantly constant null vector is given by
$\boldsymbol{\xi}=\partial_{v}$ or, equivalently, by $\boldsymbol{\xi}=du$.
Note that the functions in this line element do not depend on the coordinate
$v$, which is a consequence of the fact that $\boldsymbol{\xi}$ is a Killing
vector.

Spacetimes in which all scalars constructed with the curvature, its
derivatives and the metric vanish are called vanishing scalar invariant (VSI).
Such manifolds are quite important in quantum theories of gravity, since for
them the quantum corrections to Einstein-Hilbert action vanish. There is an
important class of VSI spacetimes that admit a covariantly constant null
vector and in Ref. [27] it was found the general metric of such spacetimes.
Particularly, it was proved that, in the CMPP classification [22], the
possible algebraic types of the Weyl tensor in the latter spaces are $III$,
$N$ or $O$ [27].

####  3.2.2 The case $\boldsymbol{\xi}$ non-null

Since $\boldsymbol{\xi}$ is a closed 1-form, it follows from the Poincaré
lemma that locally we can find some function $\lambda$ such that
$\boldsymbol{\xi}=d\lambda$. We can then choose such function to be one of the
local coordinates. Thus, let us use the coordinates $\\{x^{j},\lambda\\}$,
with $i,j\in\\{1,2,\cdots,n-1\\}$. Moreover, since
$d\boldsymbol{\xi}\wedge\boldsymbol{\xi}=0$, it follows from the Frobenius
theorem that the tangent subspaces orthogonal to $\boldsymbol{\xi}$ form an
integrable foliation. Therefore, the coordinates $\\{x^{j}\\}$ can always be
chosen in such a way that the coordinate vectors $\partial_{j}$ are orthogonal
to $\boldsymbol{\xi}$. In the latter frame, since $\boldsymbol{\xi}$ is
assumed to be non-null, we have the following line element:

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

Since $\boldsymbol{\xi}=d\lambda$ we can prove that
$g_{\lambda\lambda}=f^{-1}$, where $f=\xi^{a}\xi_{a}$. Now, using (10) in this
coordinate frame we have that:

$\partial_{a}f\,=\,2\,h\,\xi_{a}\,=\,2\,h\,\delta^{\,\lambda}_{a}\quad\Rightarrow\quad\partial_{j}f\,=\,0\quad\Rightarrow\quad
f\,=\,f(\lambda)\,.$

In the same vein, thanks to Eq. (11), it follows that the function
$h=\frac{1}{n}\nabla_{a}\xi^{a}$ depends just on the coordinate $\lambda$.
Thus, using this fact into Eq. (9) lead us to:

$h(\lambda)\,g_{ab}\,=\,\nabla_{a}\,\xi_{b}\,=\,\nabla_{a}\,\delta^{\,\lambda}_{b}\,=\,-\,\Gamma_{ab}^{\lambda}\,=\,\frac{1}{2}\,f(\lambda)\left(\,\partial_{\lambda}g_{ab}\,-\,\partial_{a}g_{b\lambda}\,-\,\partial_{b}g_{a\lambda}\,\right)\,.$

Such equation is equivalent to the following constraints:

$h(\lambda)\,=\,\frac{1}{2}\,\partial_{\lambda}f\quad;\quad\partial_{\lambda}\,g_{ij}\,=\,2\,\frac{h(\lambda)}{f(\lambda)}\,g_{ij}\,.$
(19)

These two constraints, in turn, imply the following:

$\partial_{\lambda}\,g_{ij}=\left[\,\partial_{\lambda}\ln(f)\,\right]\,g_{ij}\quad\Rightarrow\quad
g_{ij}(\lambda,x)\,=\,f(\lambda)\,\tilde{g}_{ij}(x)\,.$

Thus, we have proved that if a manifold of arbitrary dimension admits a non-
null closed conformal Killing vector $\boldsymbol{\xi}$, then locally its
metric can be put in the following form:

$ds^{2}\,=\,\frac{1}{f(\lambda)}\,(\,d\lambda\,)^{2}\,+\,f(\lambda)\,\tilde{g}_{ij}(x)\,dx^{i}\,dx^{j}\,.$
(20)

In these coordinates the vector field $\boldsymbol{\xi}$ is given by:

$\boldsymbol{\xi}\,=\,f\,\partial_{\lambda}\,\sim\,d\lambda\,.$ (21)

