# Modulational Instabity of Spin-Orbit Coupled Bose-Einstein Condensates
in Discrete Media

S. Sabari1,∗, R. TamilThiruvalluvar2,† and R. Radha1,# 1Centre for Nonlinear
Science, Department of Physics, Government College for Women(A), Kumbakonam
612001, India
2Department of Physics, Pondicherry University, Puducherry - 605014, India
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We address the impact of intra-site spin-orbit (SO) coupling and associated
inter-component Rabi coupling on the modulational instability (MI) of plane-
wave states in two-component discrete Bose-Einstein condensates (BECs).
Conditions for the onset of the MI and the respective instability are found
analytically. SO coupling allows us to produce the MI even for a small initial
wavenumber ($q<\pi/2$) for miscible states. In particular, SO coupling
introduces MI even in the absence of hopping coefficient, a concept which may
have wider ramifications in heavy atomic BECs. We have also shown how our
results of the linear stability analysis can be corroborated numerically. The
fact that we have brought out the stability criteria in different domains of
system parameters means that our model is tailor made experiments.

###### keywords:

Bose-Einstein condensates, Spin-orbit coupling, Modulational instability,
Discrete media, Linear stability analysis.

††journal: Physics Letters A

##  1 Introduction

Even though the ubiquitous nature of the spin-orbit (SO) coupling, which
involves the interaction between quantum particle’s spin and its momentum is
quite well known, it came to the fore only after the experimental realization
of Bose-Einstein condensates (BECs), a new state of mater comprising of
ultracold atoms getting piled up in the ground state like a giant matter wave.
It must be emphasized that BECs provide an ideal platform (launching pad) to
engineer SO coupling in a neutral atomic BEC by dressing two atomic spin
states with parts of laser. In fact, in the presence of laser coupling [1],
the interaction between the two dressed atomic spin states are modified
driving a quantum phase transition from a spatially spin mixed state (laser
off) to a phase separated state(stripped phase). The stripped phase which
occurs above a critical laser density initiates a phase transition by
engineering SO coupling thereby leading to the rich possibility of generating
synthetic electric and magnetic fields. Recent investigations have explored
the possibility of identifying tunable spin-orbit coupled (SOC) BECs with
various trapping potentials and stable domains of condensates have been
observed.

The SOC-BECs involving the interplay between the nonlinear two-body
interaction with the linear interaction of SOC have been investigated in the
last few years. For instance, it has been reported that the new ground-state
phase can be created in two-component nonlinear systems such as stripes, phase
separation,etc [2], tricritical points [3], different type of solitons [4, 5,
6, 7], 2D solitons with embedded vorticity and vortex lattice [8, 9, 10, 11].

The combinations of optical lattice (OL) potential with the interaction of SO
coupling was shown to posses interesting phenomena like flattening of Bloch
potential [12], atomic Zitterbewegung [13] and new topological phases [14].
Generally, BECs become fragmented on each lattice when it is trapped by deep
OL potential. Such a system has been effectively described in the tight-
binding approximation by discrete version of the corresponding Gross-
Pitaevskii (GP) equation [15, 16, 17]. Recent investigations on composite
solitons and localized modes are carried out on Rashba type SOC-BEC trapped in
deep OL [11, 18].

The modulational instability (MI) is a generic phenomenon which occurs in a
dynamical system like fluid dynamics, nonlinear optics, plasma physics due to
the interplay between nonlinearity and dispersion (or diffraction in spatial
domain) resulting in the fragmentation of carrier waves into a train of
localized waves [19]. Recently, the onset of MI in scalar [20, 21, 22] and
vector BECs [23, 24, 25] within the framework of continuous NLS type (GP)
equations has been investigated. The above scenario could drastically change
in discrete multicomponent settings [26, 27, 28, 29, 30].

It has been found that the onset of MI not only depends on the sign of
diffraction (or dispersion), but also on the sign of inter-species and intra-
species interaction strengths. Recently, MI in SOC-BECs has been investigated
in continuous media [31, 32]. However, the condition for the onset MI in
discrete media (or settings) has not yet been completely brought out,
particularly after the advent of SOC-BECs. In the light of the above
developments, it would be interesting to study the onset of MI in SOC-BEC in
discrete media. The freedom associated with the SOC-BECs by virtue of Rasbha
coupling, SO coupling, intra/inter species interaction and dispersion could
lead to the identification of several interesting stable domains from an
experimental perspective.

In view of the above, we have made an attempt to analyze the impact of SO
coupling on MI in discrete BECs. This paper is structured as follows. In sec.
2, the SO coupling model is constructed in terms of coupled discrete GP
equations and associate dispersion relations are found via linear stability
analysis. The results of semi analytical analysis are discussed with different
signs of intra- and inter component fragmentation with SO coupling interaction
in sec. 3. Finally in sec. 4, we elaborate on the salient features of the
present investigation.

##  2 Theoretical model and dispersion relation for MI

We consider a BEC comprising of atoms in different hyperfine ground states
corresponding to $|F=1,m_{F}=0\rangle$ and $|F=1,m_{F}=-1\rangle$ [1]. These
settings induce the SO coupling resulting in the formation of two dressed
states as a basis. The dynamics of such a system described by mean-field GP
equation in the presence of deep OL in spinor form is governed by

$\begin{split}\imath\hbar\frac{\partial\Psi}{\partial
t}=\left[\frac{\hat{p}^{2}}{2m}+V_{OL}+\hat{H}_{SOC}+\hat{H}_{mix}+H_{int}\right]\Psi\end{split}$
(1)

where $\Psi=(\Psi^{+},\Psi^{-})^{T}$ represents the spinor wave function in
the dressed atomic state and the term $\frac{\hat{p}^{2}}{2m}$ represents the
kinetic energy of the quasi-one-dimensional condensate along the x-direction.
The Hamiltonian term which describes the SOC interaction has the following
form,

$\hat{H}_{SOC}=-\imath\frac{\hbar^{2}}{m}\gamma\sigma_{z}\frac{\partial}{\partial
x}$ (2)

where $\gamma$ is the strength of SO coupling and $\sigma_{z}$ is z-component
Pauli’s spin matrix. The linear coupling term is represented by
$\hat{H}_{mix}=\hbar\delta\sigma_{x}$, which orginates through magnetic field
in intra-SOC system where $\delta$ is strength of Rabi coupling. $H_{int}$
accounts for the intra- and inter-component interactions in the BEC for the
mixture which are nonlinear in nature.

Such a system in deep OL leads to the replacement of the continuous GP
equation by the discrete counterpart to single component BECs in earlier
studies [15, 16, 17] and two-component BECs [27, 28, 29, 30]. Thus, we arrive
at a system of coupled discrete GP equation with intra-SOC and Rabi coupling
of the following form [11, 18],

$\begin{split}i\frac{\partial\Psi_{n}^{\pm}}{\partial
t}=&-\Gamma^{\pm}\big{(}\Psi_{n+1}^{\pm}+\Psi_{n-1}^{\pm}-2\Psi_{n}^{\pm}\big{)}\mp\imath\gamma\big{(}\Psi_{n+1}^{\pm}-\Psi_{n-1}^{\pm}\big{)}\\\
&+\big{(}g^{\pm}|\Psi_{n}^{\pm}|^{2}+G^{c\pm}|\Psi_{n}^{\mp}|^{2}\big{)}\Psi_{n}^{\pm}-\delta\Psi_{n}^{\mp}.\end{split}$
(3)

where, $\Gamma$ represents hopping coefficient of adjacent lattice sites while
$g^{\pm}$ and $G^{c\pm}$ represent intra- and inter-component collision
strengths respectively. This system conserves two quantities for infinite
lattices namely

$N=\sum_{n}\left(|\Psi_{n}^{+}|^{2}+|\Psi_{n}^{-}|^{2}\right)$

and the Hamiltonian which is

$\begin{split}H&=\sum_{n}\Big{(}-\Gamma\left(\Psi_{n}^{+\ast}\Psi_{n+1}^{+}+\Psi_{n}^{-\ast}\Psi_{n+1}^{-}\right)-\imath\gamma(\Psi_{n}^{+\ast}\Psi_{n+1}^{+}-\\\
&\Psi_{n}^{-\ast}\Psi_{n+1}^{-})+\frac{1}{4}\left(g^{+}|\Psi_{n}^{+}|^{4}+g^{-}|\Psi_{n}^{-}|^{4}\right)+\frac{1}{4}(G^{c+}|\Psi_{n}^{+}|^{2}\\\
&|\Psi_{n}^{-}|^{2}+G^{c-}|\Psi_{n}^{+}|^{2}|\Psi_{n}^{-}|^{2})-\delta\Psi_{n}^{+\ast}\Psi_{n}^{-}\Big{)}+c.c.\end{split}$
(4)

