# Analysis of the structural complexity of Crab nebula observed at
radio and infrared frequencies using a multifractal approach

Athokpam Langlen Chanu1<EMAIL_ADDRESS>(Corresponding
author) Pravabati Chingangbam2<EMAIL_ADDRESS>Fazlu Rahman2,3
<EMAIL_ADDRESS>R. K. Brojen Singh1<EMAIL_ADDRESS>Preeti Kharb4
<EMAIL_ADDRESS>1 School of Computational and Integrative Sciences,
Jawaharlal Nehru University, New Delhi 110067, India.
2 Indian Institute of Astrophysics, Koramangala II Block, Bangalore 560034,
India.
3 Department of Physics, Pondicherry University, R.V. Nagar, Kalapet, 605014,
Puducherry, India.
4 National Centre for Radio Astrophysics - Tata Institute of Fundamental
Research, Savitribai Phule Pune University Campus, Pune 411007, Maharashtra,
India.

###### Abstract

The Crab nebula is an astrophysical system that exhibits complex morphological
patterns at different observing frequencies. We carry out a systematic
investigation of the structural complexity of the nebula using publicly
available imaging data at radio and infrared frequencies. For the analysis, we
use the well-known multifractal detrended fluctuation analysis (MFDFA) in two
dimensions. We study the Crab nebula at various length scales to search for
local scaling behaviors. We find that both radio and infra-red data exhibit
long-range correlations, as expected from the underlying physics of the
supernova explosion and evolution. The correlations follow a power-law scaling
with length scales. The structural complexity is found to be multifractal in
nature, as evidenced by the dependence of the generalized Hurst exponent on
the order of the moments of the detrended fluctuation function. The nature and
strength of the multifractality vary with the observing frequency. By
repeating the analysis on shuffled data, we further probe the origin of the
multifractality in both the imaging data. For the radio data, we find that the
probability density function (PDF) is close to a Gaussian form, and hence the
multifractal behaviour is due to the differing nature of long-range
correlations of the large and small detrended fluctuation field values. In
contrast, we find that the infra-red data has a broad non-Gaussian PDF.
Consequently, its multifractal properties originate from the broad PDF as well
as the different correlations of large and small fluctuation values. Our
analysis thus provides a fresh perspective on the morphology of the Crab
nebula from a statistical physics viewpoint.

Keywords: Complex systems, Crab nebula, MFDFA, Hurst exponent, multifractal
spectrum

##  I Introduction

Natural systems are generally complex, consisting of a large number of
interlinked/integrated components of heterogenous properties Mitchell . The
individual components of such complex systems have non-linear characters
kwapien2012physical ; gershenson2005can . Their interaction can give rise to
emergent local behaviors at different scales. These local behaviors correspond
to scale dependent properties of the system gershenson2005can . An important
property of complex systems is self-organization, which can be captured by
fractal (scale independent self-similar) or multifractal (scale dependent
fractal) processes kauffman1993origins . Kantelhardt et al. Kantelhardt_2002
developed a numerical method known as the multifractal detrended fluctuation
analysis (MFDFA) to calculate various multifractal parameters for
characterizing multifractal scaling properties and detecting long-range
correlations in noisy, non-stationary time series data of complex systems. The
method is based on a generalization of the detrended fluctuation analysis
(DFA) initially given by Peng et al Peng_1994 . Gu and Zhou Gu_2006  extended
the conventional MFDFA to the two-dimensional multifractal detrended
fluctuation analysis (2D-MFDFA) to analyze higher dimensional fractals and
multifractals.

Astrophysics provides a multitude of examples of complex systems owing to the
different physical interactions and evolution time scales involved. Supernova
remnants are particularly interesting for questions related to complexity
because of their complicated morphological patterns that evolve with time. In
this paper, we focus our attention on the Crab nebula which is a well-known
remnant of a supernova that was recorded in 1054 AD Duyvendak:1942 . It is
located about 6500 light-years from the earth Trimble:1973  and has been well-
studied using multi-wavelength observations from radio frequencies to gamma
rays (see reviews Hester:2008 ; Buehler:2014 ). We choose to study the Crab
nebula from the viewpoint of complex systems because of its historical
importance and availability of imaging data with good spatial resolution in
several frequency bands. Several studies have been carried out to investigate
the morphological variation of the Crab nebula across different frequencies
(see e.g. Dubner:2017 ; Yeung_2019 ; Fang_2019 ). However, a quantitative
analysis of the morphological complexity of the Crab nebula is lacking in
literature.

The intricate morphology of the Crab nebula that can be easily discerned by
eye naturally leads to the question of its structural complexity. A complex
systems approach to the Crab nebula views the physical system as one system of
inter-connected components with their organization exhibiting emergent
behaviours instead of studying the dynamics of the individual components. To
investigate the structural complexity of the complex system of Crab nebula, we
adopt a multi-scale multifractal approach. A multifractal approach is very
well-suited to study and analyse the structural properties of the Crab nebula
since it provides a framework for detecting and identifying complex local
structures, and describing local singularities. We carry out a systematic
investigation of the structural complexity of the Crab nebula observed at
radio and infra-red frequencies using publicly available data with the
2D-MFDFA method. Our findings provide the local scaling behavior, nature of
correlations, origin of multifractality, heterogenous properties, and
organization in the structural system of the Crab nebula. Our results deepen
our understanding of the multi-scale physics operating in the Crab nebula.

The multifractal formalism is a well-established formalism for studying
multifractality in a wide variety of fields. It is used to study long-term
persistence in river precipitation kantelhardt2006long , dynamics of stock
markets jiang2019multifractal , anomalies in the dynamics of brain
islam2013multifractal , social media activities oh2022multifractal  and
species abundance distributions borda2002species . In astrophysics,
multifractal formalism is applied to study the large scale structure of the
universe Jones:1988 ; gaite2019fractal , behavior of galaxy clustering on
large scales pan2000large , the interstellar medium Elia_2018 , Gamma-ray
bursts Meredith1995 ; PhysRevA.40.5284 , flux correlations of pulsars
Eghdami_2018 , flux variability in the quasar 3C 273 Belete_2018 ,
gravitational wave signals detected by LIGO de_Freitas_2018 , to name a few.
The MFDFA, in particular, is a well-established method for studying
multifractality in data generated by various complex systems. The one-
dimensional MFDFA is used to study sunspot number fluctuations Movahed_2006 ,
investigate the statistical properties of the cosmic microwave background
radiation Movahed_2011 , and analyze the pseudorapidity distribution of ring-
like events of charged mesons Haldar . In material sciences, the 2D-MFDFA is
used to study the complexity and roughness of images of the surface of various
thin films Yadav2012 ; Yadav_2013 ; Yadav_2015 ; Hosseinabadi_2017 ; Das_2021a
, the surface texture of dental resin-based nanocomposites lu_2018  and cross-
correlations of material structures Xi_2017 . In earth sciences, the 2D-MFDFA
is used to investigate the long-range dependence of spatial patterns on urban
impervious surfaces Nie_2015 . In biomedical sciences, the 2D-MFDFA is applied
to images of tumor tissues Das_2021  to study the morphological changes with
the progression of cancer sahoo_2020 ; Sharma_2021 ; Nawn_2021 ; Reza_2015 .

This paper is organized as follows. Section II is devoted to outline the
methodology used in our analysis. We briefly introduce the concept of
multifractality, describe the 2D-MFDFA algorithm and the associated physical
interpretations. Section III contains a description of the physics of the Crab
nebula and the imaging data used in our analysis. In section IV, we present
our main analysis and results. We conclude with a summary and discuss the
implications of our results in section V.

##  II Methodology

In this section we briefly review multifractality and complexity measure,
followed by a presentation of the 2D-MFDFA algorithm and its associated
physical interpretations.

###  II.1 Complexity and multifractality

Consider a system of $N$ randomly interacting particles whose process is
defined by a random variable $y(x)$ with given $y(0)=0$. The process
$\\{y(x)\\}$ is said to be a self-affine process if the observable $y(x)$
obeys the following scaling behavior RePEc:cwl:cwldpp:1164

$\displaystyle\frac{y(cx)}{y(x)}=c^{D}\ ;\ 0<c<1,$ (1)

for any scale factor $c$. The scaling exponent $D>0$ represents the fractal
dimension or self-similarity dimension of the self-affine process. Such a
fractal system is characterized by a single scaling law at any scale of the
system. Fractal dimension is a parameter which provides a measure of
complexity llyod . Data generated by various complex systems, for instance,
chaotic spectra of atoms, records of astrophysical X-ray light sources, time
series of internet traffic, climate dynamics and records of heartbeat, exhibit
fluctuations on a broad range of scales. Such fluctuations often follow a
scaling relation of the type (1) over several orders of magnitude in both
equilibrium and non-equilibrium situations.

Now, if the self-affine process becomes scale dependent, which provides a rich
temporal heterogeneity in the structure, then we can generalize the process in
the following form

$\frac{y(cx)}{y(x)}=\Gamma(c)\ ;\ 0<c<1,$ (2)

where $y(x)$ and $\Gamma(c)$ are independent random functions. Now, for a
multifractal system, the observable $y(x)$ obeys Mandelbrot’s classical
multifractal rules RePEc:cwl:cwldpp:1164 , briefly described as follows. For
various scales $c_{1},c_{2},\dots,c_{n}$ of the multifractal system, the
scaling function $\Gamma(c)$ follows the property RePEc:cwl:cwldpp:1164

$\displaystyle\Gamma(c_{1}c_{2}\dots c_{n})$ $\displaystyle=\Gamma_{1}(c_{1})\
\Gamma_{2}(c_{2})\dots\Gamma_{n}(c_{n});$ $\displaystyle\
0<c_{1},c_{2},\dots,c_{n}<1.$ (3)

Using eq. (1) and (2), each local scale $c_{\kappa}$ has the local fractal
dimension $D_{\kappa}$ and the system shows multifractality as

$\Gamma_{\kappa}(c_{\kappa})\sim c_{\kappa}^{D_{\kappa}}.$ (4)

The multifractality becomes monofractality when $c_{1}=c_{2}=\dots=c_{n}=c$ in
eq. (3). Using eq. (4), we can write the function
$\Gamma=c^{D1+D_{2}+\dots+D_{n}}=c^{D}$, where $D=D_{1}+D_{2}+...+D_{n}$,
which is exactly eq. (1). This function thus provides homogenous fractal
structure defined by a single fractal property at various scales. Multifractal
analysis, on the otherhand, can describe the fluctuations-driven scale
dependent local patterns/clusters of a data field represented by a set of
scaling exponents corresponding to the local patterns/clusters.

###  II.2 Review of two-dimensional Multifractal Detrended Fluctuation
Analysis (2D-MFDFA)

We briefly review here the algorithm and physical interpretations of the two-
dimensional MFDFA method, following Kantelhardt_2002 ; Gu_2006

####  II.2.1 2D-MFDFA algorithm

Consider a two-dimensional compact space $S$, which is a subset of flat two-
dimensional space. For our purpose, $S$ is a rectangular region which is
subdivided into $M\times N$ equal-area square pixels. Let $d$ denote the data
field, which is a function on $S$. The $d$ is thus a two-dimensional $M\times
N$ array.

The algorithm for the 2D-MFDFA involves the following steps.

  1. 1.

Partition $S$ into $M_{s}\times N_{s}$ disjoint square subsets, with each
subset containing $s^{2}$ pixels. We refer to $s$ as the scale size. Let us
call each subset a superpixel. Let each superpixel be indexed by $I,J$, such
that $I=1,2,\ldots,M_{s}$ and $J=1,2,\ldots,N_{s}$. We assume that $M,N$ are
multiples of $s$ so that $M_{s}=M/s$ and $N_{s}=N/s$ are positive integers. We
then carry out the partitioning for different values of $s$. For a given $s$,
all superpixels contain the same number of pixels.

