# Maximum entropy principle and the form of source in non-equilibrium
statistical operator method

V Ryazanov  Institute for Nuclear Research, pr.Nauki, 47, Kiev, Ukraine
<EMAIL_ADDRESS>
###### Abstract

It is supposed that the exponential multiplier in the method of the non-
equilibrium statistical operator (Zubarev‘s approach) can be considered as a
distribution density of the past lifetime of the system, and can be replaced
by an arbitrary distribution function. To specify this distribution the method
of maximum entropy principle as in [Schönfeldt J-H, Jiminez N, Plastino A R,
Plastino A, Casas M 2007 Physica A 374 573] is used. The obtained distribution
is close to exponential one. Another approach to the maximum entropy
principle, as in [Van der Straeten E and Beck C 2008 Phys. Rev. E 78 051101],
except exponential distributions yields power-like, log-normal distributions,
as well as distributions of other kind and transitions between them.

###### pacs:

05.70.Ln, 05.40.-a

##  1 Introduction

Among possible approaches to the description of non-equilibrium systems the
Non-equilibrium Statistical Operator Method (NESOM) especially demonstrated
its efficiency [1, 2, 3]. NESOM provides a very promising technique that
implies in a far-reaching generalization the statistical methods developed by
Boltzmann and Gibbs. NESOM was initially built on intuitive and heuristic
arguments, apparently it can be incorporated within an interesting approach to
the rationalization of statistical mechanics, as contained in the maximization
of (informational statistical) entropy (MaxEnt for short) and Bayesian
methods. NESOM appears as a very powerful, concise, based on sound principles,
and elegant formalism of a broad scope to deal with systems arbitrarily far
from equilibrium. The non-equilibrium statistical operator (NSO) introduced in
[1, 2, 3] has a form

$\ln\varrho(t)=\int_{0}^{\infty}p_{qzub}(u)\ln\varrho_{q}(t-u,-u)du,\quad\ln\varrho_{q}(t,0)=-\Phi(t)-\sum_{n}F_{n}(t)P_{n};\\\
$ (1)
$\ln\varrho_{q}(t,t_{1})=\rme^{\left\\{-t_{1}H/i\hbar\right\\}}\ln\varrho_{q}(t,0)\rme^{\left\\{t_{1}H/i\hbar\right\\}};\quad\Phi(t)=\ln
Sp\exp\\{-\sum_{n}F_{n}(t)P_{n}\\}\,,$

where

$p_{q}(u)=p_{qzub}(u)=\varepsilon\rme^{-\varepsilon u}\,,\quad u=t-t_{0}\,,$
(2)

$H$ is Hamiltonian, $\ln\rho(t)$ is the logarithm of the NSO,
$\ln\rho_{q}(t_{1},t_{2})$ is the logarithm of the quasi-equilibrium (or
relevant) distribution; the first time argument indicates the time dependence
of the values of the thermodynamic parameters $F_{n}$; the second time
argument $t_{2}$ in $\ln\rho_{q}(t_{1},t_{2})$ denotes the time dependence
through the Heizenberg representation for dynamical variables $P_{n}$ on which
$\ln\rho_{q}(t,0)$ can depend [1, 2, 3].

In [1, 2, 3] $p_{q}(u)=p_{qzub}(u)=\varepsilon\exp\\{-\varepsilon u\\}$; after
the thermodynamic limiting transition $N\rightarrow\infty$,
$V\rightarrow\infty$, $N/V=const$, $\varepsilon\rightarrow 0$. From the
complete group of solutions of Liouville equation (symmetric in time) the
subset of retarded ”unilateral” in time solutions is selected by means of
introducing a source $K$ in the Liouville equation for $\ln\rho(t)$ ($L$ is
Liouville operator;
$iL=-\\{H,\varrho\\}=\sum_{k}{\displaystyle\left[\frac{\partial H}{\partial
p_{k}}\frac{\partial\varrho}{\partial q_{k}}-\frac{\partial H}{\partial
q_{k}}\frac{\partial\varrho}{\partial p_{k}}\right]}$; $p_{k}$, and $q_{k}$
are pulses and coordinates of particles; $\\{\dots\\}$ are Poisson brackets)

$\frac{\partial\ln\rho(t)}{\partial
t}+iL\ln\rho(t)=-\varepsilon(\ln\rho(t)-\ln\rho_{q}(t,0))=K_{zub}.$ (3)

In [4, 5] a convenient redefinition of the source term is proposed. Although
infinitesimally small, the source term introduced by Zubarev into the
Liouville equation is shown to influence the macroscopic behaviour of the
system in the sense that the corresponding evolution equations do not coincide
exactly with those obtained from an initial-value problem which corresponds to
a definite experimental situation and a physical set of macroobservables.

In [6, 7] it was noted that in place of the function
$p_{q}(u)=p_{qzub}(u)=\varepsilon\rme^{-\varepsilon u}$ in (1) arbitrary (but
having certain properties [8]) weigth functions $w(t,t_{0})$ can be used.
Zubarev’s nonequilibrium statistical operator does satisfy Liouville equation,
but it must be borne in mind that the group of its solutions is composed of
two subsets, one corresponding to the retarded and second one to the advanced
solutions. The presence of the weight function $w(t,t_{0})$ (Abel’s kernel (2)
in Zubarev’s approach) in the time-smoothing or quasi-average procedure that
has been introduced selects the subset of retarded solutions from the total
group of solutions of the Liouville equation. This consideration is related to
the question: how to obtain an irreversible behavior in the evolution of the
macroscopic state of the system? In the MaxEnt-NESOM approach the
irreversibility is incorporated from the outset using an ad hoc non-mechanical
hypothesis. MaxEnt-NESOM yields information on the macrostate of the system at
time $t$, when a measurement is performed, including the evolutionary history
(in the interval from the initial time of preparation $t_{0}$ up to time $t$)
by which the system came into that state (which introduced a generalization of
Kirkwood‘s time-smoothing formalism [9]). Functions $w(t,t_{0})$ are typically
kernels [1, 2, 3, 8] that appear in the mathematical theory of convergence of
integrals. In [10, 11, 12] other interpretation of the functions $w(t,t_{0})$,
denoted as $p_{q}(u)$, is given. With the change of function
$w(t,t_{0})=p_{q}(u)$ the form of source (3) in the Liouville equation also
changes. For an arbitrary function $p_{q}(u)$ it looks like (4).

In [10, 11, 12] it was noted that the function
$p_{q}(u)=p_{qzub}(u)=\varepsilon\rme^{-\varepsilon u}$ in NESOM [1, 2, 3] for
the non-equilibrium distribution function can be interpreted as the
exponential probability distribution of the lifetime $\Gamma$ of a system.
$\Gamma$ is a random variable of lifetime (time span) from the moment $t_{0}$
of its birth till the current moment $t$; $\varepsilon^{-1}=\langle
t-t_{0}\rangle=\langle\Gamma\rangle$, where $\langle\Gamma\rangle=\int
up_{q}(u)du$ is the average lifetime of the system. This time period can be
called the time period of getting information about system from its past.
Instead of the exponential distribution $p_{qzub}(u)$ (2) in (1) used in [1,
2, 3] any other sample distribution $p_{q}(u)$ could be taken; integration by
parts in time is performed at $\int p_{q}(u)du_{|u=0}=-1$; $\int
p_{q}(u)du_{|u\rightarrow\infty}=0$. If $p_{q}(u)=p_{qzub}(u)=\varepsilon
e^{-\varepsilon u}$ by (2), $\varepsilon=1/\langle\Gamma\rangle$ the
expression for NSO passes in (1) from [1, 2, 3].

The same interpretation of the distribution $p_{q}(u)$ is given in [2], where
this value is understood as the distribution of the initial moment of time
$t_{0}$. Since the random (past) lifetime is equal to $\Gamma=u=t-t_{0}$, the
distribution of the past lifetime ${u}$ coincides with the distribution of the
initial time values $t_{0}$. The moment $t_{0}$ will be the moment of the
first passage in the inverse time, if the moment $t$ is taken as initial. In
[2] the uniform distribution for an initial moment $t_{0}$ is chosen, which
after the transition from Abel integration to Cesàro integration passes to the
exponential distribution $p_{q}(u)=p_{qzub}(u)=\varepsilon\rme^{-\varepsilon
u}$. Such distribution serves as the limiting distribution of the lifetime
[13], the first-passage time of a certain level. In the general case it is
possible to choose a lot of functions for the obvious type of distribution
$p_{q}(u)$, which was noted in [10, 11, 12].

In [13] the lifetimes of the system are introduced as random moments of the
first-passage time till the moment when a random process describing system
reaches a certain limit, for example, a zero value. In [13] approximate
exponential expressions for the probability density function (with a single
parameter) and probability distribution of lifetime are obtained, and the
accuracy of these expressions is estimated.

In [14] it was noted, that the role of the form of the source term in the
Liouville equation in the NSO method has never been investigated. In [15] it
is stated that the exponential distribution is the only one which possesses
the Markovian property of the absence of afteraction, that is whatever is the
actual age of a system, the remaining time does not depend on the past and has
the same distribution as the lifetime itself. It is known [1, 2, 3] that the
Liouville equation for NSO contains the source
$K=K_{zub}=-\varepsilon[\ln\rho(t)-\ln\rho_{q}(t,0)]$ (3) which becomes
vanishingly small after taking the thermodynamic limit and setting
$\varepsilon\rightarrow 0$, which in the spirit of [10] corresponds to the
infinitely large lifetime value of an infinitely large system. For a system
with finite size this source is not equal to zero. In [8] this term enters the
modified Liouville operator and coincides with the form of Liouville equation
suggested by Prigogine [16] (the Boltzmann-Prigogine symmetry), when the
irreversibility is introduced in the theory at the microscopic level.

In [10] a new interpretation of the method of the NSO is given, in which the
operation of taking the invariant part [1, 2] or the use of an auxiliary
”weight function” (in the terminology of [6, 7, 8]) in NSO are treated as
averaging the quasi-equilibrium statistical operator over the distribution of
past lifetime of a system. This approach agrees with the approach of the
general theory of random processes, the renewal theory, and also with the
conception of Zubarev work [2] where the NSO is conceived as some averaging
over the initial moment of time.

The statistical operator depends on the information-gathering interval
$(t_{0},t)$, but it must be borne in mind that this is the formal point
consisting in that (as Kirkwood pointed out) that the description to be built
must contain all the previous history in the development of the macrostate of
the system. In [6, 7, 8] several basic steps for the construction of the NESOM
formalism are indicated: a third basic step has just been introduced, namely,
the inclusion of the past history (other terms used are retro-effects or
historicity) of the macrostate of the dissipative system. A fourth basic step
needs now to be considered, which is a generalization of Kirkwood’s time-
smoothing procedure: the one that accounts for the past history and future
dissipative evolution. The time-smoothing procedure introduces a kind of
Prigogine’s dynamical condition for dissipativity. The procedure introduces a
kind of evanescent history as the system macrostate evolves toward future from
the initial condition at time $t_{0}$. The function $w(t;t_{0})$ [6, 7, 8]
introduces the time-smoothing procedure. In principle, any kernel provided by
the mathematical theory of convergence of trigonometrical series and transform
integrals provides is acceptable for these purposes. Kirkwood, Green, Mori [9,
17, 18] and others have chosen what in mathematical terminology is Fejèr (or
Cesàro-1) kernel. Meanwhile Zubarev introduced the one consisting in Abel’s
kernel for $w$ in Eq. (1) - which apparently appears to be the best choice,
either mathematically but mostly physically: that is, taking
$w(t;t_{0})=\varepsilon\rme^{\varepsilon(t_{0}-t)}$, where $\varepsilon$ is a
positive infinitesimal value which tends to zero after the calculation of
averages has been performed, and with $t_{0}$ going to $-\infty$. Therefore a
process with fading memory is introduced. In Zubarev’s approach this fading
process occurs in an adiabatic-like form towards the remote past: as time
evolves memory decays exponentially with lifetime $\varepsilon^{-1}$ [8]. The
approach suggested in [10, 11, 12] and in the present work enables to use a
family of functions $w(t,t_{0})=p_{q}(u)$ and makes clear both their physical
sense and those physical situations in which one or another function
$w(t,t_{0})=p_{q}(u)$ can be used.

Besides the Zubarev’s form of NSO [1, 2, 3], the NSO formulation in the Green-
Mori form [17, 18] is known, where one assumes the auxiliary weight function
[6, 7, 8] to be equal to $W(t,t_{0})=1-(t-t_{0})/\tau$;
$w(t,t_{0})=dW(t,t_{0})/dt_{0}=1/\tau$; $\tau=t-t_{0}$. After averaging one
sets $\tau\rightarrow\infty$. This choice at $p_{q}(u=t-t_{0})=w(t,t_{0})$
coincides with the uniform lifetime distribution. The source in the Liouville
equation takes the form $K=\ln\rho_{q}/\tau$. In [1] this form of NSO is
compared to the Zubarev’s form. One could name many examples of explicitely
setting the function $p_{q}(u)$. Each and every definition implies some
specific form of the source term $K$ in the Liouville equation, some specific
form of the modified Liouville operator and NSO [10, 11, 12]. Thus the whole
family of NSO is defined.