Where in the above equation $\boldsymbol{\xi}$ was first written as a vector
field and then as a 1-form. By means of (8), we find that the Killing-Yano
tensor associated to such closed conformal Killing vector is given by:

$Y_{a_{1}a_{2}\cdots
a_{n-1}}\,=\,\sqrt{|f^{n}\,\tilde{g}|\,}\,(n-1)!\,\delta_{[a_{1}}^{\,1}\delta_{a_{2}}^{\,2}\cdots\delta_{a_{n-1}]}^{\,n-1}\,=\,\sqrt{|f|\,}\,\tilde{\epsilon}_{a_{1}a_{2}\cdots
a_{n-1}}\,.$

Where in the above equation $\tilde{g}$ denotes the determinant of
$\tilde{g}_{ij}$ and $\tilde{\boldsymbol{\epsilon}}$ denotes the volume-form
of the submanifold spanned by the coordinates $\\{x^{i}\\}$, whose metric is
$f\tilde{g}_{ij}$.

####  3.2.3 The special case $\boldsymbol{\xi}$ non-null and $f$ constant

A particularly simple case occurs when $f$ is a non-zero constant. In such
special case Eq. (19) implies that $h=0$, so that $\boldsymbol{\xi}$ is a
covariantly constant non-null vector field. Moreover, because of (20), we
conclude that the metric can be chosen to be

$ds^{2}\,=\,\pm\,d\lambda^{2}\,+\,\tilde{g}_{ij}(x)\,dx^{i}\,dx^{j}\quad(\textrm{$f$
constant})\,,$

with $\boldsymbol{\xi}=\partial_{\lambda}$ being the covariantly constant
vector field. In these coordinates the components of the Riemann tensor are
such that

$R_{\lambda abc}\,=\,0\quad\textrm{and}\quad R_{ijkl}\,=\,\tilde{R}_{ijkl}\,,$
(22)

where the indices $a,b,c$ are arbitrary; $i,j,k,l$ run from $1$ to $(n-1)$;
and $\tilde{R}_{ijkl}$ is the Riemann tensor associated to the
$(n-1)$-dimensional metric $\tilde{g}_{ij}$.

By means of Eq. (22), one can easily see that $g_{ab}$ is a Ricci-flat metric
if, and only if, $\tilde{g}_{ij}$ is Ricci-flat. Particularly, in 4 dimensions
if $g_{ab}$ is a vacuum solution then $\tilde{g}_{ij}$ is a 3-dimensional
Ricci-flat metric. But, since the Weyl tensor in 3 dimensions is zero, it
follows that if the Ricci tensor vanishes identically then the Riemann tensor
is zero, $\tilde{R}_{ijkl}=0$. This, in turn, implies that the Riemann tensor
of $g_{ab}$ is also zero, as can be grasped from (22). Thus, we conclude that
the only 4-dimensional manifold with vanishing Ricci tensor admitting a
covariantly constant non-null vector field is the flat space.

###  3.3 Relation with the Concept of Angular Momentum

In Classical Mechanics, if $\boldsymbol{x}$ and $\boldsymbol{p}$ are
respectively the position and the linear momentum of a particle in 3
dimensions, then the angular momentum is given by the vector
$\boldsymbol{L}=\boldsymbol{x}\times\boldsymbol{p}$. Nevertheless, since the
vectorial product is not defined in higher dimensions, in Special Relativity
the angular momentum shall be understood as an anti-symmetric tensor of rank
two [28],

$L^{ab}=2\,x^{[a}p^{b]}\,.$ (23)

Where $x^{a}$ and $p^{a}$ are the position and momentum 4-vectors. In General
Relativity the momentum of a particle is well defined, it is just the mass
times the normalized tangent vector of its world line. But in general curved
spaces the notion of position vector has no intrinsic meaning, so that one
cannot define angular momentum as in (23). However, in the present section it
will be proved that in spacetimes endowed with a Killing-Yano tensor of order
$n-1$ the associated closed conformal Killing vector can play the role of a
position vector.

Let $(M,\boldsymbol{g})$ be the flat Euclidean space. Then, one can introduce
cartesian coordinates $\\{x^{a}\\}$ in which the metric components are given
by $g_{ab}=\delta_{ab}$ and the covariant derivative is just the partial
derivative. Thus, since the Riemann tensor is identically zero in this space,
equations (9) and (15) imply that $\partial_{a}\xi_{b}=h\delta_{ab}$ and
$\partial_{a}h=0$. The general solution to these equations is
$\xi_{a}=h\,x^{a}+c_{a}$, where $h$ and $c_{a}$ are constants. This leads to
the conclusion that, in flat Euclidean spaces, a closed conformal Killing
vector is, apart from a global multiplicative scale and a constant
translation, the position vector. Therefore, in the present section we shall
extrapolate such result and interpret the closed conformal Killing vectors as
a kind of generalized position vectors, even in curved spaces.