We now explore the stability of stationary plane-wave solutions of Equ.(3)
with the following ansatz

$\Psi_{n}^{\pm}=u^{\pm}\exp[\imath\omega^{\pm}t+\imath nq^{\pm}].$ (5)

Employing the ansatz given by Equ.(5), one can obtain the following dispersion
relation

$\omega=-2\Gamma_{\pm}\cos(q)+g^{\pm}u^{\pm
2}+G^{c\pm}u_{\mp}^{2}-\frac{u^{\mp}}{u^{\pm}}\delta\pm 2\gamma\sin(q)$ (6)

Next, we slightly modify the plane wave ansatz given by Equ.(5) of the
following form

$\Psi_{n}^{\pm}=(u^{\pm}+\xi^{\pm})\exp[\imath\omega^{\pm}t+\imath nq^{\pm}]$

Subsequently, perturbed solution of the above ansatz was substituted in
Equ.(3) to extract the discrete differential equation only in linear terms of
$\xi^{\pm}$ ($\xi^{\pm\ast}$ is complex conjugate) as

$\begin{split}\imath\frac{\partial\xi_{n}^{\pm}}{\partial
t}&=\,-\Gamma^{\pm}\big{(}\xi_{n+1}^{\pm}+\xi_{n-1}^{\pm}-2\xi_{n}^{\pm}\big{)}+g^{\pm}(\xi_{n}^{\pm}+\xi_{n}^{\pm\ast})\\\
&+G^{c\pm}(\xi_{n}^{\mp}+\xi_{n}^{\mp\ast})\mp\imath\gamma(\xi_{n+1}^{\pm}-\xi_{n-1}^{\pm})-\delta\xi_{n}^{\mp}.\end{split}$
(7)

The perturbed amplitudes are taken as plane waves in discrete coordinates of
the following form

$\begin{split}\xi_{n}^{+}&=\psi 1\exp[\imath\Omega t+\imath nQ]+\psi
2\exp[-\imath\Omega t-\imath nQ]\\\ \xi_{n}^{-}&=\phi 1\exp[\imath\Omega
t+\imath nQ]+\phi 2\exp[-\imath\Omega t-\imath nQ]\end{split}$ (8)

A straightforward substitution of equation (8) in (7) yields the following
dispersion relation for eigenfrequency $\Omega$ as

$\displaystyle\begin{split}\Omega^{4}+P_{3}\Omega^{3}+P_{2}\Omega^{2}+P_{1}\Omega+P_{0}=0\end{split}$
(9)

where

$\displaystyle\begin{split}P_{0}=&\big{[}(g+\gamma)(g+2G-\gamma)+\Lambda_{+}(\gamma-G)-\Lambda_{-}(G-\gamma+\Lambda_{+})\big{]}\\\
&\big{[}(g-\gamma)(g-2G+\gamma)+\Lambda_{-}(G-\gamma)-\Lambda_{+}(\gamma-G+\Lambda_{-})\big{]}\\\
P_{1}=&2(\Lambda_{-}-\Lambda_{+})(\Lambda_{-}\Lambda_{+}-g^{2}+\gamma(\gamma-2G))\\\
P_{2}=&2g^{2}+4G\gamma-2\gamma^{2}+\Lambda_{-}^{2}+\Lambda_{+}^{2}-4\Lambda_{-}\Lambda_{+}\\\
P_{3}=&2(\Lambda_{+}-\Lambda_{-})\end{split}$ (10)

and $\Lambda_{\pm}=\theta_{\pm}+g+\gamma$ and
$\theta_{\pm}=2\cos(q)(\Gamma-\Gamma\cos(Q)\pm\gamma\sin(Q))$ The instability
growth rate is defined as $\zeta=2|\text{Im}(\Omega_{\pm})|$

##  3 Results And Discussion

###  3.1 MI in unstaggered mode

In this section, we study the onset of MI in a unstaggered mode. It has been
found that the MI gives rise to unstaggered mode at $g^{+}>0$ ($q=0$), and
staggered mode for $g^{+}<0$ ($q=\pi$). We have brought out the onset of MI
domains in the absence/presence/ of all system parameters in Table 1.

Table 1: Review of the MI in discrete SOC-BECs at unstaggered mode Hopping |  | Rabi | Possible |
---|---|---|---|---
Strength | SOC | coup. | Cases | Conjecture
|  |  |  $g>0$, $g_{12}>0$ | Unstable
|  | $\delta=0$ |  $g>0$, $g_{12}<0$ | Unstable
|  |  |  $g<0$, $g_{12}>0$ | Stable
| $\gamma=0$ |  |  $g<0$, $g_{12}<0$ | Stable
|  |  |  $g>0$, $g_{12}>0$ | Unstable
|  | $\delta=1$ |  $g>0$, $g_{12}<0$ | Unstable
|  |  |  $g<0$, $g_{12}>0$ | Stable
$\Gamma=1$ |  |  |  $g<0$, $g_{12}<0$ | Unstable
|  |  |  $g>0$, $g_{12}>0$ | Unstable
|  | $\delta=0$ |  $g>0$, $g_{12}<0$ | Unstable
|  |  |  $g<0$, $g_{12}>0$ | Stable
| $\gamma=1$ |  |  $g<0$, $g_{12}<0$ | Stable
|  |  |  $g>0$, $g_{12}>0$ | Unstable
|  | $\delta=1$ |  $g>0$, $g_{12}<0$ | Unstable
|  |  |  $g<0$, $g_{12}>0$ | Stable
|  |  |  $g<0$, $g_{12}<0$ | Unstable
|  |  |  $g>0$, $g_{12}>0$ |
|  | $\delta=0$ |  $g>0$, $g_{12}<0$ | Unstable
|  |  |  $g<0$, $g_{12}>0$ |
$\Gamma=0$ | $\gamma=1$ |  |  $g<0$, $g_{12}<0$ |
|  |  |  $g>0$, $g_{12}>0$ | Unstable
|  | $\delta=1$ |  $g>0$, $g_{12}<0$ | Unstable
|  |  |  $g<0$, $g_{12}>0$ | Stable
|  |  |  $g<0$, $g_{12}<0$ | Stable

Figure 1: (color online) Upper row: Two-dimensional (2D) plot showing the MI
gain as a function of $Q$ and Lower row: Three-dimensional (3D) surface plot
showing the MI gain as a function of $Q$ and $\gamma$. Left and right columns
correspond to $\delta=1$ and $\delta=0$, respectively. Remaining parameters
are $g=1$, $G=-1$ and $\Gamma=1$.

In Fig.1, upper two panels show the MI gain as a function of $Q$ for five
different values of SO coupling strengths, $\gamma=0.0,0.25,0.50,0.75,$ and
$1.0$. Left and right columns correspond to $\delta=1$ and $\delta=0$. As the
strength of the SO coupling increases from $0$ to $0.5$, the MI gain decreases
for both $\delta=1$ and $\delta=0$. As we increase the strength of SO coupling
further, one observes no variation in the gain for $\delta=1$ . But, for
$\delta=0$, the MI gain increases for $\gamma>0.5$. The three-dimensional (3D)
surface plot clearly illustrates this behaviour in the lower panels.

####  3.1.1 Miscible / Immiscible condensates

Next, we study the onset of MI in Miscible / Immiscible condensates.

3D surface plot shows the existence of MI domain as a function of $Q$ and
$\gamma$ for upper row: partially miscible, ($g=1$, $G=-1$), middle row:
immiscible ($g=1$, $G=-0.5$) and lower row: miscible ($g=1$, $G=-1.5$)
condensates.

Figure 2: (color online) 3D surface plot showing the MI region as a function
of $Q$ and $\gamma$ for upper row: partially miscible, ($g=1$, $G=-1$), middle
row: immiscible ($g=1$, $G=-0.5$) and lower row: miscible ($g=1$, $G=-1.5$)
conditions. Left and right columns correspond to $\delta=1$ and $\delta=0$,
respectively.

###  3.2 MI in Staggered Mode

In this section, we study the onset of MI in a staggered mode. It is worth
mentioning at this juncture that at $\gamma=0$, our results completely agree
with the Ref. [27], and could reproduce the results of one of the cases (in
the absence of three-body interaction) discussed in Ref. [28]. These results
are reproduced by our investigation.

Next, to study the impact of SO coupling on MI i.e $\gamma\neq 0$, we first
discuss the MI gain for $g>0$ and $G>0$. To facilitate our understanding and
to make the investigation straightforward, we have reviewed the possible MI
domains in table 2. We have thoroughly brought out the MI domains in the
absence/presence/ of hopping strength, SO coupling and Rabi coupling for
different possible combinations of signs of intra- and inter-component
interactions strength.