  2. 2.

Let $d_{I,J}$ denote the $s\times s$ data array within each superpixel $I,J$.
The cumulative sum $U_{I,J}$ of the data for each superpixel is defined by

$U_{I,J}(i,j)=\sum_{k_{1}=1}^{i}\sum_{k_{2}=1}^{j}d_{I,J}(k_{1},k_{2}),$ (5)

where $i,j=1,\ldots,s$ are indices for the pixels within the superpixel.

  3. 3.

To subtract the trend in each superpixel, we first fit $U_{I,J}(i,j)$ with a
linear bivariate polynomial function $U_{I,J}^{\rm fit}(i,j)$, given by

$U_{I,J}^{\rm fit}(i,j)=Ai+Bj+C,$ (6)

where $A,B,C$ are parameters determined by the fitting process. In general,
one can also use higher order polynomials for the fitting function. The
detrended fluctuation function that is defined in the steps below will differ
based on the order of the fitting function. We have chosen a linear form since
it is the simplest. Even though the numerical values of our derived quantities
are affected by the order of the fitting polynomial, the overall conclusions
of our results remain the same. We then calculate the residual

$\Delta U_{I,J}(i,j)=U_{I,J}(i,j)-U_{I,J}^{\rm fit}(i,j).$ (7)

Then, for a given $s$, the detrended fluctuation function of each superpixel,
which we denote by $G(I,J,s)$, is calculated as

$G^{2}(I,J,s)=\frac{1}{s^{2}}\sum_{i=1}^{s}\sum_{j=1}^{s}\Delta
U_{I,J}(i,j)^{2}.$ (8)

  4. 4.

Let $q$ be a non-zero real number. The detrended fluctuation function of order
$q$, denoted by $F_{q}(s)$, is calculated by averaging over all the
superpixels as

$F_{q}(s)=\left\\{\frac{1}{M_{s}N_{s}}\sum_{I=1}^{M_{s}}\sum_{J=1}^{N_{s}}[G(I,J,s)]^{q}\right\\}^{1/q}.$
(9)

For positive integer values of $q$, this expression reduces to the usual
definition of the $q^{th}$ order moment of $G(I,J,s)$.

For the case $q=0$, eq. (9) cannot be used since it diverges. Instead, we use
the following expression for $F_{0}(s)$

$\displaystyle
F_{0}(s)=\exp\left\\{\frac{1}{M_{s}N_{s}}\sum_{I=1}^{M_{s}}\sum_{J=1}^{N_{s}}\ln[G(I,J,s)]\right\\}.$
(10)

Since we use the positive square root of eq. (8), $F_{q}(s)$ is always real.

  5. 5.

For different values of $q$, the scaling relation between $F_{q}(s)$ and the
scale size $s$ can be expressed as

$F_{q}(s)\sim s^{h(q)},$ (11)

where the scaling exponent $h(q)$ is the generalized Hurst exponent. The
$h(q)$ is a measure of self-similarity symmetry and correlation present in the
data field. We restrict our attention here to data for which $F_{q}(s)$ are
increasing functions of $s$, or positive values of $h(q)$. This means that the
data of interest contains positive correlations.

####  II.2.2 Physical interpretations

As described above the 2D-MFDFA algorithm determines the scaling relation
between $F_{q}(s)$ and $s$, from which we can calculate $h(q)$. For a general
data field, $h$ can depend on both $q$ and $s$. From the behaviour of $h(q)$,
we can infer the following:

  * •

If the data contains long-range power-law correlations, then the dependence of
$F_{q}(s)$ on $s$ has a power-law form, and $h(q)$ is independent of $s$.

  * •

If $h(q)$ remains the same for varying $q$, then the data field has
monofractal scaling property. This means that the data or physical system
contains one structural element or pattern which is invariant under size scale
transformations. Then only one scaling exponent, namely $h$ describes the
self-similar scaling behaviour of the system.

  * •

If, however, $h$ has a dependence on $q$, then the data has multifractal
scaling behaviour. This implies that the data contains more than one
structural element, which are invariant under size scale transformations. This
further shows that small and large fluctuations scale differently with $s$.

  * •

For positive values of $q$, the dominant contributions to the sum on the
right-hand side of eq. 9 will come from superpixels containing large
fluctuations (or equivalently large deviations from the fitting function
$U_{I,J}^{\rm fit}$ given in step 3 in the previous subsection). So, for
$q>0$, $h(q)$ effectively describes the scaling behaviour of large
fluctuations. In contrast, for negative values of $q$, the dominant
contribution for the averaging in eq. 9 will come from superpixels containing
small values of fluctuations. Hence in the region $q<0$, $h(q)$ effectively
describes the scaling behaviour of small fluctuations.

  * •

Furthermore, given some data having multifractal properties, the value of
$h(q)$ is usually larger for $q<0$, compared to $q>0$. This can be explained
by the fact that the contributions to the averaging in eq. 9 from small
fluctuations typically vary more across the scale $s$, compared to large
fluctuations.

We next relate $h(q)$ to other measures of complexity used in the literature.
From the standard partition function-based multifractal formalism, the
classical multifractal scaling exponent $\tau$ is related to $h(q)$ as
Kantelhardt_2002

$\tau(q)=qh(q)-D_{f},$ (12)

where $D_{f}$ is the fractal dimension, which we take to be 2 for 2D images
Gu_2006 . If $h$ is independent of $q$, then $\tau$ is a linear function of
$q$. Therefore, a nonlinear shape of $\tau(q)$ is an indication of
multifractality of the system.

The H$\ddot{o}$lder exponent (also known as singularity strength), denoted by
$\alpha$, is calculated by taking the Legendre transformation of eq. (12) as

$\alpha(q)=\tau^{\prime}(q),$ (13)

where the prime denotes derivative with respect to $q$. We get

$\alpha(q)=h(q)+qh^{\prime}(q).$ (14)

$\alpha$ is a measure of the shape of $h(q)$ since it is a function of
$h^{\prime}(q)$. Different values of $\alpha$ characterise different parts of
the system roughly, since small, intermediate or large values of the
fluctuation field will contribute to different values of $\alpha$. From eq. 14
we can see that $\alpha$ can take both positive and negative real values
depending on the sign of the second term and its relative value with respect
to the first term. Note that for a monofractal system, we have
$h^{\prime}(q)=0$ since $h$ is a constant, and as a consequence, $\alpha$ has
only one value given by $\alpha=h$. Therefore, a variation of $\alpha$ with
$q$ signifies multifractality, and the range over which it varies quantifies
the strength of multifractality.

The last quantity we use is the multifractal spectrum (also called the
singularity spectrum) $f(\alpha)$, defined by

$f(\alpha)=q\alpha-\tau(q).$ (15)

This quantity is useful in extracting information about the symmetry between
small and large fluctuations.

In order to gain an intuitive understanding of the quantities we have defined
so far, let us consider two toy examples of multifractal complex systems whose
$h(q)$ are given below.

Example A.

Let $h(q)$ be given by

$h(q)=2.2-0.08\,{\rm tan}^{-1}\left(\frac{q}{2}\right).$ (16)

Example B.

Let $h(q)$ be given by

$h(q)=2.2-0.08\,{\rm
tan}^{-1}\left(\frac{q}{2}\right)+0.04\,e^{-(q+1)^{2}/2}.$ (17)

These examples have been chosen so as to capture the essence of our results in
section IV. Figure 1 shows $h$, $h^{\prime}$, $\tau$ and $\alpha$ versus $q$,
and $f$ versus $\alpha$ for the two examples. We have chosen $D_{f}=2$ for
both the examples. If $h(q)$ is a well-behaved monotonic function of $q$, such
as in example A, then all the other quantities are single-valued. However, if
$h(q)$ is not monotonic, such as in example B, then $h^{\prime}$ and $\alpha$
become multi-valued. As a consequence, the multifractal spectrum $f(\alpha)$
is not a well-behaved function.

Example A Example B

Figure 1: $h,\ h^{\prime},\tau,\ \alpha$ are plotted versus $q$, and $f$ is
plotted versus $\alpha$, for the toy examples given by eqs. 16 and 17.

In case, $f(\alpha)$ has a well-defined peak located at $\alpha=\alpha_{0}$,
then we can quantify its symmetry about $\alpha_{0}$. Let $\alpha_{\rm min}$
be the smallest value of $\alpha$, and $\alpha_{\rm max}$ denote the value of
$\alpha$ where $f(\alpha_{\rm max})=f(\alpha_{\rm min})$. Then, let $\chi$
denote the skewness parameter, given by

$\chi\equiv\frac{|\alpha_{\rm max}-\alpha_{0}|}{|\alpha_{\rm
min}-\alpha_{0}|}.$ (18)

The $f(\alpha)$ is right-skewed if $\chi>1$, left-skewed if $\chi<1$, and
symmetrical if $\chi=1$. If $\chi<1$, then it shows that the scaling of the
system is dominated by large fluctuations (smaller generalised Hurst
exponents). However, if $\chi>1$, then it shows the dominance of scaling by
small fluctuations (higher generalized Hurst exponents).

##  III The Crab nebula - physical processes and observed data

In this section, we present a brief description of the physical processes that
are relevant to understanding the Crab nebula. Then we give details of the
imaging data that we use for our analysis.

###  III.1 Physics of the Crab nebula

The Crab nebula belongs to the class of supernova remnants known as pulsar
wind nebulae (PWN). Such a nebula inherits its morphology and spectrum from
the following factors.

  1. 1.

During the supernova explosion of the progenitor star, the stellar material
gets ejected (known as ejecta). The ejecta sweeps up the interstellar medium,
creates a heated shock front, slows down and eventually freely expands into
the interstellar regions beyond the confines of the nebula.

  2. 2.

The collapse of the core of the progenitor star results in a rotating
magnetized neutron star, known as the Crab pulsar. The magnetic field causes
the charged ejecta particles to accelerate and emit synchrotron radiation.

  3. 3.

The Crab pulsar spins at the rate of 33 ms Rosat:1995 . The pulsar emits non-
thermal plasma, known as pulsar wind, which gets accelerated to high
velocities by the magnetic field. The plasma pushes into the surrounding
ejecta in the nebula and undergoes turbulent mixing. This leads to Rayleigh-
Taylor instabilities Chevalier:1975  that give shape to the Crab nebula’s
filamentary network and wispy structures.

The Crab nebula appears like a bubble whose shell expands radially over time
as the ejected material moves outward, while the interior structure consists
of an expanding network of filaments and wispy structures when viewed across a
wide range of frequencies. Freely expanding ejecta is also expected to be
present beyond the visible Crab. Here we focus on radio and infra-red
observations of the Crab nebula. The origin of these emissions are described
below.

Radio frequencies: The emission in radio frequencies is primarily the non-
thermal synchrotron radiation from the shocked pulsar wind; charged particles
are accelerated due to the magnetic field of the pulsar. Visually, one can see
the expanding network of filaments and fine-scale wispy features (Figure 1,
top panel). The source of most of the thermal emission from the filaments is
understood to be the photo-ionization by the hard continuum from the
synchrotron PWN emission (see e.g., Hester:1996, ). In addition, radiative
cooling behind the shock driven by the PWN into the freely expanding ejecta
also gives rise to radio emission Sankrit:1998 .

Infra-red (IR) frequencies: The near- and mid-IR emissions of the Crab nebula
are dominated by continuum synchrotron radiation. The emission observed in the
far-IR may be attributed to line emission or dust particles Temim:2006 , which
form when the supernova ejecta cool and condense. The dust particles get
heated by the pulsar wind and emit thermally in the far-IR.