It is possible to make different assumptions about the form of the function of
$p_{q}(u)$, getting different expressions for the source in the Liouville
equation and for non-equilibrium characteristics of the system. It is possible
to show [12] that certain choices of the function of $p_{q}(u)$ result in the
changes in non-equilibrium characteristics in the limit of infinitely large
average lifetimes as well. In [10, 11, 12] an analogy is traced to the passage
to the thermodynamic limit of systems of infinite size. So explicit form of
the function $p_{q}(u)$ is important for describing non-equilibrium systems by
the NSO method.

Setting the form of the function $p_{q}(u)$ reflects not only the internal
properties of a system, but also the influence of the environment on an open
system, the particular character of its interaction with the environment [8].
In [2] a physical interpretation of the exponential distribution for the
function $p_{q}(u)$ is given: a system evolves freely like an isolated system
governed by the Liouville operator. Besides that the system undergoes random
transitions, and the phase point representing the system switches from one
trajectory to another one with an exponential probability under the influence
of the ”thermostat”; the average intervals between successive push events
increase infinitely. This takes place if the parameter of the exponential
distribution tends to $0$ after the transition to the thermodynamic limit.
Real physical systems have finite sizes. The exponential distribution
describes completely random systems. The influence of the environment on a
system can have organized character as well, for example, this is the case of
systems in a stationary non-equilibrium state with input and output fluxes.
The character of the interaction with the environment can also vary; therefore
different forms of the function $p_{q}(u)$ can be used.

The adequate choice of the function $p_{q}(u)$ is important for correct
description of the non-equilibrium properties of statistical systems. To find
the type of function $p_{q}(u)$, it is necessary to resort to some general
principles, such as MaxEnt principle. In this work two variants of MaxEnt are
used for this purpose: the one introduced in [19] for the Liouville equation
with a source (Section 2) and that suggested in [26] for the superstatistics
(Section 3).

##  2 Maximum entropy principle for Liouville equations with source

In this paper we apply the maximum entropy principle for the determination of
the function $p_{q}(u)$. The same approach was applied in [19] for the
evolution equations with source terms. In [7, 10, 11, 12] a general form for
the source in the Liouville equation for $\ln\rho(t)$ (3) is obtained. For our
case the source term has the following form

$K=p_{q}(0)\ln\rho_{q}(t,0)+\int_{0}^{\infty}\frac{\partial p_{q}(u)}{\partial
u}\ln\rho_{q}(t-u,-u)du.$ (4)

In [19] in the Liouville equation the distribution function ${\rho}$(z,t) is
written in a form

$\rho(\overrightarrow{z},t)=Nf_{ME}(\overrightarrow{z},t)=\frac{N}{Z}\exp\\{-\sum_{i=1}^{M}\lambda_{i}A_{i}\\},$
(5)

where $A_{i}(\vec{z})$ are $M$ appropriate quantities that are functions of
the phase space point ${\vec{z}}$; the quantities ${A_{i}(\vec{z})}$
correspond to the values $P_{i}$ from (1). The partition function $Z$ is given
by

$Z=\int\exp\\{-\sum_{i=1}^{M}\lambda_{i}A_{i}\\}d^{N}z.$ (6)

The function ${f_{ME}(\vec{z},t)}$ is normalized to unity:

$\int f_{ME}(\overrightarrow{z},t)d^{N}z=1;$ (7)

$\int\rho(\overrightarrow{z},t)d^{N}z=N(t);\quad\frac{dN}{dt}=\int Kd^{N}z;$
(8) $\frac{\partial\rho}{\partial
t}+\overrightarrow{w}\nabla\rho=\frac{d\rho}{dt}=K;\quad\nabla\overrightarrow{w}=0.$

The probability distribution ${f_{ME}(\vec{z},t)}$ is the one that maximizes
the entropy $S[f]$ under the constraints imposed by normalization and relevant
mean values $\langle A_{i}\rangle=\int A_{i}\rho d^{N}z$ (or $a_{i}=\langle
A\rangle_{i}/N)$. The re-scaled mean values $a_{i}$ and the associated
Lagrange multipliers $\lambda_{i}$ are related by the Jayne’s relations [20,
21]

$\lambda_{i}=\frac{\partial S}{\partial a_{i}},\quad\ a_{i}=\frac{\langle
A_{i}\rangle}{N}=-\frac{\partial}{\partial\lambda_{i}}(\ln Z)\,,$ (9) $S=-\int
f\ln fd^{N}z=\ln Z+\sum_{i}\lambda_{i}a_{i}.$

If we choose for $\ln\rho(z,t)$ the function

$\ln\rho=\ln\varrho(t)=\int_{0}^{\infty}p_{q}(u)\ln\varrho_{q}(t-u,-u)du=\int_{-\infty}^{t}p_{q}(t-t_{0})\ln{\rho}_{q}(t_{0},t_{0}-t){dt}_{0}$
(10)

from (1), which is included in the Liouville equation (3) (without integrating
over time) with $p_{qzub}\rightarrow p_{q},K_{zub}\rightarrow K$, and,
following [1], choose

$\mathit{\lambda}_{i}=p_{q}(t-t_{0})F_{i}(t_{0}),$ (11)

then

$\displaystyle\ln{\rho}(\vec{z},t)=\int_{-\infty}^{t}p_{q}(t-t_{0})\ln{\rho}_{q}(t_{0},t_{0}-t){dt}_{0}=$
(12)
$\displaystyle-\int_{-\infty}^{t}(\sum_{i}\lambda_{i}A_{i}+p_{q}(t-t_{0})\ln
Z_{1}){dt}_{0}=\ln f^{\prime}_{ME}+\ln N\,,$

where

$\ln f^{\prime}_{ME}=-\int_{-\infty}^{t}\sum_{i}\lambda_{i}A_{i}{dt}_{0}-\ln
Z_{\lambda};$ $\ln N={\Delta}Z=\ln
Z_{\lambda}-\int_{-\infty}^{t}p_{q}(t-t_{0})\ln Z_{1}{dt}_{0};\ $ (13)
$Z_{\lambda}=\int\exp[-\int_{-\infty}^{t}\sum_{i}\lambda_{i}A_{i}{dt}_{0}]d^{N}z,\quad
Z_{1}=\int\exp[-\sum_{i}F_{i}A_{i}]d^{N}z$

($F_{i}$ are taken from (1)). The values $Z_{\lambda}$ and $Z_{1}$ in the
terminology of [1] are related to the partition functions for a non-
equilibrium and relevant statistical operator accordingly.

In [19] for the Liouville equation of the kind (3) with constant sources
equation one gets for $d\lambda_{i}/dt$

$\frac{d\lambda_{i}}{dt}=(\sum_{i=1}^{M}C_{ji}{\lambda}_{j})-\frac{1}{N}\frac{\partial}{\partial
a_{i}}\int{K\ln f_{ME}}d^{N}z,$

where the Zubarev-Peletminskiy selection rule [22, 23, 1, 7, 8]

$\vec{{w}}\vec{\nabla}A_{i}=\sum_{j=1}^{M}{C_{ij}}A_{j},\ (i=1,\dots,M),\
\frac{d\vec{z}}{dt}=\vec{w}(\vec{z});\ \frac{\partial\mathit{\rho}}{\partial
t}+\vec{\nabla}(\rho\vec{w})=K$ (14)

is used; $i,j=1,2...$; the $C_{ij}$ are c-numbers. In other representations
the quantities $A_{i}$ can depend on the space variable, that is, when
considering local densities of dynamical variables, and then the $C_{ij}$ can
depend on the space variable as well or be differential operators.

If more complex shape of the source (4) is considered, the equation for
$d\lambda_{i}/dt$ takes on the form

$\displaystyle\frac{d\lambda_{i}}{dt}=\left(\sum_{j=1}^{M}C_{ji}{\lambda}_{j}\right)-\frac{\partial}{\partial
a_{i}}\left(\frac{1}{N}\int{K\ln f_{ME}d^{N}z}\right)-$ (15)
$\displaystyle\sum_{j}{\lambda}_{j}\frac{\partial}{\partial
a_{i}}\left({\displaystyle\frac{1}{N}}\int A_{j}Kd^{N}z\right)-\ln
Z_{1}{\displaystyle\frac{\partial}{\partial
a_{i}}}\left({\displaystyle\frac{\dot{N}}{N}}\right).$

We replace the operators ${\displaystyle\frac{\partial}{\partial a_{i}}}$ and
${\displaystyle\frac{\partial}{\partial\mathit{\lambda}_{i}}}$ taking into
account (9)-(13) by the functional differentiation of the kind

$\frac{\partial}{\partial a_{i}}\rightarrow\frac{\delta}{\delta
a_{i}}=N\frac{\delta}{\delta\langle
A_{i}\rangle},\quad\frac{\partial}{\partial\lambda_{i}}\rightarrow\frac{\delta}{\delta\lambda_{i}},$

which takes off the integration over time. For example

$\frac{\delta}{\delta\langle A_{i}\rangle}\ln
Z_{\lambda}=-p_{q}(t-t_{0})\sum_{k}\frac{\partial F_{k}}{\partial\langle
A_{i}\rangle}\langle A_{k}\rangle;\quad\frac{\delta\ln
Z_{\lambda}}{\delta\lambda_{i}}=-\langle A_{i}\rangle.$

The relations (5-8) and (9) are thus hold. If to take into account that
$\int\dots\rho_{q}d^{N}z=\int\dots\rho d^{N}z=\langle\dots\rangle$ in the NSO
method, then

$\frac{\partial}{\partial a_{i}}N=\frac{\partial}{\partial
a_{i}}\left(\frac{\dot{N}}{N}\right)=0\,.$ (16)

Let us consider the integrals in the rhs of (15) of the form $\int KB(z)dz$,
$B$ being an arbitrary function of the dynamic variables $z$, and the source
term $K$ taken from (4); for Eq. (8), (14)
$K=[p_{q}(0)\ln\rho_{q}(t,0)+{\displaystyle\int_{-\infty}^{t}\frac{\partial
p_{q}(t-t_{0})}{\partial t}}\ln\rho_{q}(t_{0},t_{0}-t)dt_{0}]\rho$. Assume
that ${p_{q}(u)}$ does not depend on $z$. Integrating by parts and assuming
${p_{q}(u)_{u\rightarrow\infty}\rightarrow 0}$, we get:

$\int KBd^{N}z=-\int_{0}^{\infty}p_{q}(u)\frac{d}{du}\langle
B\ln\rho_{q}(t-u,-u)\rangle du;\quad u=t-t_{0}.$ (17)

Taking into account (16) and the fact that the operation
${\displaystyle\frac{\partial}{\partial
a_{i}}\rightarrow\frac{\mathit{\delta}}{\mathit{{\delta}a}_{i}}}$ eliminates
the integration by time, the equation (15) takes on the form

${F_{i}(t_{0})\frac{dp_{q}(t-t_{0})}{dt}=-p_{q}(t-t_{0})C_{i}-p_{q}(t-t_{0})r_{1}-p_{q}^{2}}(t-t_{0})r_{2},$
(18)

where $C_{i}=\sum_{j}C_{ji}F_{j}(t_{0})$,

$r_{1}=-\frac{\partial}{\partial\langle
A_{i}\rangle}\frac{d}{dt}\langle(\ln{\rho}(t)-\ln
N)\ln{\rho}_{q}(t_{0},t_{0}-t)\rangle;$ (19)

$r_{2}=-\sum_{j}{F_{j}(t_{0})r_{2j};\;\quad
r_{2j}=\frac{\partial}{\partial\langle A_{i}\rangle}\frac{d}{dt}\langle
A_{j}(t)\ln{\rho}_{q}(t_{0},t_{0}-t)}\rangle.$ (20)

An unknown function $p_{q}(u)$ enters the expression (19) through the terms
$\ln\rho(t)$ and $\ln N$. To get rid of this dependence, we use the averaging
theorem. For the expressions for $\ln{\rho}(t)$ and $\ln N$ in (19) we take
all terms besides $p_{q}(u)$ out of the time integration. For each of these
function however a different effective average time value should be used. The
remaining integrals over ${p_{q}(u)}$ are equal to unity. We get:

$\ln N\simeq\ln Z_{1}(c_{3})-\ln Z_{1}(c_{4}),$ (21)

$\ln{\rho}(t)\simeq\ln{\rho}_{q}(c_{1},c_{1}-t)=-(\sum_{m}F_{m}(c_{1})A_{m}(c_{1}-t)+\ln
Z(c_{1})).$ (22)

Let us make another approximation and change the order of the operations
${\partial/\partial\langle A_{i}\rangle}$ and $d/dt$ in the expressions
(19)-(20). The value $\int\limits_{t_{0}}^{t}r_{1}dt=D(t)-D(t_{0})$ enters the
expression (18), where