Table 2: Review of the MI in discrete SOC-BECs at Staggered mode Hopping |  | Rabi | Possible |
---|---|---|---|---
Strength | SOC | coup. | Cases | Conjecture
|  |  |  $g>0$, $g_{12}>0$ | Stable
|  | $\delta=0$ |  $g>0$, $g_{12}<0$ | Stable
|  |  |  $g<0$, $g_{12}>0$ | Unstable
| $\gamma=0$ |  |  $g<0$, $g_{12}<0$ | Unstable
| [27] |  |  $g>0$, $g_{12}>0$ | Stable
|  | $\delta=1$ |  $g>0$, $g_{12}<0$ | Unstable
|  |  |  $g<0$, $g_{12}>0$ | Stable
$\Gamma=1$ |  |  |  $g<0$, $g_{12}<0$ | Unstable
|  |  |  $g>0$, $g_{12}>0$ | Stable
|  | $\delta=0$ |  $g>0$, $g_{12}<0$ | Stable
|  |  |  $g<0$, $g_{12}>0$ | Unstable
| $\gamma=1$ |  |  $g<0$, $g_{12}<0$ | Unstable
|  |  |  $g>0$, $g_{12}>0$ |
|  | $\delta=1$ |  $g>0$, $g_{12}<0$ | Stable
|  |  |  $g<0$, $g_{12}>0$ |
|  |  |  $g<0$, $g_{12}<0$ |
|  |  |  $g>0$, $g_{12}>0$ |
|  | $\delta=0$ |  $g>0$, $g_{12}<0$ | Unstable
|  |  |  $g<0$, $g_{12}>0$ |
$\Gamma=0$ | $\gamma=1$ |  |  $g<0$, $g_{12}<0$ |
|  |  |  $g>0$, $g_{12}>0$ | Unstable
|  | $\delta=1$ |  $g>0$, $g_{12}<0$ | Unstable
|  |  |  $g<0$, $g_{12}>0$ | Stable
|  |  |  $g<0$, $g_{12}<0$ | Stable

Figure 3: (color online) Left and right columns correspond to $\delta=1$ and
$\delta=0$, respectively. The remaining parameters are $g=1$, $G=1$ and
$\Gamma=1$.

In Fig.3, upper two panels show the MI gain as a function of $Q$ for five
different values of SO coupling strengths, $\gamma=0.0,0.25,0.50,0.75,$ and
$1.0$. Left and right columns correspond to $\delta=1$ and $\delta=0$. As the
strength of the SO coupling increases from $0$ to $0.5$, the MI gain decreases
for both $\delta=1$ and $\delta=0$ . As we increase the strength of SO
coupling further, one observes no variation in the gain for $\delta=1$ . But,
for $\delta=0$ , the MI gain increases for $\gamma>0.5$. The three-dimensional
(3D) surface plot clearly illustrates this behaviour in the lower panels.

Figure 4: (color online) MI gain as a function of $Q$ for (top left to bottom
right) (i) $g=1$, $G=1$, (ii) $g=1$, $G=-1$,(iii) $g=-1$, $G=1$ and (iv)
$g=-1$, $G=-1$. Remaining parameters are $\delta=1$ and $\Gamma=1$.

Figure 5: (Upper and middle) The space-time evolution of the density in both
condensates (left and right) for MS and MU, $g>0$, $g_{12}<0$ and $g<0$,
$g_{12}>0$, respectively. Lower panel shows the comparative time evolution of
the maximum density of MU and MS for the same parameters corresponding to
$2^{nd}$ and $3^{rd}$ cases in Table 1.)

Next, we discuss the MI domains for other three different possible
combinations of intra- and inter-species fragmentation, i.e $g$ and $G$. When
the inter-component interaction is repulsive ($G>0$), the MI gain decreases
when one increases the strength of SO coupling and this occurs irrespective of
the signs of intra-component interaction. Also, when $G<0$, the MI gain
increases when we increase $\gamma$. Further, one does not observe MI when
$\gamma=0$ for both $G<0$ as shown in Fig.4.

Figure 5 shows the modulationally stable (MS) and modulationally unstable (MU)
behaviour for $2^{nd}$ and $3^{rd}$ cases corresponding to $g>0$, $g_{12}<0$
and $g<0$, $g_{12}>0$, in Table 1. Upper and middle panels show the 3D space-
time evolution of the density for both condensates (left and right) and for MS
and MU cases, respectively. In the upper two panels, during the time evolution
of the wave in condensates, the wave amplitudes (periodicity) exhibit small
oscillations and remain close to their initial values demonstrating the
stability of the system. On the contrary, in the lower two panels, the wave
amplitude in the condensates grows exponentially due to their instability. The
comparative evolution of the wave amplitudes in both cases is plotted in lower
panel. The maximum density shows mild oscillations nearly around its initial
value 1 for the MS case in both condensates (dashed blue and dash-dotted red
lines). Blue and red solid lines show the exponential growth of the maximum
densities for the MU mode in both the condensates.

Figure 6: The space-time evolution of the density in both the condensates
(left and right). Panels from upper to lower show the density with increasing
strengths of the SOC, $\gamma=0.1$, $\gamma=0.2$, $\gamma=0.3$, $\gamma=0.4$
and $\gamma=0.5$, respectively. Remaining parameters are $g=1.0$,
$g_{12}=-1.0$ and $\delta=1.0$ with the other parameters picked from Fig.4
(ii).

Figure 6 shows the 3D space-time evolution of the density in both the
condensates for the unstable modes. The instability increases on increasing
the strengths of the SOC. The 3D space-time evolution of the density remains
stable upto time interval $t=60$ with small disturbances in both the
condensates (left and right) for $\gamma=0.1$. However, when we increase the
strength of the SOC to $\gamma=0.3$, the 3D space-time evolution of the
densities starts to collapse around time $t=20$. Also, if we increase the
strength of the SOC to $\gamma=0.5$ further, the maximum densities increase
exponentially with large oscillations.

####  3.2.1 Miscible / Immiscible condensates

Next, we study the MI gain in Miscible / Immiscible condensates.

Figure 7 displays the MI region in 3D surface plot as a function of $Q$ and
$\gamma$ for partially miscible ($g=1$, $G=1$) condensates in upper row,
miscible ($g=1$, $G=0.5$) case in middle row and immiscible ($g=1$, $G=1.5$)
case in lower row.

Figure 7: (color online) 3D surface plot showing the MI region as a function
of $Q$ and $\gamma$ for upper panel: partially miscible, ($g=1$, $G=1$) and
lower row (left): immiscible ($g=1$, $G=0.5$) and (right): miscible ($g=1$,
$G=1.5$) conditions.

Figure 8: Comparative time evolution of the maximum density for miscible,
immiscible and semi miscible cases corresponding to the parameters in Fig. 7.

Figure 8 shows the comparative evolution of the wave amplitudes in both cases
discussed in Fig. 7 for miscible, immiscible and partially miscible cases.
Mild and large oscillations of the maximum density illustrate MS and MU cases.

Figure 9: (color online) Regions of modulational stability / instability in
the $(g,G)$ plane for all possible combinations of Rabi and SOCs.

Figure 10: (color online) Q Vs q for (a) $\Gamma=1$ and $\gamma=0$, (b)
$\Gamma=1$ and $\gamma=1$, (c) $\Gamma=0$ and $\gamma=1$. Remaining parameters
are $g=1$, $G=1$ and $\delta=1$.

In the absence of both Rabi and SOC, the MI region exists only in the
repulsive two-body inter-atomic interaction zone, as shown in Fig.9 (a). When
we add Rabi coupling alone, the MI region expands from repulsive to attractive
two-body inter-atomic as well as intra-atomic interaction zone as shown in
Fig.9 (b). We also observe that in the presence of SOC and absence of Rabi
coupling, the MI region exists only in repulsive two-body inter-atomic and
both repulsive and attractive intra-atomic interactions zone, as displayed in
Fig.9 (c). Now, when we reinforce SOC along with Rabi coupling, the MI region
exists mostly in attractive two-body inter-atomic and both repulsive and
attractive intra-atomic interaction zone as shown in Fig.9 (d).

Further, from panels (a)-(c), where the strength of the Rabi coupling is
absent, the stability zone stays in the same region with small variations.
However, from panels (a)-(b), where the strength of the SOC is absent, we
observe that the repulsive inter-atomic region which was stable earlier
becomes unstable while repulsive/attractive inter-atomic and intra-atomic
regions which were unstable now shift to stable domain. All the above occurs
at the behest of Rabi coupling. Hence, from Fig.9, it is pretty obvious that
the impact of Rabi coupling is more pronounced compared to the other system
parameters and this result underscores the dominance of Rabi coupling.