###  III.2 Observational data of the Crab nebula

Figure 2: Top: VLA image of Crab nebula at 4.76 GHz. Bottom: Spitzer image of
Crab nebula at 22 $\mu$m ($\sim 1.4\times 10^{4}$ GHz).

We focus on the data of the Crab nebula observed at radio and IR frequencies.
We have made use of publicly available imaging data for this.

  1. 1.

We obtained the radio imaging data at 4.76 GHz with an angular resolution of
0.7 arcseconds from the Very Large Array (VLA) 111VLA is operated by the
National Radio Astronomy Observatory (NRAO), USA. The National Radio Astronomy
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. data archive. We used
the Crab nebula imaging data from the VLA image archive
222https://archive.nrao.edu/archive/archiveimage.html. These data have
undergone full calibration and reduction. The image is shown in the top panel
of Figure 2. The surface brightness in the image is in units of Jy beam-1. We
applied the 2D-MFDFA algorithm to this imaging data.

  2. 2.

We obtained the IR data from NASA’s Spitzer Space Telescope
333https://www.nasa.gov/mission${\\_}$pages/spitzer/main/index.html data
archive. We used the IR image at $22~{}\mu$m ($\sim 1.4\times 10^{4}$ GHz)
from the Spitzer Heritage Archive
444https://sha.ipac.caltech.edu/applications/Spitzer/SHA/. This image is shown
in the bottom panel of Figure 2. The surface brightness in the image is in
units of MJy sr-1. The image has an angular resolution of 0.86 arcseconds. We
found that bright foreground stars were visible in the FITS image file. They
needed to be removed before the MFDFA analysis. To remove them, we calculated
the first derivative of the field and identified the pixels with high
derivative values. Bright point sources contribute to these high values in the
derivative field. These pixels were then masked and inpainted with the average
of the field values of the surrounding pixels. In this way, we could minimize
the contamination from the foreground stars and bright point objects and
prepare the image for the MFDFA analysis.

##  IV Analysis and results

We apply the 2D-MFDFA algorithm to both the radio and IR imaging data of the
Crab nebula and study their multifractal properties. Using the intensity or
field value at each pixel, we first calculate the fluctuation functions
$F_{q}$ defined by eqs. 9 and 10, and from them, we obtain the generalized
Hurst exponents $h(q)$. Then, from $h(q)$, we obtain the derived quantities -
the classical scaling exponents $\tau(q)$, the H$\ddot{o}$lder exponents
$\alpha$, the multifractal spectra $f(\alpha)$ and the skewness parameter
$\chi$.

The multifractal analysis using the 2D-MFDFA algorithm was performed using the
MATLAB$\\_$R2018a platform. The 2D-MFDFA algorithm calculates $h(q)$ by the
method of linear fitting using the inbuilt regression statistics regstats of
MATLAB. In particular, it uses the inbuilt routine tstat based on Student’s
$t$-distribution. The code also calculates the error bars for $h$ as the
standard errors from tstat.

In order to study the spatial scaling behaviors of the multifractal variables
over different spatial regions of the Crab nebula, we repeat the 2D-MFDFA
analysis on smaller partitions of the image for both radio and IR data. This
is carried out by dividing the images into $N_{a}\equiv 2^{a}$ equal-area
portions, where $a\geq 0$ is an even integer that determines the number of
partitions. We refer to $a$ as the partition scale parameter (so as not to
confuse with the size scale $s$). We use values $a=0,2,4,6$ since the number
of pixels in each partition becomes too small beyond that. The spatial scaling
behavior will reveal whether the same morphological rules are followed or not
at different parts of the system. It will also enable us to identify the
presence of heterogenous properties at different length scales.

###  IV.1 Complexity of Crab nebula at radio frequency of 4.76 GHz

The Crab nebula image at the radio frequency of $4.76$ GHz (shown in the top
panel of Fig. 2) consists of $651\times 741$ number of pixels. The smallest
partition corresponding to $a=6$ has $81\times 87$ number of pixels.

Figure 3 shows the log-log plots of the fluctuation functions $F_{q}$ as
functions of the scale size $s$, for different partition scales $a=0,2,4$
(indicated against each panel) for different values of the order of
statistical moments $q$ (shown by different colours). For all the partition
scales $a$, it is observed that $F_{q}$ has power law dependence on $s$ and
the slopes of $F_{q}$ vary for different $q$. This is a signature of the
presence of multiple local fractal behaviours at different local domains $s$,
across all $a$. The spread of the curves along $F_{q}$, i.e. $\Delta F_{q}$
for small values of $s$ is much larger than the spread in $F_{q}$ at large
values of $s$. This shows that the role of fluctuations is more evident in
smaller local patterns/clusters of the system, which provide distinct local
behaviors. However, at large values of $s$, this spread $\Delta
F_{q}\rightarrow 0$, where the system converges to a single behavior.

Figure 3:  Radio data of Crab Nebula at 4.76 GHz \- The fluctuation function
$F_{q}$ is plotted versus scale size $s$, for different values of $q$, and
different partitions of the radio image corresponding to $a=0,2,4$.

The slopes of the log-log $F_{q}$-$s$ plots in Figure 3 give the values of the
generalized Hurst exponent $h$ as a function of $q$, as defined by eq. (11).
The left column of Figure 4 shows $h$ versus $q$, ranging from -40 to 40 in
steps of one. The cases of $a=0,2,4$ and $6$ are shown in the panels from top
to bottom. The number of plots (shown in different colours) in each panel
corresponds to the number of partitions of the radio image for each $a$
(except for $a=6$ for which we show 20 plots to avoid overcrowding). The
vertical dashed lines in the panels showing $h(q)$ and $\tau(q)$ corresponds
to $q\sim 0$. We observe significant dependence of $h$ on $q$ for all $a$,
indicating distinct local behaviours at various partition scales $a$. This
shows that small and large fluctuations of the emission at the radio frequency
have very different scaling natures.

As described in section II.2.2, for positive (negative) values of $q$, $h$
describes the scaling behaviour of the segments with large (small)
fluctuations. From the left column of Figure 4, we find that the values of
$h(q)$ are larger for $q<0$ than that of $q>0$ for all partition scales $a$.
Hence, local behaviors of the patterns/clusters at various $a$ are found to be
sensitive to small fluctuations. These small fluctuations drive the local
behaviors significantly different compared to the large fluctuations in the
radio data KANTELHARDT200287 . The significant spread in $h$ as a function of
$q$, i.e. $\Delta h$ at each $a$ shows that scaling laws of the
patterns/clusters at various $a$ are distinctly different, which shows a
heterogenous distribution of the patterns/clusters in various partitions of
the system. Further, since the values of $h(q)$ for $q<0$ are larger than that
of $q>0$, the correlated memory in the patterns/clusters/system driven by
small fluctuations persists longer as compared to that driven by large
fluctuations RePEc:cwl:cwldpp:1164 ; Kantelhardt_2002 .

Visually, in the radio image of the Crab nebula (see top panel of Figure 2) we
can discern that in general large fluctuation values track the structurally
dominant network of filaments and are correlated over large length scales. In
contrast, the smaller fluctuation values track the small scale wispy
structures and are correlated over shorter length scales. Hence, our finding
that $h(q>0)<h(q<0)$ is not surprising. It encapsulates the long range
correlations of the filamentary network, and the shorter range correlations as
well as stronger scale dependence of the wispy structures. Further, we observe
scale dependent heterogenous patterns in the various partitions, which show
different local multifractal scaling laws of the patterns.

In all the plots of $h(q)$, we see that it asymptotes to constant values as
$|q|\to\infty$. This implies that small fluctuation values tend towards the
same scaling behavior, and so do large fluctuations. Let $h_{\pm}$ denote the
asymptotic values at $q\to\pm\infty$. We observe that $h_{-}$ varies
considerably large across the different partitions for $a\geq 2$, while
$h_{+}$ shows relatively less variations such that $h_{\pm}\approx
h_{\pm}(a,m)$, where $m$ is the partition index. This means that the
structural properties associated with small fluctuations vary considerably
large amongst the different partitions of the radio image, while large
fluctuations are comparable. Further, since the values of $h_{-}>h_{+}$, the
persistence of the patterns/clusters in each partition driven by small
fluctuations has longer memory as compared to large fluctuations. We thus
expect to see more heterogenous or dissimilar structures with distinct local
scaling laws across the different partitions of the radio image significantly
contributed by small fluctuations. This corroborates what we described in the
previous paragraph. We also find the largest variation of $h_{-}$ across the
different partitions for $a=4$. A closer look at the values of $h_{\pm}$ for
different partitions of the radio image will be discussed in section IV.3
where we make a comparison with the results for IR data.

Figure 4: Multifractal parameters for the radio image of the Crab nebula.
Column 1: Generalized Hurst exponents $h$ versus $q$ for $a=0,2,4,6$. Column
2: The classical scaling exponents $\tau$ versus $q$ for $a=0,2,4,6$. Column
3: Multifractal spectra $f$ versus the singular exponents $\alpha$ for
$a=0,2,4,6$. Different colors represent $N_{a}=2^{a}$ partitions for each
value of $a$. The vertical dashed lines in the panels for $h$ and $\tau$
represent the value $q\sim 0$, where the slopes make a transition.

In the middle column of Figure 4, we plot the classical scaling exponent
$\tau$ versus $q$ for different values of $a$. In all the plots, we observe
that $\tau$ does not depend on $q$ linearly. Eq. (12) tells us that if $h$ is
independent of $q$, then $\tau$ should be a linear function of $q$. But $h$ is
dependent on $q$, and it exhibits a step-like transition around $q\sim 0$ (see
the left column of Figure 4). Therefore, we expect that $\tau$ will also make
a transition around $q\sim 0$. This is indeed what we see in the plots of
$\tau$, where we observe two linear regimes with a slope change around $q\sim
0$ (indicated by the vertical dashed lines). This value marks the transition
of the scaling nature between large and small fluctuations exhibiting
multifractal properties.

Lastly, in the right column of Figure 4, we plot the multifractal spectra
$f(\alpha)$ versus the singularity strength $\alpha$ for different partition
scales $a$. Since $h(q)$ is a monotonous function, $\alpha$ is single valued.
We observe that all the curves have their maxima, denoted by $f_{\rm max}$ at
roughly $\alpha=\alpha_{0}\sim 2$. Also, visually it is clear that all $f$
curves are right-skewed around $\alpha_{0}$, showing high degree of complexity
in the structure of each partition defined by $a$ stovsic2015multifractal .
Further, the widths of $\alpha$ for a fixed value of $f(\alpha)$ of all the
partitions of each $a$ are significantly different, which shows heterogeneous
structures in the partitions with different scaling behaviors
shimizu2002multifractal . The values of the skewness parameters $\chi$ for
each $f$ are calculated using eq. (18). We will discuss $\chi$ in detail in
section IV.3.

####  IV.1.1 Origin of the multifractal nature for radio data

Figure 5: Left: PDF of the radio data. The $x$-axis is in units of Jy beam-1.
The other three panels show $h(q)$ (second from left), $\tau(q)$ (third from
left) and $f(\alpha)$ (right) for the radio data for the original and shuffled
cases.

We now probe the physical origin of the multifractal nature that is obtained
for the radio data. The observed multifractality can be due to two sources-
(1) distribution-induced multifractality caused by a broad probability density
function (PDF) of the data, and (2) correlation-induced multifractality caused
by a different nature of long-range correlations for the large and small
fluctuations. First, in order to isolate the effect of correlation, we
randomly shuffle the data to destroy all spatial correlations. We will use
superscripts ‘orig’ and ‘shuf’ to indicate quantities obtained from the
original and shuffled data respectively. Then we calculate the multifractal
quantities $h(q),\ \tau(q)$ and $f(\alpha)$ for the shuffled data. To have a
handle on what to expect of the multifractal parameters for uncorrelated
Gaussian data we have calculated $h(q),\ \tau(q)$ and $f(\alpha)$ for one
realization of an uncorrelated Gaussian random field in appendix A.