$\displaystyle D(t)=-\frac{\partial}{\partial\langle
A_{i}\rangle}\left\langle(\ln{\rho}(t)-\ln
N)\ln{\rho}_{q}(t_{0},t_{0}-t)\right\rangle$ $\displaystyle=F_{i}(t_{0})[\ln
Z(c_{3})-\ln Z(c_{4})-\ln Z(c_{1})]-$ (23) $\displaystyle F_{i}(c_{1})\ln
Z(t_{0})+\sum_{m,n}\big{(}\langle A_{m}A_{n}\rangle-\langle
A_{m}\rangle\langle A_{n}\rangle\big{)}\left[\frac{F_{n}(t_{0})}{\langle
A_{m}A_{i}\rangle-\langle A_{m}\rangle\langle A_{i}\rangle}-\right.$
$\displaystyle\left.-\frac{F_{m}(c_{1})}{\langle A_{i}A_{n}\rangle-\langle
A_{i}\rangle\langle
A_{n}\rangle}\right]-\sum_{m,n}F_{m}(c_{1})F_{n}(t_{0})\sum_{k}\frac{\langle
A_{k}A_{m}A_{n}\rangle-\langle A_{m}A_{n}\rangle\langle A_{k}\rangle}{\langle
A_{i}A_{k}\rangle-\langle A_{i}\rangle\langle A_{k}\rangle};$

$\displaystyle r_{2j}=\frac{\partial}{\partial
t}\left[\sum_{m}F_{m}(t_{0})\sum_{k}\frac{\langle
A_{k}A_{j}A_{m}\rangle-\langle A_{j}A_{m}\rangle\langle A_{k}\rangle}{\langle
A_{i}A_{k}\rangle-\langle A_{i}\rangle\langle A_{k}\rangle}+\delta_{ij}\ln
Z(t_{0})\right.$ $\displaystyle\left.-\sum_{m}\frac{\langle
A_{j}A_{m}\rangle-\langle A_{j}\rangle\langle A_{m}\rangle}{\langle
A_{i}A_{m}\rangle-\langle A_{i}A_{m}\rangle}\right]\,.$ (24)

The values of the correlators $\langle A_{i}\rangle$, $\langle
A_{i}A_{k}\rangle$, $\langle A_{j}A_{k}A_{m}\rangle$ are averaged with
$\rho(t)$ and are $t$-dependent. In deriving (23), (24) we used the relations
like

$\frac{\partial\ln Z(c)}{\partial\langle A\rangle}=\sum_{n}\frac{\partial\ln
Z(c)}{\partial F_{n}(c)}\frac{\partial F_{n}(c)}{\partial\langle
A\rangle}=\sum_{n}\frac{\langle A_{n}\rangle}{\langle AA_{n}\rangle-\langle
A\rangle\langle A_{n}\rangle};\quad A_{n}(-c)=e^{\textstyle-icL}A_{n};$

$\frac{\partial F_{m}}{\partial\langle A_{i}\rangle}=\frac{1}{\langle
A_{i}A_{m}\rangle-\langle A_{i}\rangle\langle A_{m}\rangle}.$

One can proceed with the expression (24) using the relations

$\frac{\partial F_{i}}{\partial t}=\sum_{k}\frac{\partial
F_{i}}{\partial\langle A_{k}\rangle}\frac{\partial\langle
A_{k}\rangle}{\partial t}=\sum_{k}{\frac{\partial F_{k}}{\partial\langle
A_{i}\rangle}\frac{\partial\langle A_{k}\rangle}{\partial
t};\;\;\frac{\partial F_{i}}{\partial\langle A_{k}\rangle}=\frac{\partial
F_{k}}{\partial\langle A_{i}\rangle}=\frac{\partial^{2}S}{\partial\langle
A_{k}\rangle\partial\langle A_{i}\rangle}\,;}$

$\frac{\partial\ln Z}{\partial t}=-\sum_{m}\frac{\partial F_{m}}{\partial
t}\langle A_{m}\rangle.$

The solution to (18) has the form

$p_{q}(t-t_{0})=\frac{p_{q}(0)M}{1+p_{q}(0){\displaystyle\int\limits_{t_{0}}^{t}}{\displaystyle\frac{1}{F_{i}}}r_{2}Mdt}\,.$
(25)

Since $C_{i}$ and $F_{i}(t_{0})$ does not depend on $t$,

$\displaystyle
M(t)=\exp\left\\{-\frac{C_{i}}{F_{i}}(t-t_{0})-\frac{1}{F_{i}}\int_{t_{0}}^{t}r_{1}dt\right\\}=$
$\displaystyle\exp\left\\{-\frac{C_{i}}{F_{i}}(t-t_{0})-\frac{1}{F_{i}}(D(t)-D(t_{0}))\right\\}\
,$ (26)

$M(t_{0})=1$, where $\int_{t_{0}}^{t}r_{1}dt$ is written in (23).

Integrating by parts the second term in the denominator of (25) write it in
the following form:

$p_{q}(t-t_{0})=\frac{p_{q}(0)M(t)}{1-L};$ (27)
$L=p_{q}(0)\frac{1}{F_{i}}\left[\left(\int r_{2}dt\right)_{|t_{0}}-\left(\int
r_{2}dt\right)_{|t}M(t)-\int_{t_{0}}^{t}\left(\int
r_{2}dt\right)\left(\frac{C_{i}}{F_{i}}+\frac{1}{F_{i}}\frac{dD(t)}{dt}\right)M(t)dt\right],$

where

$\displaystyle\int
r_{2}{dt}=\sum_{j}\sum_{m}F_{m}(t_{0})F_{j}(t_{0})\sum_{k}\frac{\langle
A_{k}A_{j}A_{m}\rangle-\langle A_{j}A_{m}\rangle\langle A_{k}\rangle}{\langle
A_{i}A_{k}\rangle-\langle A_{i}\rangle\langle A_{k}\rangle}+$ (28)
$\displaystyle F_{i}\ln Z(t_{0})-\sum_{m}\sum_{j}F_{j}(t_{0})\frac{\langle
A_{j}A_{m}\rangle-\langle A_{j}\rangle\langle A_{m}\rangle}{\langle
A_{i}A_{m}\rangle-\langle A_{i}\rangle\langle A_{m}\rangle};$

$\displaystyle\frac{dD(t)}{dt}=\sum_{m,n}\left\\{F_{n}(t_{0})\frac{\langle
A_{m}A_{n}\rangle-\langle A_{m}\rangle\langle A_{n}\rangle}{\langle
A_{m}A_{i}\rangle-\langle A_{m}\rangle\langle
A_{i}\rangle}\left[\frac{d}{dt}\ln(\langle A_{m}A_{n}\rangle-\langle
A_{m}\rangle\langle A_{n}\rangle)-\right.\right.$ (29)
$\displaystyle\left.-\frac{d}{dt}\ln(\langle A_{m}A_{i}\rangle-\langle
A_{m}\rangle\langle A_{i}\rangle)\right]+F_{m}(c_{1})\frac{\langle
A_{m}A_{n}\rangle-\langle A_{m}\rangle\langle A_{n}\rangle}{\langle
A_{i}A_{n}\rangle-\langle A_{i}\rangle\langle A_{n}\rangle}\times$
$\displaystyle\left[\frac{d}{dt}\ln(\langle A_{m}A_{n}\rangle-\langle
A_{m}\rangle\langle A_{n}\rangle)-\frac{d}{dt}\ln(\langle
A_{i}A_{n}\rangle-\langle A_{i}\rangle\langle A_{n}\rangle)\right]-$
$\displaystyle-F_{m}(c_{1})F_{n}(t_{0})\sum_{k}\frac{\langle
A_{k}A_{m}A_{n}\rangle-\langle A_{m}A_{n}\rangle\langle A_{k}\rangle}{\langle
A_{i}A_{k}\rangle-\langle A_{i}\rangle\langle
A_{k}\rangle}\left[\frac{d}{dt}\ln(\langle A_{k}A_{m}A_{n}\rangle-\langle
A_{m}A_{n}\rangle\langle A_{k}\rangle)-\right.$
$\displaystyle\left.\left.-\frac{d}{dt}\ln(\langle A_{i}A_{k}\rangle-\langle
A_{i}\rangle\langle A_{k}\rangle)\right]\right\\}\,.$

The expression for $M(t)$ is given in (26), and $p_{q}(0)$ is determined from
the conditions for the norm $\int_{0}^{\infty}p_{q}(u)du=1$ and for the
average lifetime $\langle\Gamma\rangle=\int_{0}^{\infty}up_{q}(u)du$.

If one either considers the stationary case or assumes a weak time dependence
in the correlators in (23)-(24), $D(t)\simeq D(t_{0}),r_{2}\simeq 0$, and
(25)-(27) take on the form

$p_{q}(t-t_{0})=p_{q}(0)\exp\\{-{\frac{C_{i}}{F_{i}}}(t-t_{0})\\}$ (30)

with $C_{i}/F_{i}=p_{q}(0)=1/\langle\Gamma\rangle$. In (30) an exponential
distribution for $p_{q}(u)$, used in [1, 2, 3] is obtained. However generally
the correlators in (23)-(24) are time dependent. Applying the full form of
(27) for concrete systems it is possible to state the conditions where the
denominator in (27) is considerably different from unity, hence the lifetime
distribution essentially deviates from the exponential one (30).

The term $L$ in the denominator of (27) is small, since $p_{q}(0)\approx
1/\langle\Gamma\rangle\ll 1$. In the denominator of (27) it stands in the
combination with 1,  $1-L\approx 1$ . Therefore one can write
$p_{q}=p_{q}(0)M[1+L+L^{2}+\dots]$ , and the distribution (27) is close to the
exponential distribution (30) used in [1, 2, 3, 7, 8]. This results agrees
with the results of [13], where the exponential distribution for the lifetime
is shown to be a limiting distribution. But the expression for
$\langle\Gamma\rangle$ is explicitely given in (30), and in (23)-(29). It was
already pointed above that the situations where L is comparable to 1 can arise
as well.

For the maximum entropy principle with the Shannon measure of the information
entropy the exponential distribution used in [1, 2, 3], is basic. Choosing
another measures for the information entropy (e.g. Tsallis and Renyi measures
[24, 25]) changes the function $f_{ME}$ (5), which yields another forms of the
lifetime distributions. For the NSO method the functions $f_{ME}$ are in the
form (5), and the information entropy is given by the Shannon measure, hence
basic distribution is the exponential one.

The expressions (25), (27) for $p_{q}(u)$ depend on $F_{i}$, the functions
$C_{i}=\sum_{j}C_{ji}F_{j}$, $r_{k},\;k=1,2$, in (18)-(19) depend on the index
$i$, the function $p_{q}(u,i)$ depends on $i$ as well. One can get an
$i$-independent function $p_{q}(u)$ by symmetrizing the distribution, for
example, using the operation

$p_{q}(u)=\left[\prod_{i=1}^{M}p_{q}(u,i)\right]^{\textstyle\frac{1}{M}}.$

Such formulation of maxent principle, as in [19], gives the distribution for
the lifetime, related to the exponential distribution which serves in this
case as the base one. Distributions of other type can be obtained using some
other form of maxent principle.

##  3 Another approach to the method of maximum entropy

Yet another approach to the determination of the type of function of
distribution of lifetime is related to the method of maximal entropy inference
(”maxent”), developed in [26] for the determination of the distribution of
superstatistics. We note a formal similarity between the superstatistics
method where the averaging is performed over the parameter $\beta$, (for
example, the inverse temperature)

$p(E)\propto\int_{0}^{\infty}f(\beta)e^{-\beta E}d\beta\,,$

and the NSO method where the averaging is performed over the past life spans
$u=t-t_{0}$,

$\ln\rho(t)=\int_{0}^{\infty}p_{q}(u)\ln\rho_{q}(t-u,-u)du;\ u=t-t_{0},$

which was already used in [27]. Therefore the maxent method of [26] can be
applied for determining the function ${p_{q}(u)}$. The analogy here is not
merely formal. The principal assumption of [26], that is the separation of the
time scales is essential for the NSO method as well [1, 2, 3]. In the approach
of superstatistics [28, 29, 30] the system is split into cells and local
fluctuations of the value $\beta$ are considered; the fluctuations of the
value $u=t-t_{0}$ affect the complete system.

Closely related is also the research of Crooks [31]. He studies general non-
equilibrium systems, without assuming that the system can be divided into
different cells that are at local equilibrium. Crooks claims that instead of
trying to obtain the probability distribution of the entire non-equilibrium
system, one has to try to estimate the “metaprobability,” the probability of
the microstate distribution. Crooks also uses the maximum entropy principle
but sets in (31) $\lambda_{3}=0$. The main difference is that Crooks does not
assume a local equilibrium in the cells, hence his approach, though being an
interesting theoretical construction, does not give a straightforward physical
interpretation to the fluctuating parameter $\beta$. However such an approach
can be applied to the fluctuations of the value $u=t-t_{0}$. In the approach
of [26] one obtains a local fluctuating temperature that coincides with the
thermodynamic temperature and which can in principle be measured. The work of
Crooks is used by Naudts [32] to describe equilibrium systems. The author
shows that some well-known results of the equilibrium statistical mechanics
can be reformulated in a very general context with the use of the concepts
introduced in [26, 29, 31].

In [26] following expression is obtained for the distribution function of the
superstatistics $f(\beta;\lambda_{i})$

$f(\beta;\lambda_{i})=\frac{Z(\beta)^{-\lambda_{1}/V}}{Z(\lambda_{i})}\exp(-\beta\lambda_{2}\frac{E(\beta)}{V}-\lambda_{3}g(\beta)),$

where $\lambda_{i}$ are Lagrange multipliers, $V$ being an arbitrary constant
(taking out a common factor out of the definition of $\lambda_{1}$ and
$\lambda_{2}$ will turn out to be useful in the following). Using the well-
known formula $S_{\beta}=\ln Z(\beta)+\beta E(\beta)$ with $Z(\lambda_{i})$
being a normalization constant that is fixed by the condition $\langle
1\rangle_{\beta}=1$.