In Fig.10, panels (a) and (b) show the effect of SOC in $(q,Q)$ plane. When
there is no SOC, the MI regions exist only for $Q\geq 0.8$. But, in the
presence of SOC, the onset of MI appears at $Q\geq 0.1$. In panel (c), we show
the effect of the SOC when the hopping strength is absent ($\Gamma=0$). In
this case, there is no unstable island inside the stable region as shown in
panel (b). In addition, from fig.10, we observe MI region when $q\geq 1.7$.

##  4 Conclusion

We have investigated the impact of intra-site SO coupling and associated
inter-component Rabi coupling on the MI of plane-wave states in two-component
discrete BECs. We have brought out the existence of MI domains under the
interplay between SOC, Rabi coupling, intra and inter species interactions. We
have also found that the SO coupling initiates MI even for a small initial
wavenumber in miscible states and also introduces MI even in the absence of
hopping coefficient, a concept which may have wider ramifications in heavy
atomic BECs. We have also shown how the results of linear stability analysis
can be corroborated numerically.

## Acknowledgements

Authors wish to thank the reviewers for their critical comments to improve the
contents of the paper. SS wishes to thank the Council of Scientific and
Industrial Research(CSIR) for financial assistance(Grant No. 03(1456)/19/EMR-
II), the Government of India. RR wishes to acknowledge the financial
assistance from DAE-NBHM(Grant No .02011/3/20/2020/NBHM(R.P)/R $\&$ D II dated
8th September 2020) and CSIR(Grant No 03(1456)/19/EMR-II).

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BASIC AUTOMORPHISM GROUPS OF CARTAN

FOLIATIONS COVERED BY FIBRATIONS

K.I. SHEINA

National Research University Higher School of Economics,
Department of Informatics, Mathematics and Computer Science,
ul. Bolshaja Pecherskaja, 25/12, Nizhny Novgorod, 603155, Russia
<EMAIL_ADDRESS>
Abstract The basic automorphism group ${A}_{B}(M,F)$ of a Cartan foliation
$(M,F)$ is the quotient group of the automorphism group of $(M,F)$ by the
normal subgroup, which preserves every leaf invariant. For Cartan foliations
covered by fibrations, we find sufficient conditions for the existence of a
structure of a finite-dimensional Lie group in their basic automorphism
groups. Estimates of the dimension of these groups are obtained. For some
class of Cartan foliations with integrable an Ehresmann connection, a method
for finding of basic automorphism groups is specified.

Keynwords: foliation; Cartan foliation; Lie group; basic automorphism;
automorphism group

2010 Mathematics Subject Classification: 53C12; 22Exx; 54H15; 53Cxx

##  1 Introduction

One of the main objects associated with a geometric structure on a smooth
manifold is its automorphism group. In the introduction to the monograph by S.
Kobayashi [11], it was emphasized that the existence of a structure of a
finite-dimensional Lie group in the group of automorphisms of a manifold with
a geometric structure is one of the central problems in differential geometry.

As is known, the solved Hilbert's 5-th problem is devoted to finding
conditions under which a topological group admits the structure of a Lie group
[16]. It is known from the numerous works of E. Cartan, R. Mayer, H. Steenrod,
K. Nomizu, S. Kabayashi, S. Ehresmann and other authors that the automorphism
groups of many geometric structures are Lie groups of transformations (see
overview [7]).

The spaces that are now called Cartan geometries were introduced by E. Cartan
in the 1920-s. The theory of Cartan geometries is presented in the monographs
of A. $\check{C}$ap, J. Slovak [5], R.V. Sharpe [14], M. Krampin and D.
Saunders [8]. Currently, Cartan geometries and Cartan foliations are studied
by many mathematicians and find application in various physical theories, see,
for example, [1], [6], [13] and [9], [17].

Let $(M,F)$ be a smooth foliation. Recall that geometry structure on the
manifold $M$ is called transverse to $(M,F)$ if it is a invariant with respect
to local holonomic diffeomorphisms. Another, equivalent definition of a
transverse geometric structure, which is represented by Cartan geometry, is
given in Section 3. Morphisms are understood as local diffeomorphisms mapping
leaves onto leaves and preserving transverse geometries (the precise
definition see in Section 3). Let us denote by $\mathfrak{C}\mathfrak{F}$ the
category of Cartan foliations.

This paper is devoted to the investigation of automorphism groups of Cartan
foliations, i.e. foliations that admit Cartan geometries as transverse
structures. The study of Cartan foliations is motivated by the fact that such
broad classes of foliations as parabolic, conformal, projective, pseudo-
Riemannian, Lorentzian, Weyl, transverse homogeneous foliations and foliations
with transverse linear connection belong to Cartan foliations. Therefore, the
investigation of Cartan foliations allows us to study the general properties
of these foliations from a single point of view, while many authors study them
separately.

Let us denote by $A(M,F)$ the group of all the automorphisms of the Cartan
foliation $(M,F)$ in the category $\mathfrak{C}\mathfrak{F}$. The group

${A}_{L}(M,F):=\\{f\in{A}(M,F)\mid f(L_{\alpha})=L_{\alpha}\,\,\,\forall
L_{\alpha}\in F\\}$

is a normal subgroup of the group ${A}(M,F)$ and called the group of leaf
automorphisms of $(M,F).$ The quotient group ${A}(M,F)/{A}_{L}(M,F)$ is called
the basic automorphism group  and denoted by ${A}_{B}(M,F).$

We study the groups of basic automorphisms ${A}_{B}(M,F)$ of Cartan foliations
$(M,F)$ covered by fibration and find sufficient conditions for the existence
of a structure of a finite-dimensional Lie group in the group ${A}_{B}(M,F)$.
J. Leslie [12] was the first who solved a similar problem for smooth
foliations on compact manifolds and considered an application to foliations
with transverse G-structures. For foliations with complete transversely
projectable affine connection, this problem was raised by I.V. Belko [2].
Foliations $(M,F)$ with effective transverse rigid geometries were
investigated by N.I. Zhukova [19] where an algebraic invariant
$\mathfrak{g}_{0}=\mathfrak{g}_{0}(M,F),$ called the structural Lie algebra of
$(M,F)$, was constructed and it was proved that $\mathfrak{g}_{0}=0$ is a
sufficient condition for the existence of a unique Lie group structure in the
basic automorphism group of this foliation. In [15], the existence of a Lie
group structure was investigated in the basic automorphism groups of Cartan
foliations modeled on inefficient Cartan geometries.

##  2 Main results

Among the Cartan foliations, foliations covered by fibrations are
distinguished.

######  Definition 1.

Let $\kappa:\widetilde{M}\to M$ be the universal covering map. We say that a
smooth foliation $(M,F)$ is covered by fibration if the induced foliation
$(\widetilde{M},\widetilde{F})$ is formed by fibres of a locally trivial
fibration $\widetilde{r}:\widetilde{M}\to B.$

The following theorem describes the global structure of Cartan foliations
covered by fibrations.

######  Theorem 1.

Let $(M,F)$ be a Cartan foliation modeled on a Cartan geometry $\xi$ covered
by the fibration $\widetilde{r}:\widetilde{M}\to B$, where
$\widetilde{\kappa}:\widetilde{M}\to M$ is the universal covering map. Then:

  1. (1)

there exists a regular covering map $\kappa:\widehat{M}\to M$ such that the
induced foliation $\widehat{F}$ is made up of fibres of the locally trivial
bundle $r:\widehat{M}\to B$ over a simply connected manifold $B$, and $\xi$
induces on $B$ a Cartan geometry $\eta$ that is locally isomorphic to $\xi$;

  2. (2)

an epimorphism $\chi:\pi_{1}(M,x)\to\Psi,\,x\in M,$ of the fundamental group
$\pi_{1}(M,x)$ onto a subgroup $\Psi$ of the automorphism Lie group
$Aut(B,\eta)$ of the Cartan manifold $(B,\eta)$ is determined;

  3. (3)

the group of deck transformations of the covering $\kappa:\widehat{M}\to M$ is
isomorphic to the group $\Psi$.

######  Definition 2.

The group $\Psi=\Psi(M,F)$ satisfying Theorem 1 is called the global holonomy
group of the Cartan foliation $(M,F)$ covered by fibration.

We give a detailed proof of the following theorem, formulated without a proof
in the work [15, Prop. 8]. Theorem 2 establishes a connection between the
basic automorphism group $A_{B}(M,F)$ of a Cartan foliation $(M,F)$ covered by
fibration and its global holonomy group $\Psi$.