According to General Relativity, free particles, i.e., particles influenced
just by the gravity, move along geodesics. Therefore, we expect from the above
paragraphs that if $p^{a}$ is an affinely parameterized geodesic vector field
representing the momentum of a free particle of mass $m$,
$p^{a}\nabla_{a}p^{b}=0$ and $p^{a}p_{a}=-m^{2}$, and $\xi^{a}$ is a closed
conformal Killing vector, then,

$L^{ab}=2\,\xi^{[a}p^{b]}$

might represent a kind of angular momentum. In particular, the tensor
$\boldsymbol{L}$ should be conserved along the geodesic. Indeed,

$p^{c}\nabla_{c}\,L_{ab}\,=\,p^{c}\left(\,\nabla_{c}\xi_{a}\,p_{b}\,-\,p_{a}\,\nabla_{c}\xi_{b}\,\right)\,=\,p^{c}\left(\,h\,g_{ca}\,p_{b}\,-\,p_{a}\,h\,g_{cb}\,\right)\,=\,0\,.$

As a consequence, the scalar $L^{ab}L_{ab}=-2\,[m^{2}f+(\xi^{a}p_{a})^{2}]$ is
also conserved. Thus, in spacetimes admitting a Killing-Yano tensor of order
$n-1$ one can define a generalization of the angular momentum of a particle.
More comments on the relation between Killing-Yano tensors and generalizations
of the angular momentum can be found in [19, 10].

###  3.4 Integrating the Equation of a Closed Conformal Killing Vector with
the Help of Symmetries

In Refs. [29, 30], Garfinkle and Glass have introduced a method that helps in
the integration of Killing tensor equation and Killing-Yano equation when the
manifold admits a non-null hyper-surface orthogonal Killing vector
$\boldsymbol{\zeta}$. In [30], such method was also applied to the conformal
Killing vector equation. The technic amounts to split these differential
equations in the direction parallel to $\boldsymbol{\zeta}$ and in the
orthogonal part, an approach analogous to the one adopted for obtaining the
time evolution of a initial data on a Cauchy surface [31]. In the present
section we shall apply this method to split Eq. (9).

Suppose that $(M,\boldsymbol{g})$ admits a non-null hyper-surface orthogonal
Killing vector $\boldsymbol{\zeta}$. Then, the manifold can be foliated by
leaves that are orthogonal to this vector field. Each leaf is a submanifold of
dimension $(n-1)$. Let us denote the covariant derivative on such submanifold
by $D_{a}$ and the tensor that projects into these leaves by
$\hat{g}_{ab}=g_{ab}-\frac{1}{N}\zeta_{a}\zeta_{b}$, where
$N=\zeta^{a}\zeta_{a}$. Then, by these hypotheses, $\boldsymbol{\zeta}$ obeys
to the following equation:

$\nabla_{a}\,\zeta_{b}\,=\,\frac{1}{N}\,N_{[a}\zeta_{b]}\,,\;\textrm{ where
}\;N_{a}\,\equiv\,D_{a}N\,=\,\hat{g}_{ab}\nabla^{b}N\,.$

We shall decompose $\boldsymbol{\xi}$ as

$\xi_{a}\,=\,A\,\zeta_{a}\,+\,B_{a}\,,\;\textrm{ where
}\;\zeta^{a}\,B_{a}\,=\,0\,.$

Now, let us separate the indices in the equation satisfied by the closed
conformal Killing vector, $\nabla_{a}\xi_{b}=h\,g_{ab}$, into the parts that
are parallel and orthogonal to $\boldsymbol{\zeta}$. Such decomposition has
three terms: (1) both indices projected into the leaves orthogonal to
$\boldsymbol{\zeta}$; (2) one index projected in the direction of
$\boldsymbol{\zeta}$ and the other index in the orthogonal direction; (3) both
indices projected along $\boldsymbol{\zeta}$. The equations obtained by means
of these projections are respectively given by:

$\left\\{\begin{array}[]{l}D_{a}\,B_{b}\,=\,h\,\hat{g}_{ab}\\\ \\\
\mathcal{L}_{\zeta}\,B_{a}\,=\,\frac{1}{2}A\,N_{a}\;;\;D_{a}\,A\,=\,-\frac{A}{2\,N}\,N_{a}\\\
\\\
\mathcal{L}_{\zeta}A\,=\,\frac{N^{\frac{1}{2}(n-1)}}{n-1}\,D_{a}\left(\,N^{\frac{1}{2}(1-n)}\,B^{a}\,\right)\,.\\\
\end{array}\right.$

Where $\mathcal{L}_{\zeta}$ denotes the Lie derivative along the vector field
$\boldsymbol{\zeta}$. Particularly, the first of these equations means that if
$\xi_{a}$ is a closed conformal Killing vector in $(M,\boldsymbol{g})$, then
$B_{a}=\hat{g}_{ab}\xi^{b}$ is a closed conformal Killing vector in the family
of submanifolds orthogonal to $\boldsymbol{\zeta}$. For examples showing how
this kind of decomposition can be helpful see [29, 30].

###  3.5 Generating Closed Conformal Killing Vectors Using Symmetries

In this section it will be proved a theorem stating that one can use a closed
conformal Killing vector and the symmetries of the space to generate other
closed conformal Killing vectors. Let $\boldsymbol{\xi}$ be a closed conformal
Killing vector and $\boldsymbol{\eta}$ some Killing vector, then we have that

$\nabla_{a}\xi_{b}\,=\,h\,g_{ab}\quad;\quad\nabla_{a}\eta_{b}\,=\,\nabla_{[a}\eta_{b]}\,.$

Now, let us define the following gradient vector field:

$\chi_{a}\,=\,\nabla_{a}\,(\xi^{b}\eta_{b})\,=\,h\,\eta_{a}\,+\,\xi^{b}\,\nabla_{[a}\eta_{b]}\,.$

Then, the covariant derivative of $\boldsymbol{\chi}$ is given by

$\nabla_{c}\,\chi_{b}\,=\,(\nabla_{c}h)\,\eta_{b}\,+\,\xi^{a}\nabla_{c}\nabla_{b}\eta_{a}\,=\,\frac{-1}{n-1}\,R_{c}^{\phantom{c}a}\xi_{a}\,\eta_{b}\,+\,\xi^{a}\,R_{abcd}\,\eta^{d}\,.$
(24)

Where in the last equality it was used Eq. (15) and the fact that the second
derivative of a Killing vector field can be expressed in terms of the Riemann
tensor and the Killing vector itself. Now, using the integrability condition
(17) on the right hand side of (24), we arrive at the following equality:

$\nabla_{c}\,\chi_{b}\,=\,-\frac{(\xi^{a}\,R_{ad}\,\eta^{d})}{n-1}\,\,g_{cb}\,.$
(25)

So, $\boldsymbol{\chi}$ is also a closed conformal Killing vector. Therefore,
we have proved the following:

Theorem - _If $\boldsymbol{\xi}$ is a closed conformal Killing vector and
$\boldsymbol{\eta}$ is a Killing vector then the vector field
$\chi_{a}=\nabla_{a}(\xi^{c}\eta_{c})$ is a closed conformal Killing vector._

Once we have found $\boldsymbol{\chi}$ we can, in principle, use it and the
Killing vector $\boldsymbol{\eta}$ to generate another closed conformal
Killing vector, $\kappa_{a}=\nabla_{a}(\chi^{c}\eta_{c})$, and so on. However,
it is worth recalling that the maximum number of independent closed conformal
Killing vectors in an $n$-dimensional manifold is $n+1$. Thus, while using the
above procedure to generate closed conformal Killing vectors we can find a
vector field that is zero or that is a linear combination, with constant
coefficients, of the closed conformal Killing vectors already known.

As a last comment, from Eq. (25) we see that if the Ricci tensor vanishes then
the covariant derivative of $\boldsymbol{\chi}$ is zero. So, we conclude that
if a Ricci-flat manifold admits a closed conformal Killing vector and a
Killing vector whose inner product between them is not constant,
$\nabla_{a}(\xi^{c}\eta_{c})\neq 0$, then the manifold admits a covariantly
constant vector field.

##  4 Examples

In this section the results obtained so far will be worked out in two
examples. It will be explicitly found all closed conformal Killing vectors of
a maximally symmetric space and it will be shown that the FLRW spacetime
admits a Killing-Yano tensor of order $n-1$.