The same approach with said limitations used for the function $p_{q}(u)$,
yields

$p_{q}(u;\lambda_{i})=\frac{Z(t-u)^{-\lambda_{1}/V}}{Z(\lambda_{i})}\exp(-\lambda_{2}\frac{\sum_{m}F_{m}(t-u)\langle
A_{m}\rangle}{V}-\lambda_{3}{g(u))},$ (31)

where $g(u)$ is an arbitrary function of $u$, which is determined by the
physical peculiarities of the behaviour of the system in one or another period
of its history. Expression (31) is obtained by the optimization of the entropy
$S(\lambda_{i})=\int p_{q}(u)\ln p_{q}(u)du$ with the constraints for entropy

$\int p_{q}(u)S(u)du=\int
p_{q}(u)\int\rho_{q}(t-u,-u)\ln\rho_{q}(t-u,-u)dzdu=$ $\int
p_{q}(u)[-\sum_{m}F_{m}(t-u)\langle A_{m}\rangle-\ln Z(t-u)]du$

and parameters $\int p_{q}(u)\sum_{m}F_{m}(t-u)\langle A_{m}\rangle du$.
Similarly to [26], one can set the functions $g(u)$ in a different fashion.
For example, for
$g(u)=u,\lambda_{1}=\lambda_{2}=0,\lambda_{3}=1/\langle\Gamma\rangle$, where
$\langle\Gamma\rangle$ is the average span of past life of a system (till the
present time moment), we get the exponential distribution used in [1, 2, 3].
Setting $g(u)=\ln u$ with appropriate corresponding values of $\lambda$ one
gets the power-like distribution for ${p_{q}(u)}$; setting $g(u)=(\ln u)^{2}$
with corresponding $\lambda$ results in the log-normal distribution. Thus
setting the function $g(u)$ and $\lambda$ accordingly it is possible to obtain
various distributions for the lifetime considered, for example, in [33].

It is possible to examine more difficult cases when the behaviour of the
system changes at different stages of its evolution, when, for example the
function $g(u)=\ln u$ yields the power-like function $p_{q}(u)$ at $u<c$, and
$g(u)$ gives an exponential shape of $p_{q}(u)$ at $u>c$.

##  4 Conclusion

For the determination of the lifetime distribution in the NSO method the
method of maximum entropy principle as in [19] is used. The obtained
distribution is close to exponential $p_{qzub}(u)$ (2), but does not coincide
with it. It is possible to find conditions at which this difference is
essential. Using other variants of the maximum entropy principle, as in [26],
it is possible to obtain other distributions except exponential one, in
particular, power-like and log-normal distributions, transitory ones between
them, as well as distributions of other classes.

In the interpretation of [2] it is the random value $t_{0}$ in $u=t-t_{0}$
that fluctuates. In [2] the limiting transition is performed for the parameter
$\varepsilon,\varepsilon\rightarrow 0$ in the exponential distribution
$p_{q}(u)=\varepsilon\exp\\{-\varepsilon u\\}$ after the thermodynamic
limiting transition. In the interpretation of [10] it corresponds to the
average lifetime of a system tending to infinity:
$\langle\Gamma\rangle=\langle t-t_{0}\rangle=1/\varepsilon\rightarrow\infty$.
But the average intervals between successive random jumps grow infinitely,
getting larger than the lifetime of a system. Therefore the source term in the
Liouville equation turns to 0. If however the distribution ${p_{q}(u)}$
changes over the interval of the lifetime, the influence of the environment
which caused this change, remains within the life span even if the lifetime
tends to infinity [12].

There are numerous experimental confirmations for such change of the lifetime
distribution $p_{q}(u)$ over the interval of the system lifetime. The examples
thereto are the transition to chaos and the transition from laminar to
turbulent flow which are accompanied by the change of the distribution of
$p_{q}(u)$. In [34, 35] the passage from Gaussian to non-Gaussian behaviour of
the distribution of the first-passage time for some time moment is
demonstrated. Besides the real systems possess finite sizes and finite
lifetime. Therefore influence of surroundings on them is always present.

Slow change of the function $g(u)=u=t-t_{0}$ corresponds in the interpretation
of [2] to the slow change of the random value $t_{0}$ on a temporal scale.
Accordingly slow change of the function $g(u)=\ln u$ corresponds to the slow
change of $\ln(t-t_{0})$. Setting other functions $g(u)$, for example,
$g(u)=(\ln u)^{2}$ and so on is explained on the same footing.

In the present work by means of two variants of the method of maximum entropy
we obtained the expressions for the distribution of the lifetime value. It is
noted that the choice of the form of the distribution function for the
lifetime value can affect the non-equilibrium behavior of a system even after
performing the thermodynamic limiting transition.

## References

## References

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These authors contributed equally to this work.

These authors contributed equally to this work.

[3]Dheeraj Kulkarni These authors contributed equally to this work.

1]Department of Mathematics, Indian Institute of Science Education and
Research, Bhopal, 462066, Madhya Pradesh, India

2]Department of Mathematics, Indian Institute of Science Education and
Research, Bhopal, 462066, Madhya Pradesh, India

[3]Department of Mathematics, Indian Institute of Science Education and
Research, Bhopal, 462066, Madhya Pradesh, India

# On Certain Rigidity results of compact regular $(\kappa,\mu)$-manifolds

Sannidhi Alape<EMAIL_ADDRESS>Atreyee Bhattacharya
<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  *

###### Abstract

In this article, we investigate the Riemannian and semi-Riemannian metrics on
the base space of the Boothby-Wang fibration of a closed regular non-Sasakian
$(\kappa,\mu)$-manifold. To this end, we study a natural class of deviations
of the projection map from being (semi-)Riemannian submersions. We consider
deviations that preserve the canonical bi-Legendrian structure on the given
$(\kappa,\mu)$-manifold. This approach gives a unified framework to analyze
rigidity results in both categories. As a consequence, in the Riemannian
category, we obtain uniqueness of Sasakian structure on the given
$(\kappa,\mu)$-manifold which orthogonalizes the canonical bi-Legendrian
structure. In the semi-Riemannian category, we obtain an explicit description
of the finitely many para-Sasakian structures which orthogonalize the
canonical bi-Legendrian structure.

###### keywords:

$(\kappa,\mu)$-spaces, Regular contact manifold, Bi-Legendrian structures

###### pacs:

[

2020 MSC Classification]53C15, 53C12, 53C25, 53D10

##  1 Introduction

Riemannian geometry of contact and symplectic manifolds has been an active
area of research linking various branches of mathematics and mathematical
physics. A particular class of contact metric manifolds, called Sasakian
manifolds, introduced by Shigeo Sasaki in [18], has garnered interest from
mathematicians and physicists alike, due to its remarkable properties. For
instance, it is known that every regular compact Sasakian manifold is
naturally “sandwiched” between two Kähler manifolds. Compact Sasakian
manifolds are also a rich source of examples of Einstein manifolds. A
generalisation of Sasakian manifolds was defined in [2], where the authors
studied contact metric manifolds $(M,\eta,\xi,\phi,g)$ satisfying the
$(\kappa,\mu)$-nullity condition

$R(X,Y)\xi=\kappa(\eta(Y)X-\eta(X)Y)+\mu(\eta(Y)hX-\eta(X)hY),$ (1.1)

where $\kappa,\mu$ are constants and 2$h$ is the Lie derivative of the tensor
field $\phi$ in the direction of the Reeb vector field $\xi$. In addition to
being natural generalizations of Sasakian manifolds, $(\kappa,\mu)$-manifolds
also generalize the class of manifolds satisfying the condition $R(X,Y)\xi=0$.
They are also invariant under $D_{a}$-homothetic transformations (see section
2.3 for details). Furthermore, $(\kappa,\mu)$-manifolds are examples of
$H$-contact manifolds ([17]) and locally $\phi$-symmetric spaces ([3]). In the
case of non-Sasakian $(\kappa,\mu)$-manifolds, the tensor field $h$ has two
eigenvalues, the eigenspaces of which define Legendrian distributions on the
underlying space. Thus, non-Sasakian $(\kappa,\mu)$-manifolds are naturally
bi-Legendrian manifolds.

One of the fundamental results relating regular contact manifolds and
symplectic manifolds is due to Boothby and Wang ([5]), wherein compact regular
contact manifolds are realised as principal $\mathbb{S}^{1}$-bundles over
symplectic manifolds with integral symplectic form and vice versa. Metric
versions of the Boothby-Wang fibration have been successfully constructed for
specific classes of contact metric manifolds. The aforementioned
correspondence between Sasakian manifolds and Kähler manifolds is one instance
of such a construction ([11]), where the projection map is a Riemannian
submersion. However, in the case of $(\kappa,\mu)$-manifolds, the Reeb vector
field not necessarily being a Killing vector field presents an obstruction for
the definition of a natural metric on the base space.

In Section 3 of this article, we investigate symplectic metric structures on
the manifold which is the base space for the Boothby-Wang fibration whose
total space is a closed regular non-Sasakian $(\kappa,\mu)$-manifold. Since
the Reeb vector field is not a Killing vector field, we notice that we cannot
have a Riemannian submersion. Hence, we consider the class of conformal
submersions. But we notice rigidity even in this case (Theorem 1.1). In fact,
we prove the following result which proves that conformal submersions are
possible only when the total space is a $K$-contact manifold.

######  Theorem 1.1.

Let $\pi:M\rightarrow B$ be the Boothby-Wang fibration from a regular closed
contact metric manifold $(M,\eta,\tilde{g},\phi)$ to a symplectic metric
manifold $(B,\omega,g,J)$. Suppose $\pi$ is a conformal submersion of the
underlying Riemannian manifolds. Then, $\pi$ is a Riemannian submersion and
$M$ is a $K$-contact manifold.

In light of the above, we consider the tensor field which quantifies the
extent to which a metric on the base space fails to define a conformal
submersion. We define the error tensor $T$ corresponding to a function $f$ on
the total space, using the equation

$\tilde{g}(TX,Y)=\tilde{g}(X,Y)-e^{2f}g(d\pi X,d\pi Y),$ (1.2)

for all $X,Y\in\Gamma(Ker(\eta))$. We focus on the error tensors for which the
Legendrian distributions are preserved. We prove that this condition forces
the absolute value of the index of the $(\kappa,\mu)$-manifold to be strictly
greater than one. To our surprise, the solutions obtained under these
assumptions point to a unique Riemannian metric on the base space. The
conclusions are summarized as follows.

######  Theorem 1.2.

Suppose $(T,f,g)$ is a solution to equation (1.2) and $hT=Th$. Then,

  * •

The Boeckx index $I_{M}$ satisfies the condition $|I_{M}|>1$.

  * •

The error tensor $T$ has two eigenvalues, with the eigenspaces coinciding with
that of $h$. The eigenvalues, and consequently $T$ itself, are determined by
the function $f$.

  * •

The metric $g$ is independent of the choice of $f$ and $T$.

The unique solution thus obtained also happens to be a Kähler metric, which
has been studied in a different context ([7]). We also establish an equivalent
formulation of the condition $hT=Th$ in terms of certain geometric properties
of the Kähler metric described above. As a consequence, we obtain rigidity of
$K$-contact structures which orthogonalize the bi-Legendrian structure
associated with the given $(\kappa,\mu)$-manifold. The result is as follows:

######  Theorem 1.3.

Let $(M,\eta,g,\phi)$ be a regular closed non-Sasakian $(\kappa,\mu)$-manifold
with $|I_{M}|>1$. There is a unique $K$-contact structure
$(M,\eta,\bar{g},\bar{\phi})$ such that the bi-Legendrian structure determined
by the eigenspaces of $h=\frac{1}{2}\mathcal{L}_{\xi}\phi$ are orthogonal.
This unique structure coincides with the canonical Sasakian structure
associated with the $(\kappa,\mu)$-manifold $(M,\eta,g,\phi)$.

In the final section, we investigate semi-Riemannian metrics associated to the
symplectic structure on the base space, motivated by the results in [7]. A
simple observation rules out the possibility of a non-trivial semi-Riemannian
submersion. We thus consider the error tensor which measures the deviation of
the projection map from being a semi-Riemannian submersion. In this case, the
error equation is given by

$\tilde{g}(T\tilde{X},\tilde{Y})=\tilde{g}(\tilde{X},\tilde{Y})-g(X,Y),$ (1.3)

where $X,Y\in\Gamma(B)$. The consequences of the error tensor preserving the
Legendrian distributions are summarized in the following result.

######  Theorem 1.4.

Suppose $(T,g)$ is a solution to equation (1.3) and $hT=Th$. Then,

  * •

The Boeckx index $I_{M}$ satisfies the condition $|I_{M}|<1$.

  * •

The solution set has cardinality $2^{n}$.

We also obtain an explicit description of the finitely many semi-Riemannian
metrics which are compatible with the symplectic structure on the base space
under the given constraints. We recover the para-Kähler structure obtained by
the authors of [7] in Theorem $3.1(ii)$ as a particular solution of the error
equation.