Suppose $F_{q}^{orig}$ and $F_{q}^{shuf}$ are the fluctuation functions
corresponding to the original and shuffled data with their respective
generalized Hurst exponents $h^{orig}$ and $h^{shuf}$. Then the ratio
$\frac{F_{q}^{orig}}{F_{q}^{shuf}}\sim s^{h^{corr}}$, where
$h^{corr}=h^{orig}-h^{shuf}$. We can analyse the origin of multifractality in
the data from $h^{corr}$ as: (i) if $h^{corr}=0$ such that
$h^{orig}=h^{shuf}$, then the cause of multifractality is the fatness of the
PDF, and (ii) if $h^{corr}=h^{corr}(q)\neq 0$ and $h^{shuf}=h^{shuf}(q)$, then
the origin of multifractality is due to both $q$-dependent long-range
correlations coming from short and long range fluctuations as well as a broad
PDF in the data, and (iii) if $h^{shuf}=2$, then the multifractality in the
data is due to only long-range correlations Kantelhardt_2002 ;
shang2008detecting .

In the left most panel of Figure 5, we show the PDF for the radio data. We
find that it is close to Gaussian. From the result in appendix A, we then
expect the shuffled data to behave close to a monofractal and hence to find
$h(q)$ to be almost constant with value 2. The second from left panel of
Figure 5 shows $h(q)$ for the two cases of original and shuffled data. We
indeed find $h^{shuf}(q)$ has weak $q$-dependence with
$h^{shuf}(q)\rightarrow{\rm constant}\rightarrow 2$ for the shuffled data.
Hence, $h^{corr}\rightarrow h^{orig}(q)-2$. Similarly, we plot $\tau(q)$ and
$f(\alpha)$ for the two cases of original and shuffled data respectively in
the third from left and right most panels of Figure 5. These are in agreement
with what we expect from Appendix A. Thus we conclude that for the radio data,
the origin of multifractality originates from long-range correlations of large
and small fluctuations in the data.

###  IV.2 Complexity of Crab Nebula at IR frequency $1.4\times 10^{4}$ GHz

Figure 6: Same as Figure 3 but for the IR image.

We now focus on the image of the Crab nebula at the infrared frequency of
$1.4\times 10^{4}$ GHz from the Spitzer Space Telescope (bottom panel of
Figure 2). It consists of $451\times 351$ number of pixels, which is smaller
than that of the radio image.

Figure 7: Same as Figure 4 but for the IR image. The behaviors of $h(q),\
\tau(q)$ and $f(\alpha)$ are quite distinct from the radio data.

Figure 6 shows the log-log plots of $F_{q}$ versus $s$ for different values of
$q$ and $a$. For all partition scales $a=0,2,4$, we observe that the slopes of
the $F_{q}$ graphs vary with different values of $q$. Thus $F_{q}$ also shows
multiple local fractal behaviors at various local domains $s$ across all
partition scales $a$, as observed in the radio data.

The left column of Figure 7 shows the variations of $h$ with $q$. As in the
case of the radio data, we observe that $h$ depends on $q$ significantly for
all $a$, indicating distinct local behaviors at various partition scales $a$.
We find that the values of $h(q)$ are larger for $q<0$ than that of $q>0$ for
all $a$. Hence, the local behaviors of the patterns/clusters at various $a$
are found to be sensitive to small fluctuations. The significant spread in $h$
as a function of $q$, i.e. $\Delta h$ at each $a$ shows that scaling laws of
the patterns/clusters at various $a$ are distinct, which shows a heterogenous
distribution of the patterns/clusters in various partitions of the system.
Further, since the values of $h(q)$ for $q<0$ are larger than that of $q>0$,
the correlated memory in the patterns/clusters/system driven by small
fluctuations persists longer as compared to that driven by large fluctuations
RePEc:cwl:cwldpp:1164 ; Kantelhardt_2002 . We again find that $h(q)$
asymptotes to constant values as $|q|\to\infty$. We will compare these values
with those of the radio data in the next subsection.

The middle column of Figure 7 shows $\tau$ versus $q$ for different partition
scales $a$. As expected, in all the plots we observe that $\tau$ has non-
linear dependence on $q$, demonstrating the multifractal nature of the data.
As before, this behavior can be explained with eq. (12). Since $h$ depends on
$q$, $\tau(q)$ has a transition around $q\sim 0$ where the value of $h$ has a
step, marking the transition of the scaling nature of large and small
fluctuations of the emission at IR frequency.

The right column of Figure 7 shows the multifractal spectra $f$ versus
$\alpha$ across different $a$. The $f(\alpha)$ curves have their maxima at
$\alpha=\alpha_{0}\sim 2$. All the curves are observed to be right-skewed
around $\alpha_{0}$ showing high degree of complexity in the structures of
each partition defined by $a$ stovsic2015multifractal . Further, the widths of
$\alpha$ for a fixed value of $f(\alpha)$ of all the partitions of each $a$
are significantly different, which shows heterogeneous structures in the
partitions with different scaling behaviors shimizu2002multifractal . We will
further discuss the skewness of $f$ in section IV.3.

We find similar behaviors of the multifractal parameters $F_{q},h,\tau,f$ in
Fig. 6 and Fig. 7 when we analyzed the original IR data that include
contamination from stars. This implies that our results are robust against
contamination by point sources.

####  IV.2.1 Origin of the multifractal nature for IR data

We now probe the physical origin of the multifractal nature that is obtained
for the IR data. As in the case of the radio data, we calculate the
multifractal quantities $h(q),\ \tau(q)$ and $f(\alpha)$ for the original and
shuffled data and compare the results.

In the left most panel of Figure 8, we show the PDF of the IR data. We find
that it is not Gaussian. The second from the left panel of Figure 8 shows
$h(q)$ for the two cases of original and shuffled data. We find that $h^{\rm
shuf}(q)$ varies with $q$, as expected from the broad PDF of the shuffled
data. The $h^{corr}\rightarrow h^{corr}(q)\neq 0$. The other two multifractal
quantities $\tau(q)$ and $f(\alpha)$ are shown in the third from the left and
the right most panels of Figure 8. We conclude that the observed
multifractality of the IR data originates from the broad PDF as well as the
long-range correlations of $q$-dependent large and small fluctuations in the
system.

Figure 8: Left: PDF of the IR data. The $x$-axis is in units of MJy
steradian-1. The other three panels show $h(q)$ (second from left), $\tau(q)$
(third from left) and $f(\alpha)$ (right) for the IR data for the original and
shuffled cases.

###  IV.3 Comparison of the two Crab nebula data at radio and IR frequencies

Figure 9: Top panels show the partitions of the radio (left) and IR (right)
images of the Crab nebula (same as Figure 2). The bottom panels show $h_{\pm}$
for each partition indexed by $i_{a}$, for $a=2,4$. The black dashed
horizontal line shows $h=2$ for reference.

In order to compare the multifractal properties of the radio and IR images of
the Crab nebula, we first examine the values of $h{\pm}$ for the different
partitions of the two images. The top panels of Figure 9 show the same radio
and IR images as Figure 2, but with the partitions numbered for the cases of
$a=2$ (orange) and $a=4$ (yellow). In the latter case, the partitions are
numbered in a nested manner so that sequential groups of four are mapped
hierarchically to the corresponding larger partition of $a=2$. Note that the
spatial regions corresponding to the same partition index for the radio and IR
images need not be the same since the fields of view of the radio and IR
observations are not identical. However, we can meaningfully compare the trend
across all the partitions.

From the bottom panels of Figure 9, we can see that for $a=2$, the radio image
has larger values of $h_{-}$ and smaller values of $h_{+}$ than the IR image.
This captures the wider variation of scaling behaviour between large and small
fluctuations values of the radio data as compared to the IR data, already
discussed earlier. This is corroborated by visual inspection of the two
images. If we closely look at the smaller partition scale structures ($a=2$)
of the Crab nebula observed at radio and IR frequencies (top panels of Figure
9), we can easily see that the radio image has more heterogenous structures
across the partitions in the form of filaments and wisps compared to the dust
emission at IR frequencies. Moreover, the filamentary network of the radio
image has longer range correlations, compared to the IR data. This can be
explained by the fact that the materials that emit synchrotron radiation at
radio frequencies, which are the charged particles of the supernova ejecta and
pulsar wind, are distributed over larger spatial scales in comparison to dust
and line emitters responsible for the IR emissions. At $a=4$, $h_{-}$ for both
radio and IR varies randomly. This implies that the structures across the
different parts of the Crab nebula are highly heterogenous when observed at
both radio and IR frequencies. These highly heterogenous structures at various
length scales make the Crab nebula a structural complex physical system.

Next, we focus on the values of $h$ in the regime around $q=0$, where it
exhibits a transition. Let the average of $h(q)$ over $q$ in a suitable regime
$-q_{c}<q<q_{c}$, with $q_{c}$ is a positive variable, be given by

$\bar{h}\equiv\frac{1}{\int_{-q_{c}}^{q_{c}}{\rm d}q}\int_{-q_{c}}^{q_{c}}{\rm
d}q\,h(q).$ (19)

In the limit $q_{c}\to\infty$, $\bar{h}$ will be dominated by the asymptotic
values, and we simply get $\bar{h}\sim(h_{-}+h_{+})/2$. In this limit, we lose
the information of the variation or shape of $h(q)$ in the intermediate $q$
values. So we focus on finite $q_{c}$ and choose $q_{c}=40$ as the value of
$q$, where $h$ becomes effectively constant for each partition scale $a$. This
is akin to weighted averaging, where the weight kernel is the top hat function
centered at $q=0$. Since $q$ takes integer values here, we need the
discretized version of eq. 19. Let $i_{q}=1,\ldots,N_{q}$ index the values of
$q$, where $N_{q}=81$ denotes the total number of $q$ values considered. Let
$i_{a}=1,\ldots,N_{a}$ index each partition of the image for a given $a$. Then
for each $i_{a}$, the discrete form of eq. 19 is

$\bar{h}_{i_{a}}\equiv\frac{1}{N_{q}}\sum_{i_{q}=1}^{N_{q}}h_{i_{a}}(q).$ (20)

For $\tau$, there are no asymptotically constant values. Hence, there is no
natural cut-off value of $q_{c}$ to define the range $-q_{c}<q<q_{c}$ over
which to average $\tau$. We will choose the same $q_{c}$ that is defined for
$h$. So we have

$\bar{\tau}_{i_{a}}\equiv\frac{1}{N_{q}}\sum_{i_{q}=1}^{N_{q}}\tau_{i_{a}}(q).$
(21)

Let $\chi_{i_{a}}$ denote the skewness parameter $\chi$ in each partition
$i_{a}$. Let $\bar{X}_{i_{a}}$ denote either of the three quantities -
$\bar{h}_{i_{a}}$, $\bar{\tau}_{i_{a}}$ or $\chi_{i_{a}}$. Then we average
them over all the partitions for each $a$, as

$\langle X\rangle\equiv\frac{1}{N_{a}}\sum_{i_{a}=1}^{N_{a}}\bar{X}_{i_{a}}.$
(22)

Figure 10: Scaling with respect to $a$ of the averaged quantities $\langle
h\rangle$ (top panel), $\langle\tau\rangle$ (middle panel) and
$\langle\chi\rangle$ (bottom panel) for radio (blue) and IR (red) data.