######  Theorem 2.

Let $(M,F)$ be a Cartan foliation covered by fibration
$r:\widehat{M}\rightarrow B$, and $B$ is the simply connected Cartan manifold.
Suppose that the global holonomy group $\Psi$ is a discrete subgroup of the
Lie group $Aut(B,\eta)$. Let $N(\Psi)$ be the normalizer of $\Psi$ in
$Aut(B,\eta)$. Then the basic automorphism group $A_{B}(M,F)$ is a Lie group
which is isomorphic to an open-closed subgroup of the Lie quotient group
$N(\Psi)/\Psi$ and $\dim(A_{B}(M,F))=\dim(N(\Psi)/\Psi).$ The structure of the
Lie group in $A_{B}(M,F)$ is unique.

The following theorem specifies a method for computing basic automorphism
groups for Cartan foliations with an integrable Ehresmann connection.

######  Theorem 3.

Let $(M,F)$ be Cartan foliation with an integrable Ehresmann connection. Then

1.There is a regular cover $\kappa:\widetilde{M}\to M$ such that
$\widetilde{M}\leavevmode\nobreak\ =L_{0}\leavevmode\nobreak\
\times\leavevmode\nobreak\ B,$ where $L_{0}$ is a manifold diffeomorphic to
any leaf with a trivial holonomy group and B is a simply connected manifold,
and the induced foliation $\widetilde{F}=\kappa^{*}F$ is formed by leaves of
the canonical projection $r:L_{0}\times B\to B$ onto the second factor, and
Cartan geometry $\eta$ is induced on $B$, with respect to which $\kappa$ is a
morphism of Cartan foliations $(M,F)$ and $(\widetilde{M},\widetilde{F})$ in
the category $\mathfrak{C}\mathfrak{F}$.

2\. The foliation $(M,F)$ is an $(Aut(B,\eta),B)$-foliation.

3\. If moreover, the normalizer $N(\Psi)$ of global holonomy group $\Psi$ is
equal to the centralizer $Z(\Psi)$ of $\Psi$ in the group $Aut(B,\eta)$, then

$A_{B}(M,F)\cong N(\Psi)/\Psi.$

Using Theorem 3, we construct an example of computing the basic automorphism
group of some conformal foliation of an arbitrary codimension $q$, where
$q\geq 3$, on a $(q+1)$-dimensional manifold in Section 7.2. Some other
examples are constructed in [15].

##  3 The category of Cartan foliations

#### The category of Cartan geometries

Let $G$ and $H$ be Lie groups with the Lie algebras $\mathfrak{g}$ and
$\mathfrak{h}$ relatively. Let $H$ be a closed subgroup of $G$. A Cartan
geometry of type $(G,H)$ on the smooth manifold $N$ is a principal $H$-bundle
$P(N,H)$ with a $\mathfrak{g}$-valued $1$-form $\omega$ on $P$ satisfying the
following conditions:
($c_{1}$) the map $\omega_{u}:T_{u}P\rightarrow\mathfrak{g}$ is an isomorphism
of vector spaces for every $u\in P$;
($c_{2}$) $\omega({A^{*}})=A$ for every $A\in\mathfrak{h}$, where $A^{*}$ is
the fundamental vector field determined by $A$;
($c_{3}$) $R^{*}_{h}\omega=Ad_{G}(h^{-1})\omega$ for every $h\in H$, where
$Ad_{G}:H\rightarrow GL(\mathfrak{g})$ is the adjoint representation of the
Lie subgroup $H$ of $G$ in the Lie algebra $\mathfrak{g}$.

The $\mathfrak{g}$-valued form $\omega$ is called a Cartan connection form.
This Cartan geometry is denoted by $\xi=(P(N,H),\omega)$. The pair $(N,\xi)$
is called a Cartan manifold.

Maximal normal subgroup $K$ of the group $G$ belonging to $H$ is called the
kernel of pair $(G,H)$. We denote the Lie algebra of the group $K$ by
$\mathfrak{k}$. The Cartan geometry $\xi=(P(M,H),\omega)$ of type $(G,H)$ is
called effective if the kernel $K$ of the pair $(G,H)$ is trivial. Further, we
assume that all Cartan geometries under consideration are effective.

Let $\xi=(P(N,H),\omega)$ and
$\xi^{\prime}=(P^{\prime}(N^{\prime},H),\omega^{\prime})$ be two Cartan
geometries with the same structure Lie group $H$. The smooth map $\Gamma:P\to
P^{\prime}$ is called a morphism from $\xi$ to $\xi^{\prime}$ if
$\Gamma^{*}\omega^{\prime}=\omega$ and $R_{a}\circ\Gamma=\Gamma\circ
R_{a}\,\,\,\forall a\in H$. The category of Cartan geometries is denoted by
${\mathfrak{C}}{\mathfrak{a}}{\mathfrak{r}}$. If $\Gamma\in
Mor(\xi,\xi^{\prime})$, then the projection $\gamma:N\to N^{\prime}$ is
defined such that $p^{\prime}\circ\Gamma=\gamma\circ p,$ where $p:P\to N$ and
$p^{\prime}:P^{\prime}\to N^{\prime}$ are the projections of the respective
$H$-bundles.

The projection $\gamma$ is called an automorphism of the Cartan manifold
$(N,\xi)$. Denote by $Aut(N,\xi)$ the full automorphism group of a Cartan
foliation $(N,\xi)$ and by $Aut(\xi)$ the full automorphism group of a Cartan
geometry $\xi$. Let $A(P,\omega):=\\{\Gamma\in Diff(P)\,|\\\
\,{\Gamma^{*}\omega=\omega}\\}$ be the automorphism group of the
parallelizable manifold $(P,\omega)$, which is known to be a Lie group, and
$\dim(A(P,\omega))\leq\dim P.$

Remark, that $Aut(\xi)=\\{\Gamma\in A(P,\omega)\,|\,\Gamma\circ
R_{a}=R_{a}\circ\Gamma\,\,\forall a\in H\\}$ is a closed Lie subgroup of the
Lie group $A(P,\omega)$. Therefore, $Aut(\xi)$ is a Lie group, and due to the
effictivity of a Cartan geometry $\xi$, there exists a Lie group isomorphism

$\sigma:A^{H}(P,\omega)\to Aut(N,\xi):\Gamma\mapsto\gamma$

mapping $\Gamma\in A^{H}(P,\omega)$ to its projection $\gamma$.

#### Cartan foliations

Let $N$ be a smooth $q$-dimensional manifold, the connectivity of which is not
assumed. Let $M$ be a smooth $n$-dimensional manifold, where $0<q<n$. Assume,
that $\xi=(P(N,H),\omega)$ is a Cartan geometry of type $(G,H)$ on the
manifold $N$. Let $p:P\to N$ be the projection the principal $H$-bundle. For
every open subset $V\subset N$, the Cartan geometry
$\xi_{V}=(P_{V}(V,H),\omega_{V})$ of the same type $(G,H)$ is induced, where
$P_{V}:=p^{-1}(V)$ and $\omega_{V}:=\omega|_{P_{V}}$. Remind that
$(N,\xi)$-cocycle on $M$ is a family
$\zeta=\\{U_{i},f_{i},\\{\gamma_{ij}\\}\\}_{ij\in J}$ satisfying the following
conditions:
1)the set $\\{U_{i}\,|\,i\in J\\}$ is a covering of the manifold $M$ by open
connected subsets $U_{i}$ of $M$, and every $f_{i}:U_{i}\to N$ is a submersion
with connected fibres;
2) if $U_{i}\cap U_{j}\neq\emptyset,\,i,j\in J$, then there exists an
isomorphism $\Gamma_{ij}:\xi_{f_{j}(U_{i}\cap U_{j})}\to\xi_{f_{i}(U_{i}\cap
U_{j})}$ of the Cartan geometries induced on open subsets ${f_{j}(U_{i}\cap
U_{j})}$ and ${f_{i}(U_{i}\cap U_{j})}$ such that the projection $\gamma_{ij}$
of the isomorphism $\Gamma_{ij}$ satisfies the equality
$f_{i}=\gamma_{ij}\circ f_{j}$ on $U_{i}\cap U_{j},\,i,j\in J$;
3) if $U_{i}\cap U_{j}\cap U_{k}\neq\emptyset$, then
$\gamma_{ij}\circ\gamma_{jk}=\gamma_{ik}$ for all $x\in U_{i}\cap U_{j}\cap
U_{k}$ and $\gamma_{ii}=id_{U_{i}}$, $i,j,k\in J$.