The paper is organized as follows. Section 2 is devoted to the preliminaries
on the Riemannian Geometry of Contact and Symplectic manifolds, and the
Boothby-Wang fibration. In Section 3, we study the Riemannian metrics
compatible with the base space of a non-Sasakian $(\kappa,\mu)$-manifold and
prove the rigidity of the solutions under certain geometric conditions. In
Section 4, we study the semi-Riemannian metrics compatible with the base space
and provide an explicit description of the finite solution space under similar
geometric conditions.

##  2 Preliminaries

###  2.1 Contact and Symplectic Manifolds

######  Definition 2.1.1.

  1. 1.

A contact form on an odd-dimensional manifold $M$ is a $1$-form $\eta$ which
satisfies the condition

$\eta\wedge(d\eta)^{n}\neq 0,$ (2.1)

where dim$(M)=2n+1$. A contact manifold is a pair $(M,Ker(\eta))$ comprising
an odd-dimensional manifold $M$ and a hyperplane field given by the kernel of
a contact form $\eta$ on $M$. The hyperplane field given by $Ker(\eta)$ is
called the contact distribution.

  2. 2.

A symplectic form on an even-dimensional manifold $B$ is a closed 2-form
$\omega$ which satisfies the condition

$\omega^{n}\neq 0,$ (2.2)

where dim$(B)=2n$. A symplectic manifold is a pair $(B,\omega)$ comprising an
even-dimensional manifold $B$ and a symplectic form $\omega$ on it.

######  Remark 2.1.2.

There could be multiple contact forms defining the same contact distribution,
each of which will be called a representative contact form for the contact
distribution. If $\eta_{1}$ and $\eta_{2}$ are two contact forms such that
$Ker(\eta_{1})=Ker(\eta_{2})$, then $\eta_{1}=f\eta_{2}$, where $f$ is a
smooth non-vanishing function on the manifold. Since we intend to work with
specific contact forms, we will use $(M,\eta)$ to denote a contact manifold.

######  Definition 2.1.3.

Given any contact manifold $(M,\eta)$, there is a unique vector field $\xi$
satisfying the conditions:

  * •

$\eta(\xi)=1$,

  * •

$d\eta(\xi,X)=0$, for any vector field $X$.

This vector field is called the Reeb vector field associated to the contact
manifold $(M,\eta)$. Throughout the article, $\xi$ will be used to denote the
Reeb vector field of the contact manifold under discussion.

For more details about contact and symplectic manifolds, we refer the reader
to [10].

We restrict our attention to a class of contact manifolds for which the Reeb
vector field is regular (a non-vanishing vector field for which there is a
flow box around each point which is pierced at most once by any integral
curve). Such contact manifolds are called regular contact manifolds. We have
the following well-known result in [5] which describes a correspondence
between closed regular contact manifolds and symplectic manifolds with
integral symplectic form.

######  Theorem 2.1.4 ([5]).

Let $(M,\eta^{\prime})$ be a closed regular contact manifold. There is a
representative contact form $\eta$ for the contact structure which generates a
free $\mathbb{S}^{1}$-action on $M$. The Reeb vector field $\xi$ associated
with this contact form has orbits isomorphic to $\mathbb{S}^{1}$ which are the
fibers of a principal $\mathbb{S}^{1}$-bundle $\pi:M\rightarrow B$ for a
symplectic manifold $(B,\omega)$ such that $\eta$ is a connection form with
curvature form $\omega$, i.e., $\pi^{*}\omega=d\eta$.

Conversely, if $(B,\omega)$ is a symplectic manifold such that the class
$[\omega]$ is integral, then the principal $\mathbb{S}^{1}$-bundle $M$
associated to $[\omega]$ is a regular contact manifold with contact form
$\eta$ such that $d\eta=\pi^{*}\omega$.

Throughout the article, $\pi$ will denote the projection map of the Boothby-
Wang fibration. Since the $\mathbb{S}^{1}$-action on $M$ which makes $M$ a
principal circle bundle over $B$ is given by the flow $\theta$ of the Reeb
vector field $\xi$ of $\eta$, we have, for all $t\in[0,1]$, $p\in M$,
$\pi(\theta_{t}(p))=\pi(p)$, where $t$ parametrizes $\mathbb{S}^{1}$. Taking
the derivative of the above equation, we get,
$d\pi_{\theta_{t}(p)}({d\theta_{t}}_{p})=d\pi_{p}$. Thus, the pushforward of
vectors in the direction of the Reeb vector field yields vectors which project
onto the same vector on the base space $B$.

######  Definition 2.1.5.

For every vector field $X$ in $B$, the lift obtained by choosing vectors in
the contact distribution is a $\xi$-invariant vector field, which will be
referred to as a horizontally lifted vector field.

Throughout this article, we use horizontally lifted vector fields for
computations. The horizontal lift of a vector field $X$ in $B$ will be denoted
by $\tilde{X}$. By the observation above, we have
$\mathcal{L}_{\xi}{\tilde{X}}=0$.

###  2.2 Contact Metric Manifolds

######  Definition 2.2.1.

  1. 1.

A contact metric manifold is a 4-tuple $(M,\eta,g,\phi)$, where $(M,\eta)$ is
a contact manifold (with Reeb vector field $\xi$), $g$ is a Riemannian metric
on $M$, and $\phi$ is a (1,1)-tensor field, which satisfies:

     * •

$\eta=i_{\xi}g$,

     * •

$\phi^{2}=-I+\eta\otimes\xi$, and $d\eta(X,Y)=g(X,\phi Y)$,

for all $X,Y\in\Gamma(M)$. In this case, $g$ is called an associated metric to
the contact form $\eta$.

  2. 2.

A symplectic metric manifold is a 4-tuple $(B,\omega,g,J)$, where $(B,\omega)$
is a symplectic manifold, $g$ is a Riemannian metric on $B$ and $J$ is an
almost complex structure on $B$, which satisfies

$g(X,JY)=\omega(X,Y),$ (2.3)

for all $X,Y\in\Gamma(M)$.

Given a contact manifold (resp. a symplectic manifold), there exists a
Riemannian metric $g$ associated with the contact form (resp. symplectic
form).

Consider a closed symplectic manifold $(B,\omega)$ such that $\omega$ is an
integral symplectic form. Let $g$ be an associalted metric on $B$ along with
an almost complex structure $J$ which makes $(B,\omega,g,J)$ a symplectic
metric manifold. Let $(M,\eta)$ be the contact manifold which is the total
space of the Boothby-Wang fibration over $(B,\omega)$. We can define a
Riemannian metric $\tilde{g}$ and a (1,1)-tensor field $\phi$ on $(M,\eta)$ as
specified below, which makes $(M,\eta,\phi,g)$ a (regular) contact metric
manifold.

$\phi X=\widetilde{Jd\pi X}\text{ and }\tilde{g}=\pi^{*}g+\eta\otimes\eta.$

Note that the horizontally lifted vector field also coincides with the
horizontal lift with the respect to the metric $\tilde{g}$, defined on $M$.
The procedure described above yields a contact metric manifold for which the
Reeb vector field is a Killing vector field, i.e., the flow of the Reeb vector
field is by isometries.

######  Definition 2.2.2.

A contact metric manifold $(M,\eta,g,\phi)$ for which the Reeb vector field is
a Killing vector field, i.e., $\mathcal{L}_{\xi}g=0$, is called a
$\boldsymbol{K}$-contact manifold.

######  Remark 2.2.3.

One can verify that $(M,\eta,g,\phi)$ is a $K$-contact manifold if and only if
the $(1,1)$-tensor field $h=\frac{1}{2}\mathcal{L}_{\xi}\phi$ identically
vanishes. The tensor field $h$ plays an important role in the study of contact
metric manifolds which are not $K$-contact.

The construction described above gives a Boothby-Wang correspondence between
regular closed $K$-contact manifolds and closed symplectic metric manifolds
with integral symplectic form. In this case, $\pi$ is a Riemannian submersion,
i.e., the differential map $d\pi$ is an isometry when restricted to the
contact distribution.

There is a natural almost complex structure on the symplectizaton of a contact
metric manifold. Requiring the almost complex structure thus obtained to be
integrable (making the manifold Kähler) gives rise to an interesting class of
contact metric manifolds, introduced by Shigeo Sasaki, in [18].

######  Definition 2.2.4.

Given any contact metric manifold $(M,\eta,g,\phi)$, one can define an almost
complex structure $J$ on the symplectization $M\times\mathbb{R}$ given by

$J\left(X,f\partial_{t}\right)=\left(\phi X-f\xi,\eta(X)\partial_{t}\right),$
(2.4)

where $X\in\Gamma(M)$, and $f\in C^{\infty}(M\times\mathbb{R})$. If $J$ is
integrable, then $(M,\eta,g,\phi)$ is called a Sasakian manifold.

The following is a result which facilitates the reformulation of the condition
specified in the definition above, in terms of the structure tensors of the
contact metric manifold.

######  Theorem 2.2.5.

For a contact metric manifold $(M,\eta,g,\phi)$, the following are equivalent:

  * •

The almost complex structure $J$ defined above, on the symplectization is
integrable.

  * •

$(\nabla_{X}\phi)Y=g(X,Y)\xi-\eta(Y)X$, for all vector fields $X,Y$.

  * •

$R(X,Y)\xi=\eta(Y)X-\eta(X)Y$, for all vector fields $X,Y$.

######  Remark 2.2.6.

Sasakian manifolds are $K$-contact but the converse only holds in dimension 3.

For a more details about Riemannian geometry of contact and symplectic
manifolds, see [1]. The following is a result asserting the existence of a
Boothby-Wang correspondence between Sasakian manifolds and Hodge manifolds
(Kähler manifolds with integral symplectic form).

######  Theorem 2.2.7 ([11]).

A Boothby-Wang bundle over a Kähler manifold with integral symplectic form is
Sasakian. Conversely, every regular compact Sasakian manifold is a Boothby-
Wang bundle over a Kähler manifold.

We refer the reader to [6], for a detailed treatment of the correspondence
between Sasakian and Kähler manifolds.

###  2.3 Contact metric $\boldsymbol{(\kappa,\mu)}$-manifolds

A generalization of Sasaskian manifolds was given in [2] where the authors
define and study a class of contact metric manifolds satisfying the nullity
condition

$R(X,Y)\xi=\kappa(\eta(Y)X-\eta(X)Y)+\mu(\eta(Y)hX-\eta(X)hY),$ (2.5)

for some real numbers $\kappa,\mu$. Contact metric manifolds satisfying the
above nullity condition are called $(\kappa,\mu)$-manifolds. The authors
proved that, in dimensions 5 and higher, the curvature of these manifolds is
completely determined by the values $\kappa$ and $\mu$. They are also
invariant under $\mathcal{D}_{a}$-homothetic transformations. By a
$\mathcal{D}_{a}$-homothetic transformation of non-zero constant $a$, of a
contact metric manifold $(M,\eta,g,\phi)$, we mean a change of the structure
tensors given by

$\tilde{\eta}=a\eta,\quad\tilde{g}=ag+a(a-1)\eta\otimes\eta,\quad\tilde{\phi}=\phi.$

One can easily check that the above changes give rise to a contact metric
manifold $(M,\tilde{\eta},\tilde{g},\tilde{\phi})$. Note that the Reeb vector
field $\tilde{\xi}$ and the tensor field $\tilde{h}$ of the manifold thus
obtained are given by

$\tilde{\xi}=\frac{1}{a}\xi,\quad\tilde{h}=\frac{1}{a}h.$

If one starts out with a contact metric manifold whose Reeb vector field
satisfies the $(\kappa,\mu)$-nullity condition (1.1), then the Reeb vector
field of the contact metric manifold obtained by the
$\mathcal{D}_{a}$-homothetic transformation by non-zero constant $a$ satisfies
the nullity condition

$\tilde{R}(X,Y)\tilde{\xi}=\left(\frac{\kappa+a^{2}-1}{a^{2}}\right)\left(\tilde{\eta}(Y)X-\eta(X)Y\right)+\left(\frac{\mu+2a-2}{a}\right)\left(\tilde{\eta}(Y)hX-\tilde{\eta}(X)hY\right).$

In other words, the nullity condition 1.1 is preserved, for constants
$\tilde{\kappa}=\frac{\kappa+a^{2}-1}{a^{2}}$ and
$\tilde{\mu}=\frac{\mu+2a-2}{a}$.

A local classification of non-Sasakian $(\kappa,\mu)$-manifolds was given by
Boeckx, in [4], in terms of the index

$I_{M}=\frac{1-\tfrac{\mu}{2}}{\sqrt{1-\kappa}}.$

It was shown that $I_{M}$ is invariant under $\mathcal{D}_{a}$-homothetic
transformations. Furthermore, any two $(\kappa,\mu)$-manifolds have the same
index if and only if they are locally isometric upto
$\mathcal{D}_{a}$-homothetic transformations.

The $(1,1)$-tensor field $h$ is symmetric with respect to the metric and has
many interesting properties in the case of $(\kappa,\mu)$-manifolds. The
eigenspaces of $h$ play an important role in the study of
$(\kappa,\mu)$-manifolds as is evident from the following result in [2].