Figure 10 shows $\langle h\rangle$ (top panel), $\langle\tau\rangle$ (middle
panel) and $\langle\chi\rangle$ (bottom panel) for the radio image (blue) and
IR image (red) as functions of partition scales $a$. The error bars shown are
the standard errors. We observe that $\langle h\rangle$, for both radio and
IR, vary with $a$, with radio having larger values than IR. This result
correlates with the value of $(h_{-}+h_{+})/2$ seen in Figure 9, with the
larger contributions coming from $h_{-}$. This implies that the radio image
has stronger small scale scaling behaviors than the IR data, showing the
presence of more fine-scale structures with well-defined local scaling laws in
the radio data than IR data.

Further, let $h_{0}$ denote the largest value of $\langle h\rangle$ chosen
from the different $a$’s. Let us normalize $\langle h\rangle$ by $h_{0}$ and
denote

$\langle h\rangle_{\rm norm}\equiv\langle h\rangle/h_{0}.$ (23)

If $\langle h\rangle_{\rm norm}\in(0.5,1)$, then this implies a strong
positive correlation in the system Kantelhardt_2002 , with long memory of
persistence in the states of the system RePEc:cwl:cwldpp:1164 ;
kantelhardt2006long . From the top panel of Figure 10, for the radio data,
$h_{0}=2.28$, which is the value of $\langle h\rangle$ at $a=0$. Using this,
we find that $\langle h\rangle_{\rm norm}$ lies in the range [0.91,1] for all
$a$ for the radio data. For the IR data, $h_{0}=4.16$, which is the value of
$\langle h\rangle$ at $a=6$. We obtain the corresponding range for $\langle
h\rangle_{\rm norm}$ to be $[0.73,1]$. Both these ranges are greater than 0.5
and close to unity. This implies strong positive correlations Kantelhardt_2002
in the Crab nebula structural system. Again, this result shows that radio data
has stronger positive correlation than IR data. Further, the decrease in
$\langle h\rangle$ as $a$ increases shows that finer structures in the smaller
partitions of the system lose their correlations and local structural
memories, which is expected. In the asymptotic limit
$\displaystyle\lim_{a\rightarrow\infty}|\langle h\rangle|\rightarrow 2$, it
shows memoryless structures whose underlying processes are Brownian motions
(see Appendix A) RePEc:cwl:cwldpp:1164 ; Kantelhardt_2002 ;
kantelhardt2006long . Again, the curve of $\langle h\rangle$ approximately
follows a power-law behaviour with $a$ as $\langle h\rangle\sim a^{-\gamma}$,
where $\gamma$ is the power-law exponent. This shows a hierarchical
organization of the heterogenous structures in the Crab Nebula structural
system s1992fractal ; ravasz2003hierarchical ; jones2005scaling . This
behaviour is similar to the Carpenter’s law as predicted in astrophysical data
analyses de1971large ; carpenter1931cluster ; jones2005scaling .

The middle panel of Figure 10 shows that the average classical scaling
exponent $\langle\tau\rangle$ (for both radio and IR) also varies with $a$,
which again implies a heterogeneity in the structure of the underlying system.
From the bottom panel of Figure 10, $\langle\chi\rangle>2$ for all $a$, which
implies the richness of complexity of the Crab nebula at both radio and
infrared frequencies. This shows the dominance of scaling of small
fluctuations or the presence of fine structures in the Crab nebula. The random
distribution of the $\langle\chi\rangle$ parameter also implies a
heterogeneous property of the structure visible in the Crab Nebula. This can
be accounted for by the non-equilibrium dynamics of the underlying system.

##  V Conclusions

We have investigated the structural complexity of the Crab Nebula at the radio
frequency of $4.76$ GHz and infrared frequency of $1.4\times 10^{4}$ GHz using
publicly available imaging data. In particular, we have analyzed the data
using the 2D-MFDFA approach. We have investigated the structural system at
various length scales for local scaling behaviors. Our results confirm that
the structure of the Crab nebula has a rich complexity characterized by
multifractal nature that varies with the frequency of observation. We have
studied the origins of multifractality in both the radio and IR data. We have
found that the multifractal property of the radio data arises from the
difference in the nature of long range correlations (and consequent scaling
with the scale parameter $s$) of large and small field values in the data. The
multifractal nature of the IR data, on the other hand, is sourced by both the
different nature of long-range correlations of large and small values and
broad PDF of the data.

We have made a systematic comparison of the multifractal properties of the
Crab nebula at radio and infrared frequencies. Not surprisingly, our results
show strong positive correlation in both radio and IR frequencies. The
relative difference between long range correlations of large and small values
is quite different between the two data which can be traced back to the
different materials which emit at these two frequencies. The variations of the
multifractal parameters, namely the generalized Hurst exponents $h$ calculated
from the fluctuation functions $F_{q}$, the classical scaling exponents $\tau$
and the skewness parameter $\chi$ across varying length scales shows a
heterogeneity in the structure across different parts of the visible Crab
Nebula. This can be accounted to the Crab nebula being a non-equilibrium,
highly dynamic structural system.

What we have done in this paper is essentially quantified how the physics of
the evolution of the Crab nebula as a supernova remnant manifests in the
language of complex systems. The 2D-MFDFA approach we have used in the study
is found to be a good method to characterize the complexity of an
astrophysical system with complex morphological features. Our multifractal
results inform us that the rich fine structures in the Crab nebula ruled by
local scaling behaviors show a self-organization at different length scales of
the structural system. Long range correlations manifest as self-similar
fractals and are identified as signatures of self-organized criticality
bak1988self ; bak1989physics . Our results thus deepen our understanding of
the multi-scale physics that is operating in the Crab nebula.

This paper represents a proof of concept of the insights that can be obtained
from viewing the Crab nebula as a complex system. However, it has some
limitations. First, our analysis considers the Crab as a two-dimensional
object and ignores its three-dimensional structure. It will be interesting to
study the full 3D structure using reconstructions such as those given by
Martin:2021 . Secondly, the Crab nebula is a rapidly evolving system on
astrophysical time scales. The data that we analyze here are observations
taken at specific times, and we do not address here the question of the time
evolution of the properties related to complexity. In order to extract the
full potential of the method, our work can be extended in the following
directions. Given the wealth of data available for this astrophysical object,
an analysis across the full range of frequencies will be valuable and will be
carried out in the future. It opens up a slew of questions that will enhance
our understanding of supernova remnants, and we plan to carry out a systematic
investigation in the near future.

## Authors’ contributions

The conceptualization of the present work is done by ALC, PC and RKBS. PC and
FR processed the data of the Crab nebula for multifractal analysis. ALC and FR
performed the multifractal analysis. ALC, PC and FR prepared the associated
figures. PK vetted the observed data. All authors wrote, discussed and
approved the final manuscript.

## Acknowledgements

We thank Wei-Xing Zhou for providing the 2D-MFDFA code for our multifractal
analysis and for helpful discussion. ALC acknowledges support from the Indian
Institute of Astrophysics (IIA), Bengaluru to visit the institute where a
major part of this work was carried out. ALC also acknowledges the Department
of Science and Technology (DST), Government of India, for providing financial
support (Order No: DST/INSPIRE Fellowship/2018/IF180043) under the INSPIRE
program. The work of PC is supported by the Science and Engineering Research
Board of the Department of Science and Technology, India, under the MATRICS
scheme, bearing project reference no MTR/2018/000896. RKBS acknowledges the
support from the Science and Engineering Research Board of the Department of
Science and Technology, Government of India, under the MATRICS scheme, bearing
project reference no MTR/2021/000766. PK acknowledges the support of the
Department of Atomic Energy, Government of India, under the project
12-R&D-TFR-5.02-0700.

##  Appendix A Multifractal parameters for uncorrelated Gaussian random data
on two dimensions

To serve as a standard reference with which to compare the multifractal
parameters of the Crab nebula at different frequencies it is useful to know
the expected behavior of the parameters for an uncorrelated Gaussian random
field. For this purpose we simulated one realization of such a field having
zero mean and variance one on a $512\times 512$ grid. Then we calculated the
multifractal parameters using the same 2D-MFDFA algorithm that is used for the
Crab nebula data.

The resulting plots of $h(q)$, $\tau(q)$ and $f(\alpha)$ are shown in Figure
11. From the left panel we see that $h(q)$ is constant $\sim 2$, which implies
a monofractal behavior as expected from the uncorrelated and Gaussian natures
of the field. The linearity of the plot for $\tau(q)$ (middle), and narrow
range for $f(\alpha)$ (right), follow immediately from the plot of $h(q)$. The
very slight variation of $\alpha$ on the $x-$axis of the third panel arises
from the statistical fluctuation of the data since we have used only one
realization of the Gaussian random field.

Figure 11: Multifractal parameters $h(q)$ (left), $\tau(q)$ (middle) and
$f(\alpha)$ (right) for one realization of an uncorrelated two dimensional
Gaussian random field having zero mean and unit variance.

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Almost Kenmotsu manifolds admitting certain vector fields

Dibakar Dey and Pradip Majhi

Department of Pure Mathematics,

University of Calcutta, 35 Ballygunge Circular Road,

Kolkata - 700019, West Bengal, India,

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract: In the present paper, we characterize almost Kenmotsu manifolds
admitting holomorphically planar conformal vector (HPCV) fields. We have shown
that if an almost Kenmotsu manifold $M^{2n+1}$ admits a non-zero HPCV field
$V$ such that $\phi V=0$, then $M^{2n+1}$ is locally a warped product of an
almost Kaehler manifold and an open interval. As a corollary of this we obtain
few classifications of an almost Kenmotsu manifold to be a Kenmotsu manifold
and also prove that the integral manifolds of $\mathcal{D}$ are totally
umbilical submanifolds of $M^{2n+1}$. Further, we prove that if an almost
Kenmotsu manifold with positive constant $\xi$-sectional curvature admits a
non-zero HPCV field $V$, then either $M^{2n+1}$ is locally a warped product of
an almost Kaehler manifold and an open interval or isometric to a sphere.
Moreover, a $(k,\mu)^{\prime}$-almost Kenmotsu manifold admitting a HPCV field
$V$ such that $\phi V\neq 0$ is either locally isometric to
$\mathbb{H}^{n+1}(-4)$ $\times$ $\mathbb{R}^{n}$ or $V$ is an eigenvector of
$h^{\prime}$. Finally, an example is presented.

Mathematics Subject Classification 2010: 53D15, 53C25.

Keywords: Almost Kenmotsu manifold, Holomorphically Planar conformal vector
field, Almost Kaehler manifold, Totally umbilical submanifolds.

##  1\.  Introduction

In the present time, the study of existence of Killing vector fields in
Riemannian manifolds is a very interesting topic as they preserves a given
metric and determine the degree of symmetry of the manifold. Conformal vector
fields whose flow preserves a conformal class of metrics are very important in
the study of several kind of almost contact metric manifolds.

A smooth vector field $V$ on a Riemannian manifold $(M,g)$ is said to be
conformal vector field if there exist a smooth function $f$ on $M$ such that

$\displaystyle\pounds_{V}g=2fg,$ (1.1)

where $\pounds_{V}g$ is the Lie derivative of $g$ with respect to $V$. The
vector field $V$ is called homothetic or Killing accordingly as $f$ is
constant or zero. Moreover, $V$ is said to be closed conformal vector field if
the metrically equivalent $1$-form of $V$ is closed. If the conformal vector
field $V$ is gradient of some smooth function $\lambda$, then $V$ is called
gradient conformal vector field. The geometry of conformal vector fields have
been investigated in ([5], [6]).