Two $N$-cocycles are called equivalent if there exists an $N$-cocycle
containing both of these cocycles. Let
$[\\{U_{i},f_{i},\\{\gamma_{ij}\\}\\}_{ij\in J}]$ be the equivalence class of
$N$-cocycles on the manifold $M$ containing the cocycle
$\zeta=\\{U_{i},f_{i},\\{\gamma_{ij}\\}\\}_{ij\in J}$. Denote by $\Sigma$ the
set of fibres of all the submersions $f_{i}$ of this equivalence class. Note,
that $\Sigma$ is the base of some new topology $\tau$ on $M$. The path-
connected components of the topological space $(M,\tau)$ form a partition
$F:=\\{L_{\alpha}\,|\,\,\alpha\in\mathfrak{J}\\}$ of the manifold $M$. The
pair $(M,F)$ is called a Cartan foliation of codimension $q$ modeled on the
Cartan geometry $\xi=(P(N,H),\omega)$ which is called a transverse Cartan
geometry for $(M,F)$. Subsets $L_{\alpha},\,\alpha\in\mathfrak{J}$ are called
leaves of this foliation. It is said that $(M,F)$ is given by the
$(N,\xi)$-cocycle $\zeta$.

#### Morphisms in the category of Cartan foliations

Let $(M,F)$ and $(M^{\prime},F^{\prime})$ are Cartan foliations defined by an
$(N,\xi)$-cocycle $\zeta=\\{U_{i},f_{i},\\{\gamma_{ij}\\}\\}_{ij\in J}$ and an
$(N^{\prime},\xi^{\prime})$-cocycle
$\zeta^{\prime}=\\{U^{\prime}_{r},f^{\prime}_{r},\\{\gamma^{\prime}_{rs}\\}\\}_{i^{\prime}j^{\prime}\in
J^{\prime}}$, respectively. All objects belonging to $\zeta^{\prime}$ are
distinguished by prime. Let $f\colon M\to M^{\prime}$ be a smooth map which is
a local isomorphism in the foliation category
${\mathfrak{F}\mathfrak{o}\mathfrak{l}}.$ Hence for any $x\in M$ and $y:=f(x)$
there exist neighborhoods $U_{k}\ni x$ and $U^{\prime}_{s}\ni y$ from $\zeta$
and $\zeta^{\prime}$ respectively, a diffeomorphism $\varphi\colon V_{k}\to
V^{\prime}_{s},$ where $V_{k}:=f_{k}(U_{k})$ and
$V^{\prime}_{s}:=f^{\prime}_{s}(U^{\prime}_{s}),$ satisfying the relations
$f(U_{k})=U^{\prime}_{s}$ and $\varphi\circ f_{k}=f^{\prime}_{s}\circ
f|_{U_{k}}$. Further we shall use the following notations:
$P_{k}:=P|_{V_{k}},\,\,\,P^{\prime}_{s}:=P^{\prime}|_{V^{\prime}_{s}}$ and
$p_{k}:=p|_{P_{k}},\,\,$ $p^{\prime}_{s}:=p|_{P^{\prime}_{s}}$

We say that $f$ preserves transverse Cartan geometry if every such
diffeomorphism $\varphi\colon V_{k}\to V^{\prime}_{s}$ is an isomorphism of
the induced Cartan geometries $(V_{k},\xi_{V_{k}})$ and
$(V^{\prime}_{s},\xi^{\prime}_{V^{\prime}_{s}})$. This means the existence of
isomorphism $\Phi:P_{k}\to P^{\prime}_{s}$ in the category
$\mathfrak{C}\mathfrak{a}\mathfrak{r}$ with the projection $\varphi$ such that
the following diagram

$\textstyle{{P_{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{k}}$$\scriptstyle{\Phi}$$\textstyle{M\supset{U_{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{k}}$$\scriptstyle{f|_{U_{k}}}$$\textstyle{V_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{P^{\prime}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}_{s}}$$\textstyle{M^{\prime}\supset{U^{\prime}_{s}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}_{s}}$$\textstyle{V^{\prime}_{s}}$

is commutative. We emphasize that the indicated above isomorphism
$\Phi:P_{k}\rightarrow P^{\prime}_{s}$ is unique if the transverse Cartan
geometries are effective. The introduced concept is well defined, i. e., it
does not depend of the choice of neighborhoods $U_{k}$ and $U^{\prime}_{k}$
from the cocycles $\zeta$ and $\zeta^{\prime}.$

######  Definition 3.

By a morphism of two Cartan foliations $(M,F)$ and $(M^{\prime},F^{\prime})$
we mean a local diffeomorphism $f:M\to M^{\prime}$ which transforms leaves to
leaves and preserves transverse Cartan structure. The category
$\mathfrak{C}\mathfrak{F}$ objects of which are Cartan foliations, morphisms
are their morphisms, is called the category of Cartan foliations.

##  4 Ehresmann connections for foliations

R. A. Blumenthal and J. J. Hebda [3] introduced the notion of Ehresmann
connection for foliation $(M,F)$ as a natural generalization of Ehresmann
connection for submersions.

Let $(M,F)$ be a foliation of codimension $q$ and $\mathfrak{M}$ be a smooth
$q$-dimensional distribution on $M$ that is transverse to the foliation $F,$
i. e. $T_{x}M=\mathfrak{M}_{x}\oplus T_{x}F\,\,\,\forall x\in M$. The
piecewise smooth integral curves of the distribution $\mathfrak{M}$ are said
to be horizontal, and the piecewise smooth curves in the leaves are said to be
vertical. A piecewise smooth mapping $H$ of the square $I_{1}\times I_{2}$ to
$M$ is called a vertical-horizontal homotopy if the curve $H|_{\\{s\\}\times
I_{2}}$ is vertical for any fixed $s\in I_{1}$ and the curve
$H|_{I_{1}\times\\{t\\}}$ is horizontal for any fixed $t\in I_{2}.$ In this
case, the pair of paths $(H|_{I_{1}\times\\{0\\}},H|_{\\{0\\}\times I_{2}})$
is called the base of $H.$ It is well known that there exists at most one
vertical-horizontal homotopy with a given base.

A distribution $\mathfrak{M}$ is called an Ehresmann connection for a
foliation $(M,F)$ (in the sense of R. A. Blumenthal and J. J. Hebda [3]) if,
for any pair of paths $(\sigma,h)$ in $M$ with a common initial point
$\sigma(0)=h(0),$ where $\sigma$ is a horizontal curve and $h$ is a vertical
curve, there exists a vertical-horizontal homotopy $H$ with the base
$(\sigma,h).$

#### A simple foliation with an Ehresmann connection

Let $f:M\to N$ be a submersion with connected fibers. Recall that the
foliation $F=\\{p^{-1}(z),\,z\in N\\}$ formed by the fibers of the submersion
is called a simple foliation. Let $(M,F)$ be an arbitrary smooth foliation
with the Ehresmann connection. It easy to show that, the existens of a
covering $\widehat{k}:\widehat{M}\to M$ such that the lifted foliation is
simple implies that the foliation $(M,F)$ is covered by fibration.

##  5 Classes of foliations covered by fibrations

####  $(G,X)$-foliations with an Ehresmann connection

Let $X$ be a smooth connected manifold and $G$ be the Lie group of
diffeomorphisms of $X$. Recall that the action of a group $G$ on a manifold
$X$ is called quasi-analytically if for any open subset $U\subset X$ and an
element $g\in G$ the equality $g|_{U}=id_{U}$ implies $g=e$, where $e=id_{X}$.

Assume that a Lie group $G$ of diffeomorphisms of a manifold $X$ acts on $N$
quasi-analytically. A foliation $(M,F)$ defined by an $X$-cocycle
$\\{U_{i},f_{i},\\{\gamma_{ij}\\}\\}_{i,j\in J}$ is called a $(G,X)$-foliation
if for any $U_{i}\cap U_{j}\neq\emptyset$, $i,j\in J$, there is an element
$g\in G$ such that $\gamma_{ij}=g|_{f_{j}(U_{i}\cap U_{j})}$. If, moreover,
$(X,\xi)$ is a Cartan manifold and the group $G$ is a subgroup of the
automorphism Lie group $Aut(X,\xi)$, then $(M,F)$ is a Cartan
$(G,X)$-foliation. It follows from [7, Section VI] that every Cartan
$(G,X)$-foliations with Ehresmann connections is a foliation covered by
fibration.

#### Cartan foliation with a vanishing transverse curvature

Let $(M,F)$ be a Cartan foliation of type $(G,H)$ with an Ehresmann
connection. As is known [1, Section VI], if the transverse curvature of
$(M,F)$ vanishes, then foliation $(M,F)$ is covered by fibration.
Consequently, all the obtained results are valid for Cartan foliations with
zero transverse curvature that admiting an Ehresmann connection.

#### Conformal foliations of codimension $q,q\geq 3$

According to [20, Thm. 5], any non-Riemannian conformal foliation of
codimension $q\geq 3$ with an Ehresmann connection is covered by fibration.

#### Foliations with an integrable Ehresmann connection

Recall that an Ehresmann connection $\mathfrak{M}$ for a foliation $(M,F)$ is
called integrable if the distribution $\mathfrak{M}$ is integrable i.e. if
there exists the foliation such that $TF^{t}=\mathfrak{M}$. Accoding to
Kashiwabara's theorem [10], foliations with an integrable Ehresmann connection
are covered by fibrations.