######  Theorem 2.3.1 ([2]).

Let $(M,\eta,g,\phi)$ be a $(\kappa,\mu)$-manifold. Then,

$h^{2}=-(1-\kappa)\phi^{2}.$

Thus, $\kappa\leq 1$. Moreover, $\kappa=1$ if and only if $M$ is Sasakian. If
$M$ is non-Sasakian, then $h$ has three eigenvalues 0, $\lambda$ and
$-\lambda$, where $\lambda=\sqrt{1-\kappa}$. The tangent bundle of $M$ admits
a decomposition into three mutually orthogonal and integrable distributions
$\mathscr{D}_{h}(\lambda),\,\mathscr{D}_{h}(-\lambda)$ and
$\mathscr{D}_{h}(0)$ given by the eigenspaces of $h$ corresponding to the
eigenvalues $\lambda,\,-\lambda$ and 0, respectively.

To conclude this subsection, we state some formulae involving the structure
tensors of a contact metric manifold.

######  Theorem 2.3.2.

Let $(M,\eta,g,\phi)$ be a contact metric manifold. Then,

  * •

The tensor field $h$ anti-commutes with $\phi$, i.e., $h\phi+\phi h=0$. Thus,
$\phi v$ is an eigenvector of $h$ for every eigenvector $v$ of $h$,

  * •

$\mathcal{L}_{\xi}g(X,Y)=2g(hX,\phi Y)$, for vector fields $X,Y$ in $M$,

  * •

For a non-Sasakian $(\kappa,\mu)$-manifold, $\mathcal{L}_{\xi}h=(2-\mu)\phi
h+2(1-\kappa)\phi$.

###  2.4 Bi-Legendrian manifolds

######  Definition 2.4.1.

Let $(M^{2n+1},\eta)$ be a contact manifold. A Legendrian distribution on $M$
is an $n$-dimensional distribution $L$ of $Ker(\eta)$ such that $d\eta(X,Y)=0$
for all $X,Y\in\Gamma(L)$. Further, if $L$ is integrable, then it defines a
Legendrian foliation.

A contact manifold equipped with a pair of complementary Legendrian
distributions is called a bi-Legendrian manifold. If $(M^{2n+1},\eta,g,\phi)$
is a non-Sasakian $(\kappa,\mu)$-manifold, then the subspaces
$\mathscr{D}_{h}(\lambda)$ and $\mathscr{D}_{h}(-\lambda)$ define orthogonal
Legendrian distributions of $Ker(\eta)$. Thus, any non-Sasakian
$(\kappa,\mu)$-manifold is naturally endowed with a pair of complementary
integrable Legendrian distributions of dimension $n$ each, making it a bi-
Legendrian manifold. Legendrian manifolds have been studied by Pang, Liberman
and Jayne (see [16], [13] and [12]). This was used in [8] to study the
interplay between $(\kappa,\mu)$-manifolds and bi-Legendrian manifolds. The
authors proved the following result which gives a characterization of
$(\kappa,\mu)$-manifolds in terms of the bi-Legendrian structure.

######  Theorem 2.4.2.

[8] A contact metric manifold $(M,\eta,g,\phi)$ is a $(\kappa,\mu)$-manifold
if and only if it admits an orthogonal bi-Legendrian structure $(L_{1},L_{2})$
such that the corresponding biligendrian connection $\bar{\nabla}$ satisfies
$\bar{\nabla}\phi=0$ and $\bar{\nabla}h=0$. The biligendrian structure
$(L_{1},L_{2})$ coincides with the eigenspaces of $h$.

Thus, it is natural to look at the bi-Legendrian structure on the given
$(\kappa,\mu)-$manifold and investigate error tensors for which the
decomposition is preserved. More generally, throughout the article we will
impose conditions which preserve the bi-Legendrian structure of the contact
manifold.

###  2.5 Semi-Riemannian metrics and Symplectic Geometry

Semi-Riemannian metrics are generalizations of Riemannian metrics obtained by
relaxing the condition of positive-definiteness. Instead, we only require that
the product is non-degenerate. Compatibility between a semi-Riemannian metric
and a symplectic manifold is given as follows.

######  Definition 2.5.1.

An almost para-Kähler manifold is a $4$-tuple $(B,\omega,g,F)$ comprising a
symplectic manifold $(B,\omega)$, a semi-Riemannian metric $g$ and a
$(1,1)$-tensor field $F$ such that $F^{2}=I$ and $\omega(X,Y)=g(X,FY)$. If
$\nabla F=0$, where $\nabla$ is the Levi-Civita connection of $g$, then
$(B,\omega,g,F)$ is said to be a para-Kähler manifold.

For a detailed account of para-Kähler geometry, we refer the reader to [9].

Since we are interested in geometric properties of submersions, we recall the
definition of a semi-Riemannian submersion here, which was introduced by
O’Neill in [15].

######  Definition 2.5.2.

A submersion between semi-Riemannian manifolds $(M,\tilde{g})$ and $(B,g)$ is
said to be a semi-Riemannian submersion if

  * •

The fibers $\\{\pi^{-1}(b)\\}_{b\in B}$ are semi-Riemannian submanifolds of
$M$, and,

  * •

The derivative map $d\pi$ preserves the scalar product of vectors normal to
the fibers.

##  3 Riemannian metrics on the base space

In this section, we investigate the properties of a compatible metric on the
base space of the Boothby-Wang fibration whose total space is a closed regular
non-Sasakian $(\kappa,\mu)$-contact manifold. The projection map of the
Boothby-Wang fibration $\pi$ cannot be a Riemannian submersion. This is
because, a Riemannian submersion would force $\mathcal{L}_{\xi}g=0$, which in
turn forces the $(\kappa,\mu)$-contact manifold to be $K$-contact and hence,
Sasakian. Next, we show that the projection map $\pi$ being a conformal
submersion (a submersion for which the differential map preserves angles when
restricted to the contact distribution) also forces the
$(\kappa,\mu)$-structure to be Sasakian. To this end, we prove Theorem 1.1.

###### Proof of Theorem 1.1.

Let $p\in M$. We can construct an orthonormal (with respect to
$\tilde{g}_{p})$ basis of $Ker(\eta_{p})$, of the form
$\\{e_{1},e_{2},\dots,e_{n},\phi e_{1},\phi e_{2},\dots,\phi e_{n}\\}$ as
follows. Choose any unit vector $e_{1}$. Since $\tilde{g}(v,w)=\tilde{g}(\phi
v,\phi w)$, $\phi e_{1}$ is a unit vector as well. Since
$d\eta(v,w)=\tilde{g}(v,\phi w)$, $\phi e_{1}$ is orthogonal to $e_{1}$.
Considering the subspace orthogonal to the span of $\\{e_{1},\phi e_{1}\\}$ in
$Ker(\eta_{p})$, we can continue the procedure described above and obtain the
necessary basis in $n$ steps. Suppose $\pi$ is a conformal submersion. There
exists a function $f$ such that

$\tilde{g}(V,W)=e^{2f}g(d\pi V,d\pi W),$

for all vector fields $V,W\in\Gamma(Ker(\eta))$. For the Boothby-Wang
fibration, we have $\pi^{*}(\omega)=d\eta$. For vectors $v,w\in
Ker(\eta_{p})$,

$\displaystyle\tilde{g}(v,\phi w)=d\eta(v,w)$
$\displaystyle=\pi^{*}\omega(v,w)$ $\displaystyle=\omega(d\pi v,d\pi w)$
$\displaystyle=g(d\pi v,J\left(d\pi w\right))$
$\displaystyle=\frac{1}{e^{2f}}\tilde{g}(v,\widetilde{Jd\pi w}).$

Substituting $w=e_{1}$ and varying $v$ in the basis constructed above, we get

$\begin{gathered}\tilde{g}(\phi e_{1},\widetilde{Jd\pi e_{1}})=e^{2f},\\\
\tilde{g}(v,\widetilde{Jd\pi e_{1}})=0\text{ for all
}v\in\\{e_{1},e_{2},\dots,e_{n},\phi e_{2},\dots,\phi e_{n}\\}.\end{gathered}$
(3.1)

In other words,

$\widetilde{Jd\pi e_{1}}=e^{2f}\phi e_{1}.$

Since $\pi$ is a conformal submersion, the image set of a basis will again be
a basis of the corresponding tangent space. Let
$\\{f_{1},f_{2},\dots,f_{2n}\\}$ be the basis of $T_{\pi p}B$ which is the
image of the basis constructed above, with elements in the order specified.
With this notation, we have,

$Jf_{1}=e^{2f}f_{n+1}.$

Similarly, we can prove that,

$Jf_{n+1}=-e^{2f}f_{1}.$

Since $J$ is an almost complex structure, we have $J^{2}f_{1}=-f_{1}$. Using
the identities above, we get $e^{4f}=1$. In other words, $f=0$. Thus, for all
$V,W\in\Gamma(Ker(\eta))$, we have $\tilde{g}(V,W)=g(d\pi V,d\pi W)$, which
proves both the statements in the theorem. ∎

The construction of the basis of the form $\\{e_{1},e_{2},\dots,e_{n},\phi
e_{1},\phi e_{2},\dots,\phi e_{n}\\}$ can be done locally, in a coordinate
neighbourhood. Such a basis is called a local $\phi$-basis. For the remainder
of the article, $\\{f_{1},f_{2},\dots,f_{2n}\\}$ will denote the (ordered)
image of a local $\phi$-basis, under the derivative map $d\pi$.

As a special case of Theorem 1.1 when restricted to $(\kappa,\mu)$-manifolds,
we obtain the following result.

######  Corollary 3.1.

Suppose $\pi:M\rightarrow B$ is the projection map of a regular compact
$(\kappa,\mu)$-manifold $(M,\eta,\tilde{g},\phi)$ onto a symplectic manifold
$(B,\omega,g,J)$. If $\pi$ is a conformal submersion, then $M$ is Sasakian and
$B$ is Kähler.

###### Proof.

From the previous lemma, we know that $\pi$ is a Riemannian submersion and $M$
is a $K$-contact manifold. Since every $(\kappa,\mu)$-manifold which is also
$K$-contact is Sasakian, we have the first part of the result. By Hatakeyama’s
result ([11]), $B$ is Kähler. ∎

Since the above result asserts that the projection map cannot be a conformal
submersion in case of a non-Sasakian compact regular $(\kappa,\mu)$-manifold,
we investigate the metrics with non-trivial deviation from being a conformal
metric. To this end, we consider the bilinear form, corresponding to any
associated metric $g$ on $(B,\omega)$ and $f\in C^{\infty}(M)$, given by

$A(X,Y)=\tilde{g}(X,Y)-e^{2f}g(d\pi X,d\pi Y),$

for all $X,Y\in\Gamma(Ker(\eta))$. There exists a unique (1,1)-tensor field
$T$ for which the above equation can be reformulated as

$\tilde{g}(TX,Y)=\tilde{g}(X,Y)-e^{2f}g(d\pi X,d\pi Y),$ (3.2)

for all $X,Y\in\Gamma(Ker(\eta))$.

Henceforth, we will call $T$ the error tensor. We will try to solve for
triples $(T,g,f)$ which satisfy (1.2). We begin by exploring the properties of
the error tensor.

######  Lemma 3.2.

The (1,1)-tensor field $T$ defined above has the following properties:

  1. [i)]

  2. 1.

$T$ is symmetric with respect to $\tilde{g}$,

  3. 2.

Every eigenvalue of $T$ is strictly less than 1,

  4. 3.

$\mathcal{L}_{\xi}T=2\phi hT-2\phi h-2(\xi f)(I-T)$.

###### Proof.

  1. [$i)$]

  2. 1.

Symmetry of $T$ follows from the symmetry of $g$ and $\tilde{g}$.

  3. 2.

Since $T$ is a symmetric tensor field, all of its eigenvalues are real.
Suppose $v$ is a unit eigenvector of $T$ corresponding to an eigenvalue $\nu$.
From equation (1.2),

$\displaystyle\nu=\tilde{g}(Tv,v)$
$\displaystyle=\tilde{g}(v,v)-e^{2f}g(d\pi(v),d\pi(v))$
$\displaystyle=1-e^{2f}\|d\pi(v)\|^{2}$

Thus,

$1-\nu=e^{2f}\|d\pi(v)\|^{2}>0.$

  4. 3.