A vector field $V$ on a contact metric manifold $M^{2n+1}(\phi,\xi,\eta,g)$ is
said to be holomorphically planar conformal vector field if it satisfies

$\displaystyle\nabla_{X}V=aX+b\phi X$ (1.2)

for any vector field $X$, where $a$, $b$ are smooth functions on $M$. As a
generalization of closed conformal vector fields Sharma [16] introduced the
notion of holomorphically planar conformal vector ( in short, HPCV ) fields on
almost Hermitian manifold. In [12], Ghosh and Sharma characterize an almost
Hermitian manifolds admitting a HPCV field. They shows that if $V$ is strictly
non-geodesic non-vanishing HPCV field on an almost Hermitian manifold, then
$V$ is homothetic and almost analytic. Further Sharma [17] shows that among
all complete and simply connected $K$-contact manifolds only the unit sphere
admits a non-Killing HPCV field and a $(k,\mu)$-contact manifold admitting a
non-zero HPCV field is either Sasakian or locally isometric to $E^{3}$ or
$E^{n+1}\times S^{n}(4)$. In [11], Ghosh studied HPCV fields in the framework
of contact metric manifolds under certain conditions and proved that a contact
metric manifold with pointwise constant $\xi$-sectional curvature admitting a
non-closed HPCV field $V$ is either $K$-contact or $V$ is homothetic.

Motivated by the above studies we consider HPCV fields in the framework of a
special type of almost contact metric manifolds, called almost Kenmotsu
manifolds. The paper is organized as follows :

In section 2, we present some preliminary notions on almost Kenmotsu manifolds
existing in the literature. Section 3 deals with HPCV fields on almost
Kenmotsu manidolds and section 4 is associated to the study of HPCV fields on
$(k,\mu)^{\prime}$-almost Kenmotsu manifolds.

##  2\. Preliminaries

An almost contact structure on a $(2n+1)$-dimensional smooth manifold
$M^{2n+1}$ is a triplet $(\phi,\xi,\eta)$, where $\phi$ is a $(1,1)$-tensor,
$\xi$ is a global vector field and $\eta$ is a $1$-form satisfying ([1], [2]),

$\phi^{2}=-I+\eta\otimes\xi,\;\;\eta(\xi)=1,$ (2.1)

where $I$ denote the identity endomorphism. Here also $\phi\xi=0$ and
$\eta\circ\phi=0$ hold; both can be derived from (2.1) easily.
If a manifold $M$ with a $(\phi,\xi,\eta)$-structure admits a Riemannian
metric $g$ such that

$g(\phi X,\phi Y)=g(X,Y)-\eta(X)\eta(Y),$

for any vector fields $X$, $Y$ on $M^{2n+1}$, then $M^{2n+1}$ is said to be an
almost contact metric manifold. The fundamental 2-form $\Phi$ on an almost
contact metric manifold is defined by

$\Phi(X,Y)=g(X,\phi Y)$

for any vector fields $X$, $Y$ on $M^{2n+1}$. The condition for an almost
contact metric manifold being normal is equivalent to vanishing of the
$(1,2)$-type torsion tensor $N_{\phi}$, defined by

$N_{\phi}=[\phi,\phi]+2d\eta\otimes\xi,$

where $[\phi,\phi]$ is the Nijenhuis tensor of $\phi$ [1]. Recently in ([8],
[9], [10]), almost contact metric manifold such that $\eta$ is closed and
$d\Phi=2\eta\wedge\Phi$ are studied and they are called almost Kenmotsu
manifolds. For more details on almost Kenmotsu manifolds we refer the reader
to go through the references ([4], [9], [10]). Obviously, a normal almost
Kenmotsu manifold is a Kenmotsu manifold. Also Kenmotsu manifolds can be
characterized by

$\displaystyle(\nabla_{X}\phi)Y=g(\phi X,Y)\xi-\eta(Y)\phi X,$

for any vector fields $X$, $Y$. Let the distribution orthogonal to $\xi$ is
denoted by $\mathcal{D}$, then $\mathcal{D}=Im(\phi)=Ker(\eta)$. Since $\eta$
is closed, $\mathcal{D}$ is an integrable distribution.

The study of nullity distributions is a very interesting topic on almost
contact metric manifolds. The notion of $k$-nullity distribution was
introduced by Gray [13] and Tanno [18] in the study of Riemannian manifolds.
Blair, Koufogiorgos and Papantonio [3] introduced the generalized notion of
the $k$-nullity distribution, named the $(k,\mu)$-nullity distribution on a
contact metric manifold. In [8], Dileo and Pastore introduce the notion of
$(k,\mu)^{\prime}$-nullity distribution, another generalized notion of the
$k$-nullity distribution, on an almost Kenmotsu manifold
$(M^{2n+1},\phi,\xi,\eta,g)$, which is defined for any $p\in M^{2n+1}$ and
$k,\mu\in\mathbb{R}$ as follows:

$\displaystyle N_{p}(k,\mu)^{\prime}=\\{Z\in T_{p}M:R(X,Y)Z$ $\displaystyle=$
$\displaystyle k[g(Y,Z)X-g(X,Z)Y]$ (2.2)
$\displaystyle+\mu[g(Y,Z)h^{\prime}X-g(X,Z)h^{\prime}Y]\\},$

where $h^{\prime}=h\circ\phi$.

Let $M^{2n+1}$ be an almost Kenmotsu manifold with structure
$(\phi,\xi,\eta,g)$. The Levi-Civita connection satisfies $\nabla_{\xi}\xi=0$
and $\nabla_{\xi}\phi=0$. We denote by $h=\frac{1}{2}\pounds_{\xi}\phi$ and
$l=R(\cdot,\xi)\xi$ on $M^{2n+1}$. The tensor fields $l$ and $h$ are symmetric
operators and satisfy the following relations [15]:

$\displaystyle h\xi=0,\;l\xi=0,\;tr(h)=0,\;tr(h\phi)=0,\;h\phi+\phi h=0,$
(2.3)

We also have the following formulas given in ([8] \- [10])

$\displaystyle\nabla_{X}\xi=X-\eta(X)\xi-\phi hX,$ (2.4)
$R(X,Y)\xi=\eta(X)(Y-\phi hY)-\eta(Y)(X-\phi hX)+(\nabla_{Y}\phi
h)X-(\nabla_{X}\phi h)Y,$ (2.5) $\displaystyle(\nabla_{X}\phi)Y-(\nabla_{\phi
X}\phi)\phi Y=-\eta(Y)\phi X-2g(X,\phi Y)\xi-\eta(Y)hX$ (2.6)

for any $X,\;Y$ on $M^{2n+1}$. The $(1,1)$-type symmetric tensor field
$h^{\prime}=h\circ\phi$ is anticommuting with $\phi$ and $h^{\prime}\xi=0$.
Also it is clear that ([8], [20])

$h=0\Leftrightarrow h^{\prime}=0,\;\;h^{\prime
2}=(k+1)\phi^{2}(\Leftrightarrow h^{2}=(k+1)\phi^{2}).$ (2.7)

##  3\. HPCV fields on almost Kenmotsu manifolds

In this section we characterize almost Kenmotsu manifolds admitting a
holomorphically planar conformal vector field $V$. Before proving our main
theorems we first state and prove the following lemma.

######  Lemma 3.1.

Let $M^{2n+1}$ be an almost Kenmotsu manifold admitting a HPCV field $V$. Then
the following relation

$\displaystyle\phi Va=4nb\eta(V)+(\xi b)\eta(V)-Vb$

holds on $M^{2n+1}$.

Proof: Differentiating (1.2) covariantly along any vector field $Y$, we have

$\displaystyle\nabla_{Y}\nabla_{X}V=a(\nabla_{Y}X)+(Ya)X+b(\nabla_{Y}\phi
X)+(Yb)\phi X.$ (3.1)

Interchanging $X$ and $Y$ in the above equation, we get

$\displaystyle\nabla_{X}\nabla_{Y}V=a(\nabla_{X}Y)+(Xa)Y+b(\nabla_{X}\phi
Y)+(Xb)\phi Y.$ (3.2)

Replacing $X$ by $[X,Y]$ in (1.2) yields

$\displaystyle\nabla_{[X,Y]}V=a(\nabla_{X}Y)-a(\nabla_{Y}X)+b\phi(\nabla_{X}Y)-b\phi(\nabla_{Y}X).$
(3.3)

Now using $R(X,Y)Z=[\nabla_{X},\nabla_{Y}]Z-\nabla_{[X,Y]}Z$ gives

$\displaystyle R(X,Y)V$ $\displaystyle=$ $\displaystyle(Xa)Y-(Ya)X+(Xb)\phi
Y-(Yb)\phi X$ (3.4) $\displaystyle+b[(\nabla_{X}\phi)Y-(\nabla_{Y}\phi)X].$

Putting $X=\phi X$ and $Y=\phi Y$ in (3.4) we get

$\displaystyle R(\phi X,\phi Y)V$ $\displaystyle=$ $\displaystyle(\phi Xa)\phi
Y-(\phi Ya)\phi X+(\phi Xb)[-Y+\eta(Y)\xi]$ (3.5) $\displaystyle-(\phi
Yb)[-X+\eta(X)\xi]+b[(\nabla_{\phi X}\phi)\phi Y-(\nabla_{\phi Y}\phi)\phi
X].$

Now adding equations (3.4) and (3.5) and using (2.6) we have

$\displaystyle R(X,Y)V+R(\phi X,\phi Y)V$ $\displaystyle=$
$\displaystyle(Xa)Y-(Ya)X+(Xb)\phi Y-(Yb)\phi X$ (3.6) $\displaystyle+(\phi
Xa)\phi Y-(\phi Ya)\phi X-(\phi Xb)Y$ $\displaystyle+(\phi Xb)\eta(Y)\xi+(\phi
Yb)X-(\phi Yb)\eta(X)\xi$ $\displaystyle+b[-\eta(Y)\phi X-2g(X,\phi
Y)\xi-\eta(Y)hX$ $\displaystyle+\eta(X)\phi Y+2g(\phi X,Y)\xi+\eta(X)hY].$

Taking inner product of (3.6) with $V$ and then substituting $X=\phi X$ and
$Y=\phi Y$ yields

$\displaystyle(\phi Xa)g(\phi Y,V)-(\phi Ya)g(\phi X,V)+(\phi
Xb)[-g(Y,V)+\eta(Y)\eta(V)]$ $\displaystyle-(\phi
Yb)[-g(X,V)+\eta(X)\eta(V)]+[-X+\eta(X)\xi](a)[-g(Y,V)+\eta(Y)\eta(V)]$
$\displaystyle-[-Y+\eta(Y)\xi](a)[-g(X,V)+\eta(X)\eta(V)]-[-X+\eta(X)\xi](b)g(\phi
Y,V)$ $\displaystyle+[-Y+\eta(Y)\xi](b)g(\phi X,V)-4bg(X,\phi Y)\eta(V)=0.$
(3.7)

Now replacing $Y$ by $\phi Y$ in the foregoing equation we obtain

$\displaystyle-g(\phi Da,X)[-g(Y,V)+\eta(Y)\eta(V)]+g(Da,Y)g(\phi
X,V)-\eta(Y)(\xi a)g(\phi X,V)$ $\displaystyle+g(\phi
Db,X)g(Y,V)+g(Db,Y)[-g(X,V)+\eta(X)\eta(V)]-\eta(Y)(\xi b)[-g(X,V)$
$\displaystyle+\eta(X)\eta(V)]+g(Da,X)g(Y,V)-\eta(X)(\xi
a)g(Y,V)+g(Da,Y)[-g(X,V)$
$\displaystyle+\eta(X)\eta(V)]+g(Db,X)[-g(Y,V)+\eta(Y)\eta(V)]-\eta(X)(\xi
b)[-g(Y,V)$ $\displaystyle+\eta(Y)\eta(V)]-g(Db,Y)g(\phi
X,V)+4bg(X,Y)\eta(V)-4b\eta(X)\eta(Y)\eta(V)=0.$ (3.8)

Contracting $X$ and $Y$ in (3.8) we have

$\displaystyle-2\phi Va-2Vb+2(\xi b)\eta(V)+8nb\eta(V)=0,$

which implies

$\displaystyle\phi Va=4nb\eta(V)+(\xi b)\eta(V)-Vb.$ (3.9)

This completes the proof.