#### Suspended foliations

The construction of a suspension foliation was proposed by A. Haefliger and
described in detail in [18]. Note that suspension foliations form a class of
foliations with integrable Ehresmann connection and are covered by fibrations.

#### Cartan foliation of codimension $q=1$

Any smooth one-dimensional distribution is integrable, so a Cartan foliation
$(M,F)$ of codimension $q=1$ with an Ehresmann connection is covered by
fibration.

##  6 Proof of Theorems 1 and 2

###  6.1  Regular covering maps

######  Definition 4.

Let $f:M\rightarrow B$ be a submersion. It is said that $\widehat{h}\in
Diff(M)$ lying over $h\in Diff(B)$ relatively $f$ if $h\circ
f=f\circ\widehat{h}$. In this case $\widehat{h}$ is called a lift of $h$ with
respect to $f:M\rightarrow B$.

Let $\widetilde{\kappa}:\widetilde{{K}}\to K$ be the universal covering map,
where $K$ and $\widetilde{{K}}$ are smooth manifolds. By analogy with Theorem
28.10 in [4], it is easy to show that for any $h\leavevmode\nobreak\
\in\leavevmode\nobreak\ Diff(K)$ there exists
$\widetilde{h}\leavevmode\nobreak\ \in\leavevmode\nobreak\
Diff(\widetilde{K})$ lying over $h$. For an arbitrary covering map the same
statement is incorrect, in general. It is not difficult to prove the following
criterion for the existence of lifts of arbitrary diffeomorphisms with respect
to regular covers.

######  Proposition 1.

Let ${\kappa}:\widehat{K}\to K$ be a smooth regular covering map with the deck
transformation group $\Gamma$. A diffeomorphism $\widehat{h}\in
Diff(\widehat{K})$ lies over some diffeomorphism $h\in Diff(K)$ if and only if
it satisfies the equality $\widehat{h}\circ\Gamma=\Gamma\circ\widehat{h}$.

###  6.2 Proof of Theorem 1

Suppose that a Cartan foliation $(M,F)$ modeled on an effective Cartan
geometry $\xi=(P(N,H),\omega)$ is covered by a fibration
$\widetilde{r}:\widetilde{M}\to B$, where $\widetilde{\kappa}:\widetilde{M}\to
M$ is the universal covering map. The fibration
$\widetilde{r}:\widetilde{M}\to B$ has connected fibres and simply connected
space $\widetilde{M}$. Therefore, due to the application of the exact
homotopic sequence for this fibration we obtain that the base manifold $B$ is
also simply connected.

For an arbitrary point $b\in B$ take $y\in\widetilde{r}^{-1}(b)$ and
$x=\widetilde{\kappa}(y)$. Without loss generality, we assume that there is a
neighbourhood $U_{i}$, $x\in U_{i}$, from the $(N,\xi)$-cocycle
$\\{U_{i},f_{i},\\{\gamma_{ij}\\}\\}_{i,j\in J}$ which defines $(M,F)$ and a
neighbourhood $\widetilde{U}_{i}$, $y\in\widetilde{U}_{i}$, such that
$\widetilde{\kappa}|_{\widetilde{U}_{i}}:\widetilde{U}_{i}\to U_{i}$ is a
diffeomorphism.

Let $\widetilde{V}_{i}:=\widetilde{r}(\widetilde{U}_{i})$. Then there exists a
diffeomorphism $\phi:\widetilde{V}_{i}\to V_{i}$ satisfying the equality
$\phi\circ\widetilde{r}|_{\widetilde{U}_{i}}=f_{i}\circ\widetilde{\kappa}|_{\widetilde{U}_{i}}$.
The diffeomorphism $\phi$ induces the Cartan geometry
$\eta_{\widetilde{V}_{i}}$ on $\widetilde{V}_{i}$ such that $\phi$ becomes the
isomorphism $(\widetilde{V}_{i},\eta_{\widetilde{V}_{i}})$ and
$(V_{i},\xi_{V_{i}})$ in the category $\mathfrak{C}\mathfrak{a}\mathfrak{r}$
of Cartan geometries. The direct check shows that by this way we define the
Cartan geometry $\eta$ on $B$, and
$\eta|_{\widetilde{V}_{i}}=\eta_{\widetilde{V}_{i}}$, $i\in J$. Thus, the
statement $(1)$ is proved.

Let us fix points $x_{0}\in M$ and
$y_{0}\in\widetilde{\kappa}^{-1}(x_{0})\in\widetilde{M}$. Then the fundamental
group $\pi_{1}(M,x_{0})$ acts on the universal covering space $\widetilde{M}$
as the deck transformation group $\widetilde{G}\cong\pi_{1}(M,x_{0})$ of
$\widetilde{\kappa}$. Since $\widetilde{G}$ preserves the inducted foliation
$(\widetilde{M},\widetilde{F})$ formed by fibres of the fibration
$\widetilde{r}:\widetilde{M}\to B$, then every
$\widetilde{\psi}\in\widetilde{G}$ defines $\psi\in Diff(B)$ satisfying the
relation $\widetilde{r}\circ\widetilde{\psi}=\psi\circ\widetilde{r}$. The map
$\chi:\widetilde{G}\to\Psi:\widetilde{\psi}\to\psi$ is a group epimorphism and
the statement (2) is proved.

Observe that $\widetilde{G}$ is a subgroup of the automorphism group
$Aut(\widetilde{M},\widetilde{F})$ of $(\widetilde{M},\widetilde{F})$ in the
category $\mathfrak{C}\mathfrak{F}$. Therefore $\Psi$ is a subgroup of the
automorphism group $Aut(B,\eta)$ in the category of Cartan geometries
$\mathfrak{C}\mathfrak{a}\mathfrak{r}$. The kernel $ker(\chi)$ of $\chi$
determines the quotient manifold $\widehat{M}:=\widetilde{M}/ker(\chi)$ with
the quotient map $\widehat{\kappa}:\widetilde{M}\to\widehat{M}$ and the
quotient group $\widehat{G}:=\widetilde{G}/ker(\chi)$ such that
$M\cong\widehat{M}/\widehat{G}$. The quotient map $\kappa:\widehat{M}\to M$ is
the required regular covering map, with $\widehat{G}$ acts on $\widehat{M}$ as
a deck transformation group. The map $\widehat{G}\to\Psi:\widetilde{\psi}\cdot
ker(\chi)\mapsto\chi(\widetilde{\psi})$, $\widetilde{\psi}\in\widetilde{G}$,
is a group isomorphism. Thus the statment (3) is proved.

###  6.3 The associated foliated bundle

Let $(M,F)$ be a Cartan foliation modeled on Cartan geometry
$\xi=(P({\mathcal{N}},H),\omega)$ of type $(G,H)$. Then there exists a
principal $H$-bundle with the projection $\pi:\mathcal{R}\to M$, the
$H$-invariant foliation $(\mathcal{R},\mathcal{F})$ and the
$\mathfrak{g}$-valued $H$-equivariant $1$-form $\beta$ on ${\mathcal{R}}$
which satisfy the following conditions:

  1. (i)

$\beta(A^{*})=A$ for any $A\in\mathfrak{h}$;

  2. (ii)

the mapping $\beta_{u}:T_{u}{\mathcal{R}}\to\mathfrak{g}$ $\forall
u\in{\mathcal{R}}$ is surjective, and $ker(\beta_{u})=T_{u}{\mathcal{F}}$;

  3. (iii)

the foliation $({\mathcal{R}},\mathcal{F})$ is transversely parallelizable;

  4. (iv)

the Lie derivative $L_{X}{\beta}$ is equal to zero for every vector field $X$
tangent to the foliation $({\mathcal{R}},\mathcal{F}).$

######  Definition 5.

The principal $H$-bundle ${\mathcal{R}}(M,H)$ is called the foliated bundle
over the Cartan foliation $(M,F)$. The foliation $({\mathcal{R}},\mathcal{F})$
is called the lifted foliation for the Cartan foliation $(M,F).$

If the lifted foliation $(\mathcal{R},\mathcal{F})$ is formed by fibres of the
locally trivial fibration $\pi_{b}:\leavevmode\nobreak\
{\mathcal{R}}\leavevmode\nobreak\ \to\leavevmode\nobreak\ W$, then
$W=\mathcal{R}/\mathcal{F}$ is a smooth manifold , and a $\mathfrak{g}$-valued
$1$-form $\beta$ such that $\pi_{b}^{*}\widetilde{\beta}:=\beta$ and locally
free action of the Lie group $H$ on $W$ are induced. In this case,
$(W,\widetilde{\beta})$ is a parallelizable manifold and
$A(W,\widetilde{\beta})$ is the Lie group of its automorphisms that acts
freely on $W$. Further, as above, by ${A}^{H}(W,\widetilde{\beta})$ we denote
the closed Lie subgroup of $A(W,\widetilde{\beta})$ formed by transformations
commuting with the induced action of the Lie group $H$ on $W$. ∎