###  4.1 Maximally Symmetric Spaces

A maximally symmetric space is a manifold possessing the maximum number of
isometries, which is $\frac{1}{2}n(n+1)$ in $n$ dimensions. In the case of
Euclidean signature, one can find a coordinate system $\\{x^{a}\\}$ in which
the line element is:

$ds^{2}\,=\,\frac{1}{(1\,+\,\kappa\,r^{2})^{2}}\,\left[\,(dx^{1})^{2}\,+\,(dx^{2})^{2}\,+\,\cdots\,+\,(dx^{n})^{2}\,\right]\,,$
(26)

where

$r^{2}\,=\,\left[\,(x^{1})^{2}\,+\,(x^{2})^{2}\,+\,\cdots\,+\,(x^{n})^{2}\,\right]\,=\,x^{a}\,x^{a}\,,\;\;\textrm{
and }\;\;\kappa\,=\,0,\,\pm 1\,.$

The case $\kappa=0$ represents the flat-space, which was already treated in
Sec. 3.3. So, in what follows let us just consider the non-trivial cases
$\kappa=\pm 1$. The metric of maximally symmetric spaces with other
signatures, like de Sitter and Anti-de Sitter spacetimes, can be obtained from
(26) by means of analytical continuations of the form $x^{a}\rightarrow
ix^{a}$. The independent Killing vectors can be chosen to be

$\boldsymbol{\eta}^{ij}\,=\,x^{i}\,\partial_{j}\,-\,x^{j}\,\partial_{i}\quad;\quad\boldsymbol{\eta}^{i}\,=\,(1-\kappa\,r^{2})\partial_{i}\,+\,2\,\kappa\,x^{i}\,x^{a}\partial_{a}\,.$
(27)

Where in the above expressions $i$ and $j$ are not tensorial indices, they are
just labels running from $1$ to $n$ with $i\neq j$. The vector fields
$\boldsymbol{\eta}^{ij}$ are obvious symmetries of the metric (26), since they
generate rotations, which keep $r$ invariant. In order to see that
$\boldsymbol{\eta}^{i}$ are also Killing vectors, one shall compute the
Christoffel symbols of such metric,

$\Gamma_{ab}^{c}\,=\,\frac{2\kappa}{1\,+\,\kappa\,r^{2}}\,\left(\,x^{c}\,\delta_{ab}\,-\,x^{a}\,\delta_{bc}\,-\,x^{b}\,\delta_{ac}\,\right)\,,$

and note that

$\nabla_{a}\,\eta^{i}_{b}\,=\,\frac{8\,\kappa}{(1\,+\,\kappa\,r^{2})^{3}}\,\delta^{i[a}\,x^{b]}\quad\Rightarrow\quad\nabla_{(a}\,\eta^{i}_{b)}\,=\,0\,.$

Now, let us look for the closed conformal Killing vectors. A natural guess is
that there exists some function $F(r^{2})$ such that its gradient is a closed
conformal Killing vector $\boldsymbol{\xi}^{0}$:

$\boldsymbol{\xi}^{0}\,=\,d\,[\,F(r^{2})\,]\,=\,F^{\prime}\,d(r^{2})\,=\,2\,F^{\prime}\,x^{a}\,dx^{a}\,,$

where the prime denotes the derivative with respect to $r^{2}$. Indeed,
imposing that $\nabla_{a}\xi^{0}_{b}\propto g_{ab}$ lead to

$(1\,+\,\kappa\,r^{2})\,F^{\prime\prime}\,+\,2\,\kappa\,F^{\prime}\,=\,0\quad\Rightarrow\quad
F(r^{2})\,=\,\frac{A}{1\,+\,\kappa\,r^{2}}\,+\,B\,,$

where $A$ and $B$ are constants. Choosing $A=-1/2\kappa$ we find that

$\boldsymbol{\xi}^{0}\,=\,x^{a}\,\partial_{a}\,.$

Using the fact that such space is an Einstein manifold, $R_{ab}=\Lambda
g_{ab}$ with $\Lambda=4\kappa(n-1)$, one can readily verify that the
integrability conditions (15) and (17) are satisfied. The other closed
conformal Killing vectors can now be easily found with the help of the theorem
presented in Sec. 3.5. Indeed, according to such theorem the 1-forms