Taking the Lie derivative of equation (1.2) along $\xi$, for horizontally
lifted vector fields $\tilde{X},\tilde{Y}$, we get, using the properties
mentioned in theorem 2.3.2,

$\displaystyle 0$
$\displaystyle=\mathcal{L}_{\xi}\tilde{g}(\tilde{X},\tilde{Y})-2(\xi
f)e^{2f}g(X,Y)-\mathcal{L}_{\xi}g(T\tilde{X},\tilde{Y})$
$\displaystyle=2\tilde{g}(h\tilde{X},\phi\tilde{Y})-2(\xi
f)\left(\tilde{g}(\tilde{X}-T\tilde{X},\tilde{Y})\right)-2\tilde{g}(hT\tilde{X},\phi\tilde{Y})-\tilde{g}((\mathcal{L}_{\xi}T)\tilde{X},\tilde{Y})$
$\displaystyle=\tilde{g}\left((\phi hT-\phi h-(\xi
f)(I-T))\tilde{X},\tilde{Y}\right)-\frac{1}{2}\tilde{g}((\mathcal{L}_{\xi}T)\tilde{X},\tilde{Y}).$

Since the above identity holds pointwise for all vector fields $\tilde{X}$ and
$\tilde{Y}$, we get

$\mathcal{L}_{\xi}T=2\phi hT-2\phi h-2(\xi f)(I-T).$

∎

Since the bi-Legendrian structure defined by the eigenspaces of $h$ play an
important role in the study of $(\kappa,\mu)$-manifolds, we consider error
tensors which preserve the bi-Legendrian structure. More precisely, we analyse
the solutions to (1.2) in which the error tensor maps the Legendrian
distributions $\mathscr{D}_{h}(\lambda)$ and $\mathscr{D}_{h}(-\lambda)$ to
themselves. We begin by stating the following fact which can be verified
easily.

######  Lemma 3.3.

Let $T$ be the error tensor as in (1.2). Then, the tensor field $h$ commutes
with $T$ if and only if
$T(\mathscr{D}_{h}(\lambda))\subseteq\mathscr{D}_{h}(\lambda)$ and
$T(\mathscr{D}_{h}(-\lambda))\subseteq\mathscr{D}_{h}(-\lambda)$.

We have already seen that the metric $g$ on the base space cannot be in the
conformal class of $\tilde{g}$. However, the condition $hT=Th$ ensures that
the metric $g$ respects the orthogonality of vectors coming from
$\mathscr{D}_{h}(\lambda)$ and $\mathscr{D}_{h}(-\lambda)$ in the following
sense.

######  Lemma 3.4.

Let $(T,f,g)$ be a solution to equation (1.2). Then the following are
equivalent:

  1. 1.

$hT=Th$.

  2. 2.

$g(d\pi X,d\pi Y)=0$ whenever
$X\in\mathscr{D}_{h}(\lambda),Y\in\mathscr{D}_{h}(-\lambda)$ .

###### Proof.

Equation (1.2) can be rearranged to obtain

$g(d\pi X,d\pi Y)=\frac{\tilde{g}(X-TX,Y)}{e^{2f}},$ (3.3)

for all $X,Y\in\Gamma(Ker(\eta)).$ Suppose $hT=Th$. Let
$X\in\mathscr{D}_{h}(\lambda)$ and $Y\in\mathscr{D}_{h}(-\lambda)$. From Lemma
3.3, $T$ preserves the eigenspaces of $h$. Thus,
$TX\in\mathscr{D}_{h}(\lambda)$ and consequently, $X-TX$ is
$\tilde{g}$-orthogonal to $Y$. Using equation (3.3), we obtain $g(d\pi X,d\pi
Y)=0$.

For the converse, suppose statement 2. in the statement of Lemma 3.4 holds.
For $X\in\mathscr{D}_{h}(\lambda)$ and $Y\in\mathscr{D}_{h}(-\lambda)$, we
have, by equation (3.3), $\tilde{g}(TX,Y)=0$. In other words,
$TX\in\left(\mathscr{D}_{h}(-\lambda)\right)^{\perp}=\mathscr{D}_{h}(\lambda)$.
A similar argument can be used to show that $TX\in\mathscr{D}_{h}(-\lambda)$
$X\in\mathscr{D}_{h}(-\lambda)$ and thus $T$ preserves the eigenspaces of $h$.
By Lemma 3.3, this is equivalent to $hT=Th$. ∎

Thus, we consider solutions to equation (1.2) for which $hT=Th$. Since $h$ and
$T$ are commuting diagonalizable operators, they are simultaneously
diagonalizable and share a common eigenbasis. The following result shows that
there is a common eigenbasis which is a (local) $\phi$-basis.

######  Lemma 3.5.

Let $T$ be the error tensor for the Boothby-Wang fibration of a closed regular
non-Sasakian $(\kappa,\mu)$-manifold. If $hT=Th$, then the Boeckx Index
$I_{M}$ must satisfy $|I_{M}|>1$. For $|I_{M}|>1$, the tensors $h$ and $T$
have a common orthonormal eigenbasis at every point, of the form
$\\{e_{1},e_{2},\dots,e_{n},\phi e_{1},\phi e_{2},\dots,\phi e_{n}\\}$, where
$e_{i}\in\mathscr{D}(\lambda)$ for all $i\in\\{1,2,\dots,n\\}$.

###### Proof.

Since $T|_{\mathscr{D}(\lambda)}$ is a symmetric operator, it has an
eigenbasis $\\{e_{1},e_{2},\dots,e_{n}\\}$ of orthonormal vectors. For every
$i\in\\{1,2,\dots,n\\}$, let $\lambda_{i}$ be the eigenvalue corresponding to
the eigenvector $e_{i}$. We claim that $\\{e_{1},e_{2},\dots,e_{n},\phi
e_{1},\phi e_{2},\dots,\phi e_{n}\\}$ is the required basis. In order to prove
this claim, it is sufficient to show that $\phi e_{i}$ is an eigenvector of
$T$ for all $i\in\\{1,2,\dots,n\\}$. To see this, we consider the equation
$hT=Th$ and take its Lie derivative along $\xi$ to get

$\displaystyle 0=$ $\displaystyle\mathcal{L}_{\xi}(Th)-\mathcal{L}_{\xi}(hT)$
$\displaystyle=$
$\displaystyle(\mathcal{L}_{\xi}T)h+T(\mathcal{L}_{\xi}h)-(\mathcal{L}_{\xi}h)T-h(\mathcal{L}_{\xi}T)$
$\displaystyle=$ $\displaystyle(2\phi hT-2\phi h-2(\xi f)(I-T))h+T((2-\mu)\phi
h+2(1-\kappa)\phi)$ $\displaystyle-((2-\mu)\phi h+2(1-\kappa)\phi)T-h(2\phi
hT-2\phi h-2(\xi f)(I-T)).$

Rearranging the above equation and evaluating it at $e_{i}$ using
$he_{i}=\lambda e_{i}$ and $Te_{i}=\lambda_{i}e_{i}$, we get,

$(I_{M}+1)T(\phi e_{i})=(\lambda_{i}(I_{M}-1)+2)$ (3.4)

For $I_{M}=-1$, we get $\lambda_{i}=1$, which is a contradiction to the fact
that the eigenvalues of $T$ are strictly lesser than 1. For all other values
of $I_{M}$, equation (3.4) can be rewritten as

$T(\phi e_{i})=\left(\frac{\lambda_{i}(I_{M}-1)+2}{1+I_{M}}\right)\phi e_{i}.$
(3.5)

Thus, $\phi e_{i}$ is an eigenvector of $T$, as required. Let $\lambda_{n+i}$
denote the eigenvalue corresponding to the eigenvector $\phi e_{i}$ of $T$. By
equation (3.5), we have

$(I_{M}-1)\lambda_{i}-(1+I_{M})\lambda_{n+i}+2=0.$ (3.6)

Equation (3.6) is an equation of a straight line in the variables
$\lambda_{i}$ and $\lambda_{n+i}$. The line has slope
$\left(\frac{1+I_{M}}{I_{M}-1}\right)$ and passes through the point $(1,1)$.
Since we need the eigenvalues to be strictly lesser than $1$, we need the
slope of the above line to be strictly positive. Equivalently, $|I_{M}|>1.$ ∎

As a part of the proof of Lemma 3.5, we have also proved the following result,
which we state separately for clarity.

######  Corollary 3.6.

Let $\\{e_{1},e_{2},\dots,e_{n},\phi e_{1},\phi e_{2},\dots,\phi e_{n}\\}$ be
the $\phi$-basis constructed in Lemma 3.5. For every $i\in\\{1,2,\dots,n\\}$,
let $\lambda_{i}$ be the eigenvalue of $T$ corresponding to $e_{i}$ and let
$\lambda_{n+i}$ be the eigenvalue of $T$ corresponding to $\phi e_{i}$. Then,

$(I_{M}-1)\lambda_{i}-(1+I_{M})\lambda_{n+i}+2=0.$ (3.7)

Now, we prove a result which shows that the eigenvalues
$\left\\{\lambda_{i}\right\\}_{i=1}^{2n}$ mentioned above satisfy another set
of equations. This allows us to determine the error tensor $T$ in terms of the
function $f$ in equation (1.2).

######  Lemma 3.7.

Let $\lambda_{i}$ be the eigenvalues of $T$ mentioned in Corollary 3.6. If
$(T,f,g)$ is a solution to equation (1.2), then

$(1-\lambda_{i})(1-\lambda_{n+i})=e^{4f}$ (3.8)

###### Proof.

Since $\pi$ is the projection map of the Boothby-Wang fibration, we have
$\pi^{*}\omega=d\eta$. For vectors $v,w\in Ker(\eta)$,

$\displaystyle\tilde{g}(v,\phi w)=d\eta(v,w)$
$\displaystyle=\pi^{*}\omega(v,w)$ $\displaystyle=\omega(d\pi(v),d\pi(w))$
$\displaystyle=g(d\pi(v),J\left(d\pi(w)\right))$
$\displaystyle=\frac{1}{e^{2f}}\left(\tilde{g}(v-Tv,\widetilde{Jd\pi(w)})\right).$
(3.9)

Consider the basis constructed in the proof of Lemma 3.5. Substituting
$w=e_{1}$ and varying $v$ in the elements of the chosen basis, we get,

$\tilde{g}(v,\widetilde{Jd\pi(e_{1})})=0\text{ for all
}v\in\\{e_{1},e_{2},\dots,e_{n},\phi e_{2},\dots,\phi e_{n}\\}.$

For $v=\phi e_{1}$, we have

$\displaystyle 1=\tilde{g}(\phi e_{1},\phi e_{1})$
$\displaystyle=\frac{1}{e^{2f}}\left(\tilde{g}(\phi e_{1}-T\phi
e_{1},\widetilde{Jd\pi(e_{1})}\right)$
$\displaystyle=\left(\frac{1-\lambda_{n+1}}{e^{2f}}\right)\tilde{g}(\phi
e_{1},\widetilde{Jd\pi(e_{1})})$

Thus,
$\widetilde{Jd\pi(e_{1})}=\left(\frac{e^{2f}}{1-\lambda_{n+1}}\right)\phi
e_{1}$. Let $\\{f_{1},f_{2},\dots,f_{2n}\\}$ be the image of the chosen basis
under $d\pi$, with elements in the same order. We have,
$\widetilde{Jf_{1}}=\left(\frac{e^{2f}}{1-\lambda_{n+1}}\right)\phi e_{1}$.
Applying the derivative map of $\pi$, we get
$Jf_{1}=\left(\frac{e^{2f}}{1-\lambda_{n+1}}\right)f_{n+1}$. By varying both
$v$ and $w$ in equation (3.9) from the elements of the chosen basis and
performing similar computations, we can conclude that

$Jf_{i}=\left(\frac{e^{2f}}{1-\lambda_{n+i}}\right)f_{n+i},\qquad
Jf_{n+i}=-\left(\frac{e^{2f}}{1-\lambda_{i}}\right)f_{i},$ (3.10)

for all $i\in\\{1,2,\dots,n\\}$. Since $J$ is an almost complex structure, we
have, for all $i\in\\{1,2,\dots,n\\}$,

$\displaystyle-f_{i}=$ $\displaystyle J^{2}f_{i}$ $\displaystyle=$
$\displaystyle
J\left(\left(\frac{e^{2f}}{1-\lambda_{n+i}}\right)f_{n+i}\right)$
$\displaystyle=$
$\displaystyle\left(\frac{e^{2f}}{1-\lambda_{n+i}}\right)\left(-\frac{e^{2f}}{1-\lambda_{i}}f_{i}\right)$

Therefore, we can conclude that the eigenvalues of $T$ in this case satisfy
the equation

$(1-\lambda_{i})(1-\lambda_{n+i})=e^{4f}.$ (3.11)

∎

The intersection of the solution sets to equations (3.6) and (3.8) consists of
two points. But due to the condition given in Lemma 3.2 2, only one of the
solutions is permissible. See Figure 1 for a representative depiction.

$\lambda_{i}$$\lambda_{n+i}$ Figure 1: The graphs of curves determining the
eigenvalues of $T$ for $|I_{M}|>1$.

Thus, the above results determine the error tensors which arise as solutions
to equation (1.2) and commute with the tensor field $h$. More precisely, we
have the following result.