######  Theorem 3.2.

If an almost Kenmotsu manifold $M^{2n+1}$ admits a non-zero HPCV field $V$
such that $\phi V=0$, then $M^{2n+1}$ is locally a warped product of an almost
Kaehler manifold and an open interval.

###### Proof.

Let $M^{2n+1}$ be an almost Kenmotsu manifold admitting a non-zero HPCV field
$V$ such that $\phi V=0$. Operating $\phi$ on it we get

$\displaystyle V=\eta(V)\xi.$ (3.10)

Now using (3.10) and $\phi V=0$ in Lemma 3.1 we have $4nb\eta(V)=0$, which
implies either $b=0$ or $\eta(V)=0$. If $\eta(V)=0$, then from (3.10) we have
$V=0$, which is a contradiction to our hypoyhesis. Thus we get $b=0$.
Differentiating (3.10) covariantly along any vector field $X$ and using $b=0$,
$\phi V=0$, (1.2) and (2.4) we obtain

$\displaystyle
aX=a\eta(X)\xi+g(X,V)\xi-2\eta(X)\eta(V)\xi+\eta(V)X-\eta(V)\phi hX.$ (3.11)

Contracting $X$ and using (2.3) in (3.11) yields $a=\eta(V)$. Substituting the
value of $a$ in (3.11) we get

$\displaystyle g(X,V)\xi-\eta(X)\eta(V)\xi-\eta(V)\phi hX=0.$ (3.12)

Replacing $X$ by $\phi X$ in the above equation and using the hypothesis $\phi
V=0$ and $\eta(V)\neq 0$ we infer that $hX=0$ for any vector field $X$ on
$M^{2n+1}$. The rest of the proof follows from Theorem 2 of [10]. ∎

Proposition 1 of [10] says that ” In an almost Kenmotsu manifold $M^{2n+1}$,
the integral manifolds of $\mathcal{D}$ are totally umbilical submanifolds of
$M^{2n+1}$ if and only if $h$ vanishes ”. Hence, we can state the following:

######  Corollary 3.3.

Let $M^{2n+1}$ be an almost Kenmotsu manifold admitting a non-zero HPCV field
$V$ such that $\phi V=0$. Then the integral manifolds of $\mathcal{D}$ are
totally umbilical submanifolds of $M^{2n+1}$.

######  Corollary 3.4.

If a locally symmetric almost Kenmotsu manifold $M^{2n+1}$ admits a non-zero
HPCV field $V$ such that $\phi V=0$, then $M^{2n+1}$ is a Kenmotsu manifold.

The above Corollary follows directly from Theorem 3 of [10].

Proposition 2.1 of [19] states that ” Any 3-dimensional almost Kenmotsu
manifold is Kenmotsu if and only if $h$ vanishes ”. Thus we arrive to the
following:

######  Corollary 3.5.

A $3$-dimensional almost Kenmotsu manifold $M^{2n+1}$ admitting a non-zero
HPCV field $V$ such that $\phi V=0$ is a Kenmotsu manifold.

######  Theorem 3.6.

Let $M^{2n+1}$ be a complete almost Kenmotsu manifold admitting a non-zero
HPCV field $V$. If $M^{2n+1}$ has positive constant $\xi$-sectional curvature,
then either $M^{2n+1}$ is locally a warped product of an almost Kaehler
manifold and an open interval or isometric to a sphere.

###### Proof.

If the sectional curvature $K(\xi,X)=c$ of an almost Kenmotsu manifold is a
positive constant, then we can easily obtain the following:

$\displaystyle R(\xi,X)\xi=-c[X-\eta(X)\xi].$ (3.13)

Now putting $X=\xi$ in (3.4) we have

$\displaystyle R(\xi,Y)V=(\xi a)Y-(Ya)\xi+(\xi b)\phi Y+b\phi Y+bhY.$ (3.14)

Taking inner product of (3.14) with $\xi$ we get

$\displaystyle g(R(\xi,Y)V,\xi)=(\xi a)\eta(Y)-(Ya).$ (3.15)

Again using (3.13) we have

$\displaystyle g(R(\xi,Y)V,\xi)=-g(R(\xi,Y)\xi,V)=c[g(Y,V)-\eta(Y)\eta(V)].$
(3.16)

Hence from (3.15) and (3.16) we obtain

$\displaystyle Da-(\xi a)\xi+cV-c\eta(V)\xi=0.$ (3.17)

Taking inner product of (3.14) with $V$ we get

$\displaystyle(\xi a)V-\eta(V)(Da)-(\xi b)\phi V-b\phi V+bhV=0.$ (3.18)

Eliminating $Da$ from (3.17) and (3.18) we have

$\displaystyle-(\xi a)\phi^{2}V-c\eta(V)\phi^{2}V-(\xi b)\phi V-b\phi
V+bhV=0.$ (3.19)

Now differentiating (3.17) covariantly along any vector field $X$ and then
taking inner product of the resulting equation with $Y$ we infer

$\displaystyle g(\nabla_{X}Da,Y)-(\xi a)[g(X,Y)-\eta(X)\eta(Y)-g(\phi
hX,Y)]-(X(\xi a))\eta(Y)$ $\displaystyle+c[ag(X,Y)+bg(\phi
X,Y)]-c\eta(Y)[g(X,V)-\eta(X)\eta(V)+a\eta(X)]$
$\displaystyle-c\eta(V)[g(X,Y)-\eta(X)\eta(Y)-g(\phi hX,Y)]=0.$ (3.20)

Antisymmetrizing the above equation and using the symmetry of the Hessian
operator, that is,
$\operatorname{Hess}_{a}(X,Y)=g(\nabla_{X}Da,Y)=g(\nabla_{Y}Da,X)$ we obtain

$\displaystyle(Y(\xi a))\eta(X)-(X(\xi a))\eta(Y)+2bcg(\phi X,Y)$
$\displaystyle-c\eta(Y)g(X,V)+c\eta(X)g(Y,V)=0.$ (3.21)

Replacing $X$ by $\phi X$ and $Y$ by $\phi Y$ in (3.21) we get $2bcg(\phi
X,Y)=0$, which implies $b=0$ as $c$ is non-zero constant by hypothesis. Then
from (3.18) we have

$\displaystyle(\xi a)V=(Da)\eta(V).$ (3.22)

Also from (3.19) we obtain

$\displaystyle[(\xi a)+c\eta(V)]\phi^{2}V=0,$ (3.23)

which implies either $\phi^{2}V=0$ or $(\xi a)=-c\eta(V)$.
Case 1: If $\phi^{2}V=0$, then we have $V=\eta(V)\xi$ and this implies $\phi
V=0$. Thus from Theorem 3.2 we infer that $M^{2n+1}$ is locally a warped
product of an almost Kaehler manifold and an open interval.
Case 2: If $(\xi a)=-c\eta(V)$, then from (3.22) we have

$\displaystyle(Da+cV)\eta(V)=0.$ (3.24)

Now if $\eta(V)=0$, then from (3.22) we have $\xi a=0$ as $V$ is non-zero.
Hence from (3.17) we get $Da=-cV$. Thus in either cases we obtain $Da=-cV$.
Differentiating this covariantly along any vector field $X$ and using (1.2) we
have $\nabla_{X}Da=-caX$. We are now in a position to apply Obata’s theorem
[14]: ” In order for a complete Riemannian manifold of dimension $n\geq 2$ to
admit a non-constant function $\lambda$ with $\nabla_{X}D\lambda=-c^{2}\lambda
X$ for any vector $X$, it is necessary and sufficient that the manifold is
isometric with a sphere $S^{n}(c)$ of radius $\frac{1}{c}$ ” to conclude that
the manifold is isometric to the sphere $S^{2n+1}(\sqrt{c})$ of radius
$\frac{1}{\sqrt{c}}$. ∎

##  4\. HPCV fields on a class of almost Kenmotsu manifolds

In this section, we study HPCV fields on almost Kenmotsu manifolds with $\xi$
belonging to the $(k,\mu)^{\prime}$-nullity distribution. Let
$X\in\mathcal{D}$ be the eigen vector of $h^{\prime}$ corresponding to the
eigen value $\lambda$. Then from (2.7) it is clear that $\lambda^{2}=-(k+1)$,
a constant. Therefore $k\leq-1$ and $\lambda=\pm\sqrt{-k-1}$. We denote by
$[\lambda]^{\prime}$ and $[-\lambda]^{\prime}$ the corresponding eigenspaces
related to the non-zero eigenvalue $\lambda$ and $-\lambda$ of $h^{\prime}$,
respectively. Before proving our main theorem in this section we recall some
results:

######  Lemma 4.1.

(Prop. $4.1$ of [8]) Let $(M^{2n+1},\phi,\xi,\eta,g)$ be an almost Kenmotsu
manifold such that $\xi$ belongs to the $(k,\mu)^{\prime}$-nullity
distribution and $h^{\prime}\neq 0$. Then $k<-1,\;\mu=-2$ and Spec
$(h^{\prime})=\\{0,\lambda,-\lambda\\}$, with $0$ as simple eigen value and
$\lambda=\sqrt{-k-1}$. The distributions $[\xi]\oplus[\lambda]^{\prime}$ and
$[\xi]\oplus[-\lambda]^{\prime}$ are integrable with totally geodesic leaves.
The distributions $[\lambda]^{\prime}$ and $[-\lambda]^{\prime}$ are
integrable with totally umbilical leaves. Furthermore, the sectional curvature
are given by the following:

  * (a)

$K(X,\xi)=k-2\lambda$ if $X\in[\lambda]^{\prime}$ and
$K(X,\xi)=k+2\lambda$ if $X\in[-\lambda]^{\prime}$,

  * (b)

$K(X,Y)=k-2\lambda$ if $X,Y\in[\lambda]^{\prime}$;
$K(X,Y)=k+2\lambda$ if $X,Y\in[-\lambda]^{\prime}$ and
$K(X,Y)=-(k+2)$ if $X\in[\lambda]^{\prime}$, $Y\in[-\lambda]^{\prime}$,

  * (c)

$M^{2n+1}$ has constant negative scalar curvature $r=2n(k-2n).$

######  Lemma 4.2.

(Lemma $4.1$ of [8]) Let $(M^{2n+1},\phi,\xi,\eta,g)$ be an almost Kenmotsu
manifold with $h^{\prime}\neq 0$ and $\xi$ belongs to the
$(k,-2)^{\prime}$-nullity distribution. Then, for any
$X,\;Y\in\chi(M^{2n+1})$,

$\displaystyle(\nabla_{X}h^{\prime})Y=-g(h^{\prime}X+h^{\prime
2}X,Y)\xi-\eta(Y)(h^{\prime}X+h^{\prime 2}X)$ (4.1)

######  Lemma 4.3.

(Prop. $4.2$ of [8]) Let $(M^{2n+1},\phi,\xi,\eta,g)$ be an almost Kenmotsu
manifold such that $h^{\prime}\neq 0$ and $\xi$ belonging to the
$(k,-2)^{\prime}$-nullity distribution. Then for any
$X_{\lambda},Y_{\lambda},Z_{\lambda}\in[\lambda]^{\prime}$ and
$X_{-\lambda},Y_{-\lambda},Z_{-\lambda}\in[-\lambda]^{\prime}$, the Riemann
curvature tensor satisfies:

$\displaystyle R(X_{\lambda},Y_{\lambda})Z_{-\lambda}=0,$ $\displaystyle
R(X_{-\lambda},Y_{-\lambda})Z_{\lambda}=0,$ $\displaystyle
R(X_{\lambda},Y_{-\lambda})Z_{\lambda}=(k+2)g(X_{\lambda},Z_{\lambda})Y_{-\lambda},$
$\displaystyle
R(X_{\lambda},Y_{-\lambda})Z_{-\lambda}=-(k+2)g(Y_{-\lambda},Z_{-\lambda})X_{\lambda},$
$\displaystyle
R(X_{\lambda},Y_{\lambda})Z_{\lambda}=(k-2\lambda)[g(Y_{\lambda},Z_{\lambda})X_{\lambda}-g(X_{\lambda},Z_{\lambda})Y_{\lambda}],$
$\displaystyle
R(X_{-\lambda},Y_{-\lambda})Z_{-\lambda}=(k+2\lambda)[g(Y_{-\lambda},Z_{-\lambda})X_{-\lambda}-g(X_{-\lambda},Z_{-\lambda})Y_{-\lambda}].$

From (2.2) we have

$\displaystyle
R(X,Y)\xi=k[\eta(Y)X-\eta(X)Y]+\mu[\eta(Y)h^{\prime}X-\eta(X)h^{\prime}Y],$
(4.2)

where $k,\;\mu\in\mathbb{R}.$ Also we get from (4.2)

$\displaystyle
R(\xi,X)Y=k[g(X,Y)\xi-\eta(Y)X]+\mu[g(h^{\prime}X,Y)\xi-\eta(Y)h^{\prime}X].$
(4.3)

######  Theorem 4.4.

A $(k,\mu)^{\prime}$-almost Kenmotsu manifold with $h^{\prime}\neq 0$
admitting a HPCV field $V$ such that $\phi V\neq 0$ is either locally
isometric to the Riemannian product of an $(n+1)$-dimensional manifold of
constant sectional curvature $-4$ and a flat $n$-dimensional manifold or $V$
is an eigenvector of $h^{\prime}$.

###### Proof.

Substituting $X=\xi$ in (3.4) we have

$\displaystyle R(\xi,Y)V=(\xi a)Y-(Ya)\xi+(\xi b)\phi Y+b\phi Y+bhY.$ (4.4)

Taking inner product of (4.4) with $\xi$ we obtain

$\displaystyle g(R(\xi,Y)V,\xi)=(\xi a)\eta(Y)-(Ya).$ (4.5)

Making use of (4.2) we get

$\displaystyle g(R(\xi,Y)V,\xi)$ $\displaystyle=$
$\displaystyle-g(R(\xi,Y)\xi,V)$ (4.6) $\displaystyle=$
$\displaystyle-k\eta(Y)\eta(V)+kg(Y,V)-2g(h^{\prime}Y,V).$

Equations (4.5) and (4.6) together implies

$\displaystyle-k\eta(Y)\eta(V)+kg(Y,V)-2g(h^{\prime}Y,V)=(\xi a)\eta(Y)-(Ya),$
(4.7)

which implies

$\displaystyle-k\eta(V)\xi+kV-2h^{\prime}V=(\xi a)\xi-Da.$ (4.8)

Now taking inner product of (4.4) with $V$ gives

$\displaystyle(\xi a)g(Y,V)-(Ya)\eta(V)+(\xi b)g(\phi Y,V)+bg(\phi
Y,V)+bg(hY,V)=0,$

which implies

$\displaystyle(\xi a)V-(Da)\eta(V)-(\xi b)\phi V-b\phi V+bhV=0.$ (4.9)

Eliminating $Da$ from (4.8) and (4.9) we have

$\displaystyle-(\xi a)\phi^{2}V-k\eta(V)\phi^{2}V-2\eta(V)h^{\prime}V-(\xi
b)\phi V-b\phi V+bhV=0.$ (4.10)

Differentiating (4.8) covariantly along any vector field $X$ and using (1.2),
(2.4), Lemma 4.2 and the value of $\mu$ from Lemma 4.1 we infer

$\displaystyle-k[g(X-\eta(X)\xi-\phi hX,V)+g(\xi,aX+b\phi
X)]\xi-k\eta(V)[X-\eta(X)\xi-\phi hX]$ $\displaystyle+k[aX+b\phi
X]-2[-g(h^{\prime}X+h^{\prime 2}X,V)\xi-\eta(V)(h^{\prime}X+h^{\prime
2}X)+h^{\prime}(aX+b\phi X)]$ $\displaystyle=(\xi a)[X-\eta(X)\xi-\phi
hX]+(X(\xi a))\xi-\nabla_{X}Da.$

Taking inner product of the foregoing equation with $Y$ we obtain

$\displaystyle-k[g(X,V)-\eta(X)\eta(V)-g(\phi
hX,V)+a\eta(X)]\eta(Y)-k\eta(V)[g(X,Y)$ $\displaystyle-\eta(X)\eta(Y)-g(\phi
hX,Y)]+k[ag(X,Y)+bg(\phi X,Y)]-2[-g(h^{\prime}X$ $\displaystyle+h^{\prime
2}X,V)\eta(Y)-\eta(V)g(h^{\prime}X+h^{\prime 2}X,Y)+g(ah^{\prime}X-bhX,Y)]$
$\displaystyle\;\;\;=(\xi a)[g(X,Y)-\eta(X)\eta(Y)-g(\phi hX,Y)]+(X(\xi
a))\eta(Y)-g(\nabla_{X}Da,Y).$ (4.11)

Antisymmetrizing the above equation and using the symmetry of the Hessian
operator, that is,
$\operatorname{Hess}_{a}(X,Y)=g(\nabla_{X}Da,Y)=g(\nabla_{Y}Da,X)$ we obtain

$\displaystyle-k[g(X,V)\eta(Y)-g(Y,V)\eta(X)-g(\phi hX,V)\eta(Y)+g(\phi
hY,V)\eta(X)]$ $\displaystyle+2kbg(\phi X,Y)-2[-g(h^{\prime}X+h^{\prime
2}X,V)\eta(Y)+g(h^{\prime}Y+h^{\prime 2}Y,V)\eta(X)]$ $\displaystyle=(X(\xi
a))\eta(Y)-(Y(\xi a))\eta(X).$ (4.12)

Putting $X=\phi X$ and $Y=\phi Y$ in the previous equation we infer that
$2kbg(\phi X,Y)=0$, which implies $b=0$ as $k<-1$. Hence from (4.9) we have

$\displaystyle(\xi a)V=(Da)\eta(V).$ (4.13)

Now letting $Y\in[\lambda]^{\prime}$ in (4.7) yields

$\displaystyle(k-2\lambda)g(Y,V)=-(Ya),$

which implies

$\displaystyle Da=(2\lambda-k)V\;\;\;\mathrm{and}\;\;\;(\xi
a)=(2\lambda-k)\eta(V).$ (4.14)

Now using $b=0$ and the value of $(\xi a)$ from (4.14) in (4.10) we have

$\displaystyle 2(\lambda+1)\eta(V)(h^{\prime}V+\phi^{2}V)=0,$ (4.15)

which implies either $\lambda=-1$ or $\eta(V)=0$ or $h^{\prime}V=-\phi^{2}V$.
Case 1: If $\lambda=-1$, then from $\lambda^{2}=-k-1$ we obtain $k=-2$. Now
letting $X,\;Y,\;Z\in[\lambda]^{\prime}$ and noticing that $k=-2$,
$\lambda=-1$, from Lemma 4.3 we have

$\displaystyle R(X_{\lambda},Y_{\lambda})Z_{\lambda}=0,$

and

$\displaystyle
R(X_{-\lambda},Y_{-\lambda})Z_{-\lambda}=-4[g(Y_{-\lambda},Z_{-\lambda})X_{-\lambda}-g(X_{-\lambda},Z_{-\lambda})Y_{-\lambda}],$

for any $X_{\lambda},Y_{\lambda},Z_{\lambda}\in[\lambda]^{\prime}$ and
$X_{-\lambda},Y_{-\lambda},Z_{-\lambda}\in[-\lambda]^{\prime}$. Also noticing
$\mu=-2$ it follows from Lemma 4.1 that $K(X,\xi)=-4$ for any
$X\in[-\lambda]^{\prime}$ and $K(X,\xi)=0$ for any $X\in[\lambda]^{\prime}$.
Again from Lemma 4.1 we see that $K(X,Y)=-4$ for any
$X,Y\in[-\lambda]^{\prime}$ and $K(X,Y)=0$ for any $X,Y\in[\lambda]^{\prime}$.
As is shown in [8] that the distribution $[\xi]\oplus[\lambda]^{\prime}$ is
integrable with totally geodesic leaves and the distribution
$[-\lambda]^{\prime}$ is integrable with totally umbilical leaves by
$H=-(1-\lambda)\xi$, where $H$ is the mean curvature tensor field for the
leaves of $[-\lambda]^{\prime}$ immersed in $M^{2n+1}$. Here $\lambda=-1$,
then the two orthogonal distributions $[\xi]\oplus[\lambda]^{\prime}$ and
$[-\lambda]^{\prime}$ are both integrable with totally geodesic leaves
immersed in $M^{2n+1}$. Then we can say that $M^{2n+1}$ is locally isometric
to $\mathbb{H}^{n+1}(-4)$ $\times$ $\mathbb{R}^{n}$.
Case 2: If $\eta(V)=0$, then from (4.14) we have $(\xi a)=0$. Then from (4.8)
we have $Da=2h^{\prime}V-kV$. Now equating the value of $Da$ from this and
(4.14) we get $h^{\prime}V=\lambda V$. This shows that $V$ is an eigenvector
of $h^{\prime}$.
Case 3: If $h^{\prime}V=-\phi^{2}V=V-\eta(V)\xi$, then applying $h^{\prime}$
on both side of it we have $h^{\prime 2}V=h^{\prime}V$. Hence using (2.7) we
obtain $-(k+2)(V-\eta(V)\xi)=0$. Now $V-\eta(V)\xi\neq 0$ as $\phi V\neq 0$ by
hypothesis. Therefore, we have $k=-2$. Now from $\lambda^{2}=-k-1$, we obtain
$\lambda^{2}=1$. Without loss of generality we assume that $\lambda=-1$. Then
by the same argument as in Case 1 we get $M^{2n+1}$ is locally isometric to
$\mathbb{H}^{n+1}(-4)$ $\times$ $\mathbb{R}^{n}$. This completes the proof. ∎

### Example

In [7], the author present an example of a $5$-dimensional
$(k,\mu)^{\prime}$-almost Kenmotsu manifold with $k=-2$ and $\mu=-2$. Then by
the same argument as in Case 1 of Theorem 4.4, $M^{5}$ is locally isometric to
$\mathbb{H}^{3}(-4)$ $\times$ $\mathbb{R}^{2}$.
Let
$X=\alpha_{1}\xi+\alpha_{2}e_{2}+\alpha_{3}e_{3}+\alpha_{4}e_{4}+\alpha_{5}e_{5}$
be any vector field on $M^{5}$ and let $V=e_{4}$. Then,
$\nabla_{X}V=0=aX+b\phi X$, where $a=b=0$. Hence, $V=e_{4}$ is an example of a
HPCV field, where $\phi V=\phi e_{4}=-e_{2}\neq 0$. Hence, Theorem 4.4 is
verified.

Acknowledgement: The author Dibakar Dey is supported by the Council of
Scientific and Industrial Research, India (File no: 09/028(1010)/2017-EMR-1)
in the form of Senior Research Fellowship.

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