$\boldsymbol{\xi}^{i}\,=\,d\,\left[\xi^{0}_{a}\,\eta^{i\,a}\,\right]\,=\,d\,\left[\frac{x^{i}}{1\,+\,\kappa\,r^{2}}\right]\,\sim\,(1+\kappa\,r^{2})\partial_{i}\,-\,2\,\kappa\,x^{i}\,x^{a}\partial_{a}$

are closed conformal Killing vectors, where the symbol $\sim$ means that the
tensors are equal if we use the one-to-one relation between 1-forms and vector
fields provided by the metric. Since $(n+1)$ is the maximum number of
independent closed conformal Killing vectors in an $n$-dimensional manifold,
and since $\\{\boldsymbol{\xi}^{0},\boldsymbol{\xi}^{i}\\}$ are linearly
independent as vector fields, it follows that we have found all closed
conformal Killing vectors of a maximally symmetric space. As a consequence,
taking the Hodge dual of the 1-forms
$\\{\boldsymbol{\xi}^{0},\boldsymbol{\xi}^{i}\\}$, one can find all Killing-
Yano tensors of order $n-1$. Moreover, using such Killing-Yano tensors along
with Eqs. (12) and (13) we can construct $\frac{1}{2}(n+1)(n+2)$ independent
Killing tensors of order two.

###  4.2 FLRW cosmological model

The so-called FLRW cosmological model amounts to assume that at large scales
the universe can be described as a Lorentzian manifold that is foliated by
spatial sections that are homogeneous and isotropic, so that the space-like
leaves are maximally symmetric submanifolds. In the case of $n-1$ spatial
dimensions, the FLRW line element can be written as

$ds^{2}\,=\,-dt^{2}\,+\,\frac{a^{2}(t)}{(1\,+\,\kappa\,r^{2})^{2}}\,\left[\,(dx^{1})^{2}\,+\,(dx^{2})^{2}\,+\,\cdots\,+\,(dx^{n-1})^{2}\,\right]\,.$
(28)

Now, let us see that such metric can be put in the standard form shown in Eq.
(20). Indeed, defining a new time-like coordinate

$\lambda\,\equiv\,\int\,a(t)\,dt\,,$

we have that the line element is given by

$ds^{2}\,=\,-\frac{1}{a^{2}(t(\lambda))}\,d\lambda^{2}\,+\,\frac{a^{2}(t(\lambda))}{(1\,+\,\kappa\,r^{2})^{2}}\,\left[\,(dx^{1})^{2}\,+\,(dx^{2})^{2}\,+\,\cdots\,+\,(dx^{n-1})^{2}\,\right]\,.$

Note that this form of the line element is exactly equal to the one shown in
(20), with $f=-\,a^{2}$. Then, from Eq. (21) we conclude that

$\boldsymbol{\xi}\,=\,d\lambda\,=\,a(t)\,dt$

is a closed conformal Killing vector. Indeed, one can prove that the covariant
derivative of this vector field is given by

$\nabla_{b}\,\xi_{c}\,=\,-\,\dot{a}\,g_{bc}\,,$

where the dot denotes the derivative with respect to the coordinate $t$. From
the above equation, we conclude that $h=-\,\dot{a}$, which agrees with Eq.
(10) and the fact that $f=-\,a^{2}$. By means of (12), we find that the
components of the Killing tensor associated to $\boldsymbol{\xi}$ are given,
in the coordinates $\\{t,x^{i}\\}$, by

$K_{bc}\,=\,a^{2}(t)\,\left[\,\delta^{\,t}_{b}\,\delta^{\,t}_{c}\,+\,g_{bc}\,\right]\,.$

Note that the metric (28) is invariant under rotations performed in the
spatial leaves spanned by the coordinates $\\{x^{i}\\}$, so that the vector
fields $\boldsymbol{\eta}^{ij}=x^{i}\,\partial_{j}-x^{j}\,\partial_{i}$ are
Killing vectors of the FLRW spacetime. Where the indices $i$ and $j$ are
labels for the spatial coordinates that run from 1 to $n-1$. Thus, in
principle, one could use these Killing vectors along with the theorem of Sec.
3.5 in order to generate new closed conformal Killing vectors in addition to
$\boldsymbol{\xi}$. However, since the Killing vectors
$\boldsymbol{\eta}^{ij}$ are orthogonal to the closed conformal Killing vector
$\boldsymbol{\xi}=a\,dt$, it follows that in this case the mentioned theorem
does not provide a new symmetry.

## Acknowledgments

I want to thank CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior - Brazil) for the financial support.

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