######  Lemma 3.8.

Suppose $(T,f,g)$ is a solution to equation (1.2) and $hT=Th$. Then,

$T|_{\mathscr{D}(\lambda)}=\left(\frac{1-\sqrt{\frac{I_{M}+1}{I_{M}-1}}e^{2f}}{\lambda}\right)h|_{\mathscr{D}(\lambda)}\qquad
T|_{\mathscr{D}(-\lambda)}=\left(\frac{1-\sqrt{\frac{I_{M}-1}{I_{M}+1}}e^{2f}}{\lambda}\right)h|_{\mathscr{D}(-\lambda)}$
(3.12)

###### Proof.

Under the hypothesis of the statement, we know that the eigenvalues of the
error tensor satisfy equations (3.6) and (3.8). Solving the equations
simultaneously gives two solutions for each eigenvalue of $T$. Out of the two
solutions, only one of them is lesser than 1. Thus, there is only one
acceptable solution. A straightforward computation shows that the acceptable
solution set for the eigenvalues $\left\\{\lambda_{i}\right\\}_{i=1}^{2n}$ of
$T$ are given by

$\lambda_{i}=1-\sqrt{\frac{I_{M}+1}{I_{M}-1}}e^{2f}\text{ and
}\lambda_{n+i}=1-\sqrt{\frac{I_{M}-1}{I_{M}+1}}e^{2f}.$

Thus, the error tensor has two eigenspaces of dimension $n$ each, which
coincide with the eigenspaces of the tensor field $h$. Therefore, we have the
desired expressions for $T$. ∎

The function $f$ in equation (1.2) determines $T$ completely. We now prove a
result which establishes rigidity of the metric $g$ appearing in the solutions
to equation (1.2).

######  Lemma 3.9.

Suppose $(T,f,g)$ is a solution to equation (1.2) and $hT=Th$. Then,

$g(X,Y)=\frac{\lvert
I_{M}\rvert}{\sqrt{I_{M}^{2}-1}}\left(\tilde{g}(\tilde{X},\tilde{Y})+\tilde{g}\left(\tilde{X},\left(\frac{h}{1-\tfrac{\mu}{2}}\right)\tilde{Y}\right)\right),$
(3.13)

for all vector fields $X,Y$ on $B$.

###### Proof.

In the proof of Lemma 3.7, we have obtained a basis $\\{f_{i}\\}_{i=1}^{2n}$
and an expression for the almost complex structure $J$ in terms of this basis
(equation (3.10)). Combining this with the values of the eigenvalues of $T$
obtained in the proof of Lemma 3.8, we get, for $i\in\\{1,2,\dots,n\\}$,

$Jf_{i}=\sqrt{\frac{I_{M}+1}{I_{M}-1}}f_{n+i}\text{ and
}Jf_{n+i}=\sqrt{\frac{I_{M}-1}{I_{M}+1}}f_{i}.$

Since the expression for $J$ in the above basis is independent of $f$ and $T$,
we conclude that any solution to equation (1.2) comprises a unique metric $g$.
In order to obtain an expression for the unique metric, we may substitute
$f\equiv 0$ in equation (1.2) and simplify in order to obtain the desired
result. ∎

The implications of the imposition of the condition $hT=Th$ are summarized in
Theorem 1.2 and proved above (Lemma 3.5, Lemma 3.8 and Lemma 3.9).

######  Remark 3.10.

The metric $g$ described in the third part of Theorem 1.2 can be checked to be
a Kähler metric. Naturally, it corresponds to a Sasakian metric on $M$, by
[11]. This Sasakian metric has also been studied, in a different context, in
[7] where the authors describe it as the canonical Sasakian structure
associated to the $(\kappa,\mu)$-structure on $M$.

######  Remark 3.11.

In [14], the authors describe the canonical structure of the base space of a
simply connected, complete $(\kappa,\mu)$-manifold. For $|I_{M}|>1$, the base
space admits the structure of the complexification of a sphere. The structure
tensors of the space obtained by the authors coincide with tensors obtained as
the unique solution obtained here.

Theorem 1.2 can be combined with Lemma 3.4 to obtain the following result.

######  Corollary 3.12.

There is a unique Riemannian metric $g$ on the base space of a Boothby-Wang
fibration of a compact regular $(\kappa,\mu)$-manifold which satisfies the
condition

$g(d\pi X,d\pi Y)=0\text{ whenever
}X\in\mathscr{D}_{h}(\lambda),Y\in\mathscr{D}_{h}(-\lambda).$ (3.14)

Furthermore, $g$ defines a Kähler structure on the base space.

The above corollary, along with the correspondence between Kähler and Sasakian
manifolds can be used to obtain Theorem 1.3.

###### Proof of Theorem 1.3.

Suppose $(M,\eta,\bar{g},\bar{\phi})$ is a $K$-contact structure as specified
in the above statement. The metric $\bar{g}$ induces a Riemannian metric $g$
on the base space of the Boothby-Wang fibration of $M$ (See Section 2.2 for
details). In this case, the corresponding projection map $\pi$ is a Riemannian
submersion. Thus, the Riemannian metric $g$ satisfies the condition specified
in equation (3.14). Thus, we conclude that $g$ is the unique Kähler metric
obtained as a solution to equation (1.2). This forces $\bar{g}=\pi^{*}g$. The
second statement follows from remark 3.10. ∎

##  4 Semi-Riemannian metrics on the base space

We have seen that there are no associated metrics on the base space (under the
Boothby-Wang fibration) of a regular closed $(\kappa,\mu)$-manifold with
Boeckx index $|I_{M}|>1$ for which the associated error tensor preserves the
Legendrian distributions given by the eigenspaces of $h$. So we look at the
possibilities of semi-Riemannian metrics on the base space associated with the
symplectic structure and corresponding error tensors. As before, we start out
with the assumption that $\pi:M\rightarrow B$ is a semi-Riemannian submersion.
Suppose the metric on $B$ has vectors of negative length. Then, we get vectors
with negative length on the total space, as well, which is not possible. Thus,
$B$ is forced to be a Riemannian manifold and we land in the previous case. In
light of this observation, we define the error tensor, as before, by the
equation,

$\tilde{g}(T\tilde{X},\tilde{Y})=\tilde{g}(\tilde{X},\tilde{Y})-g(X,Y),$ (4.1)

where $X,Y\in\Gamma(B)$. Since the analysis follows a path similar to the one
in the previous section, we will skip the proofs of the statements which
follow directly from similar computations. Symmetry of the tensor field $T$
follows immediately. A formula for the Lie derivative of $T$ can also be
obtained by carrying out computations similar to the ones in the proof of
Lemma 3.2. However, the eigenvalues of the error tensor being bounded above by
$1$ in the case of Riemannian metrics was a consequence of positive
definiteness, which is dropped here. Instead, we have the following result.

######  Lemma 4.1.

The error tensor $T$ defined in equation (1.3) can not have an eigenvalue
equal to 1.

###### Proof.

Suppose $\tilde{X}$ is an eigenvector of $T$ corresponding to the eigenvalue
$1$. Then, by equation (1.3), we get
$\tilde{g}(\tilde{X},\tilde{Y})=\tilde{g}(\tilde{X},\tilde{Y})-g(X,Y)$, for
all vectors $\tilde{Y}\in\Gamma(Ker(\eta))$. Thus, we have $g(X,Y)=0$, for all
vectors $Y\in\Gamma(B)$, contradicting the non-degeneracy of the tensor $g$.
Therefore, $T$ can not have an eigenvalue equal to $1$. ∎

Following an analysis similar to the one in the case of Riemannian metrics on
the base space, we can conclude that the error tensor $T$ commutes with the
tensor field $h$ if and only if the condition specified in (3.14) holds.
Hence, we investigate solutions $(T,g)$ to equation (1.3) for which $hT=Th$.
Since the proof of Lemma 3.5 does not depend on the positive-definiteness of
the metric tensor on the base, we can conclude that $h$ and $T$ share a common
eigenbasis which is a local $\phi$-basis and the eigenvalues
$\\{\lambda_{i}\\}_{1}^{2n}$ of $T$ satisfy the equation

$(I_{M}-1)\lambda_{i}-(1+I_{M})\lambda_{n+i}+2=0.$ (4.2)

The equation of the hyperbola given by (3.8) was derived using the fact that
$J$ is an almost complex structure on the base space. However, this is not the
case here since the base space is equipped with a $(1,1)$-tensor field $F$
which satisfies $F^{2}=1$. The proof of Lemma 3.7 can be modified to obtain
the following result.

######  Lemma 4.2.

Let $(T,g)$ be a solution to equation (1.3). Then, the eigenvalues
$\\{\lambda_{i}\\}_{1}^{2n}$ satisfy the relations

$(1-\lambda_{i})(1-\lambda_{n+i})=-1,$ (4.3)

for all $i\in\\{1,2,\dots,n\\}$.

The line has slope equal to $\left(\frac{1+I_{M}}{I_{M}-1}\right)$, and passes
through the point $(1,1)$. The line and the hyperbola have non-empty
intersection only if the slope of the line is negative (see Figure 2 for a
particular case). This is equivalent to the condition $|I_{M}|<1$. Under this
assumption on the index, the points of intersections of the line (3.6) and
hyperbola (4.3) are given by

$p_{1}\coloneqq\left(1+\sqrt{\frac{1+I_{M}}{1-I_{M}}},1-\sqrt{\frac{1-I_{M}}{1+I_{M}}}\right)\text{
and
}p_{2}\coloneqq\left(1-\sqrt{\frac{1+I_{M}}{1-I_{M}}},1+\sqrt{\frac{1-I_{M}}{1+I_{M}}}\right).$
(4.4)

$\lambda_{i}$$\lambda_{n+i}$ Figure 2: The graphs of curves determining the
eigenvalues of $T$

Thus, for every $i\in\\{1,2,\dots,n\\}$, the ordered pair
$(\lambda_{i},\lambda_{n+i})$ is either $p_{1}$ or $p_{2}$. Therefore, there
are $2^{n}$ solutions for the (ordered) set of eigenvalues
$\\{\lambda_{i}\\}_{i=1}^{2n}$ of $T$. In other words, there are $2^{n}$ many
$(1,1)$-tensor fields that can appear in the solutions to the equation 1.3.
Since these tensor fields are defined using a $\phi$-basis which is defined in
a neighbourhood, each of the solutions is a smooth $(1,1)$-tensor field. Thus,
we obtain $2^{n}$ semi-Riemannian metrics on $B$ given by the equation
$g(X,Y)=\tilde{g}(\tilde{X}-T\tilde{X},\tilde{Y})$. These semi-Riemannian
metrics define almost para-Kähler structures on the base space. These tensors
can described locally, in terms of the basis $\\{f_{i}\\}_{i=1}^{2n}$, which
is the image of the chosen $\phi$-basis, as follows. Let $S$ be a subset of
$\\{1,2,\dots,n\\}$. Let $a_{0}=\sqrt{\frac{1+I_{M}}{1-I_{M}}}$. Define a
$(1,1)$-tensor field $F_{S}$ on $B$ by the equations

$\displaystyle F_{S}f_{i}$ $\displaystyle=\begin{cases}a_{0}f_{n+i}\>\>\text{,
if }i\in S\\\ -a_{0}f_{n+i}\>\>\text{, if }i\in\\{1,2,\dots,n\\}\setminus
S\end{cases}$ $\displaystyle F_{S}f_{n+i}$
$\displaystyle=\begin{cases}\frac{1}{a_{0}}f_{i}\>\>\text{, if }i\in S\\\
-\frac{1}{a_{0}}f_{i}\>\>\text{, if }i\in\\{1,2,\dots,n\\}\setminus
S\end{cases}$

Also, define $(0,2)$-tensors $g_{S}$ by the equations

$\displaystyle g_{S}(f_{i},f_{i})$
$\displaystyle=\begin{cases}-a_{0}\>\>\text{, if }i\in S\\\ a_{0}\>\>\text{,
if }i\in\\{1,2,\dots,n\\}\setminus S\end{cases}$ $\displaystyle
g_{S}(f_{n+i},f_{n+i})$ $\displaystyle=\begin{cases}\frac{1}{a_{0}}\>\>\text{,
if }i\in S\\\ -\frac{1}{a_{0}}\>\>\text{, if }i\in\\{1,2,\dots,n\\}\setminus
S\end{cases}$ $\displaystyle g_{S}(f_{i},f_{j})$ $\displaystyle=0\>\>\text{,
if }i\neq j$

One can check that $(B,\omega,g_{S},F_{S})$ is an almost para-Kähler structure
for every $S$. The semi-Riemannian metrics corresponding to these almost para-
Kähler structures are precisely the ones that arise as solutions to (1.3). The
above analysis proves Theorem 1.4.

######  Remark 4.3.

If $(\lambda_{i},\lambda_{n+i})$ is chosen to be $p_{1}$ for every $i$, the
solution corresponds to the choice $S=\\{1,2,\dots,n\\}$ in the above
description. The para-contact structure corresponding to this almost para-
Kähler structure was studied in [7] and described as the canonical para-
Sasakian structure associated with a non-Sasakian $(\kappa,\mu)$-manifold with
index $|I_{M}|<1$. Thus, the corresponding semi-Riemannian metric on the base
space defines a para-Kähler structure.

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## Statements and Declarations

  * •

Funding: Author Sannidhi Alape was supported by CSIR grant
09/1020(0143)/2019-EMR-I, DST, Government of India. Authors Atreyee
Bhattacharya and Dheeraj Kulkarni declare that no funds, grants, or other
support were received by them for the preparation of this manuscript.

  * •

Competing interests: The authors have no relevant financial or non-financial
interests to disclose.

  * •

Data availablility statement: No data was used to derive the results in this
article.

  * •

Authors’ contributions: All authors contributed to the study conception and
design. The mathematical ideas contained in this article were contributed by
Sannidhi Alape, Atreyee Bhattacharya and Dheeraj Kulkarni. The first draft of
the manuscript was prepared jointly by all authors. All authors commented on
previous versions of the manuscript. All authors read and approved the final
manuscript.