###  6.4 Proof of Theorem 2

Suppose that a Cartan foliation $(M,F)$ is covered by fibration. By definition
1, the induced foliation $(\widetilde{M},\widetilde{F})$ on the space of the
universal covering $\widetilde{\kappa}:\widetilde{M}\to M$ is defined by a
locally trivial fibration $\widetilde{r}:\widetilde{M}\to B$. Due to Theorem
1, the regular covering map $\kappa:\widehat{M}\rightarrow M$ and locally
trivial fibration $r:\widehat{M}\rightarrow B$ are defined, where $B$ is a
simply connected manifold with the inducted Cartan geometry $\eta$. Let $\Psi$
be the global holonomy group of the foliation $(M,F)$, then $\Psi$ is
isomorphic to the deck transformations group $G$ of the covering
$\kappa:\widehat{M}\rightarrow M$. Since the manifold $\widetilde{M}$ is
simply connected, there exists the universal covering map
$\widehat{\kappa}:\widetilde{M}\to\widehat{M}$ satisfying the equality
$\kappa\circ\widehat{\kappa}=\widetilde{\kappa}$. Let $\widetilde{G}$, $G$ and
$\widehat{G}$ be the deck transformation groups of the covering maps
$\widetilde{\kappa}$, $\kappa$ and $\widehat{\kappa}$ respectively, with
$\Psi\cong G\cong\widetilde{G}/\widehat{G}$.

Let us consider the following preimages of the $H$-bundle ${\mathcal{R}}$
respectively $\widetilde{\kappa}$ and $\kappa$

$\widetilde{\mathcal{R}}:=\\{(\widetilde{x},u)\in\widetilde{M}\times{\mathcal{R}}\,|\,\widetilde{\kappa}(\widetilde{x})=\pi(u)\\}=\widetilde{\kappa}^{*}{\mathcal{R}}\,\,\,and$
$\widehat{\mathcal{R}}:=\\{(\widehat{x},u)\in\widehat{M}\times{\mathcal{R}}\,|\,\kappa(\widehat{x})=\pi(u)\\}=\kappa^{*}{\mathcal{R}}.$

Remark that the maps

$\widetilde{\theta}:\widetilde{{\mathcal{R}}}\to{\mathcal{R}}:(\widetilde{x},u)\mapsto(\widetilde{\kappa}(\widetilde{x}),u)\,\,\,\,\,\forall(\widetilde{x},u)\in\widetilde{{\mathcal{R}}},$
$\theta:\widehat{{\mathcal{R}}}\to{\mathcal{R}}:(\widehat{x},u)\mapsto(\kappa(\widehat{x}),u)\,\,\,\,\,\forall(\widehat{x},u)\in\widehat{{\mathcal{R}}},\\\
$
$\widehat{\theta}:\widetilde{{\mathcal{R}}}\to\widehat{{\mathcal{R}}}:(\widetilde{x},u)\mapsto(\widehat{\kappa}(\widetilde{x}),u)\,\,\,\,\,\forall(\widetilde{x},u)\in\widetilde{{\mathcal{R}}},$

are regular covering maps with the deck transformation groups
$\widetilde{\Gamma}$, $\Gamma$ and $\widehat{\Gamma}$, respectively, which are
isomorphic to the relevant groups $\widetilde{G}$, $G$ and $\widehat{G}$, i.e.
$\widetilde{\Gamma}\cong\widetilde{G}$, $\Gamma\cong G$ and
$\widehat{\Gamma}\cong\widehat{G}$.

Let $(\widetilde{\mathcal{R}},\widetilde{\mathcal{F}})$ and
$(\widehat{\mathcal{R}},\widehat{\mathcal{F}})$ be the corresponding lifted
foliations for $(M,F)$. Since $(\widetilde{M},\widetilde{F})$ and
$(\widehat{M},\widehat{F})$ are simple foliations, then
$(\widetilde{\mathcal{R}},\widetilde{\mathcal{F}})$ and
$(\widehat{\mathcal{R}},\widehat{\mathcal{F}})$ are also simple foliations,
which are formed by locally trivial fibrations
$\widetilde{\pi}_{b}\leavevmode\nobreak\ :\leavevmode\nobreak\
\widetilde{\mathcal{R}}\leavevmode\nobreak\ \to\leavevmode\nobreak\
\widetilde{W}$ and $\widehat{\pi}_{b}\leavevmode\nobreak\
:\leavevmode\nobreak\ \widehat{\mathcal{R}}\leavevmode\nobreak\
\to\leavevmode\nobreak\ \widehat{W}$. Hence
$\mathfrak{g}_{0}\leavevmode\nobreak\
(\widetilde{\mathcal{R}},\widetilde{\mathcal{F}})\leavevmode\nobreak\
=\leavevmode\nobreak\ 0$,
$\mathfrak{g}_{0}(\widehat{\mathcal{R}},\widehat{\mathcal{F}})=0$, and
$\widetilde{W}=\widetilde{\mathcal{R}}/\widetilde{\mathcal{F}}$,
$\widehat{W}=\widehat{\mathcal{R}}/\widehat{\mathcal{F}}$ are manifolds.

Since the fibrations $\widetilde{r}:\widetilde{M}\to B$ and $r:\widehat{M}\to
B$ have the same base $B$, each leaf of the foliation
$(\widetilde{M},\widetilde{F})$ is invariant respectively the group
$\widehat{G}$, i.e. $\widehat{G}\subset A_{L}(\widetilde{M},\widetilde{F}).$
Therefore $\widehat{\Gamma}\subset
A_{L}(\widetilde{\mathcal{R}},\widetilde{\mathcal{F}})$ and the leaf spaces
$\widetilde{\mathcal{R}}/\widetilde{\mathcal{F}}=\widetilde{W}$ and
$\widehat{\mathcal{R}}/\widehat{\mathcal{F}}=\widehat{W}$ are coincided, i.e.
$\widetilde{W}=\widehat{W}.$ Consequently, basic automorphism groups
$A_{B}(\widetilde{\mathcal{R}},\widetilde{\mathcal{F}})$ and
$A_{B}(\widehat{\mathcal{R}},\widehat{\mathcal{F}})$ may be identified.
Further we put
$A_{B}(\widetilde{\mathcal{R}},\widetilde{\mathcal{F}})=A_{B}(\widehat{\mathcal{R}},\widehat{\mathcal{F}}).$

According to the conditions of Theorem 2, $\Psi$ is a discrete subgroup of the
Lie group $Aut(B,\eta)$. Let $N(\Psi)$ be the normalizer of $\Psi$ in the Lie
group $Aut(B,\eta)\cong A^{H}(W,\widetilde{\beta})$. Hence, $N(\Psi)$ is a
closed Lie subgroup of the Lie group $Aut(B,\eta)$ and the quotient group
$N(\Psi)/\Psi$ is also a Lie group.

Let $\pi:{\mathcal{R}}\to M$ be the projection of the foliated bundle over
$(M,F)$. Due to the discreteness of the global holonomy group $\Psi$, the
lifted foliation $({\mathcal{R}},{\mathcal{F}})$ is formed by fibres of some
locally trivial fibration $\pi_{b}:{\mathcal{R}}\to W$, which is called the
basic fibration.

Observe that there exists a map $\tau:\widehat{W}\rightarrow W$ satisfying the
equality $\tau\circ\widehat{\pi}_{b}=\theta\circ\pi_{b}$. It is easy to show
that $\tau:\widehat{W}\to W$ is a regular covering map with the deck
transformations group ${\Phi}$, ${\Phi}\subset
A^{H}(\widehat{W},\widehat{\beta})$, which is naturally isomorphic to each of
the groups $\Psi$, $G$ and $\Gamma$.

Denote by $\eta=(P(B,H),\omega)$ the Cartan geometry with the projection
$p:P\to B$ onto $B$ determined in the proof of Theorem 1. Remark that
$\widehat{W}={P}$ is the space of the $H$-bundle of the Cartan geometry
$\eta$.

Since $\kappa:\widehat{M}\to M$,
$\theta:\widehat{\mathcal{R}}\to{\mathcal{R}}$ and $\pi:{\mathcal{R}}\to M$
are morphisms of the following foliations
$\kappa:(\widehat{M},\widehat{F})\to(M,F)$,
$\theta:(\widehat{\mathcal{R}},\widehat{\mathcal{F}})\to({\mathcal{R}},{\mathcal{F}})$
$\pi:({\mathcal{R}},{\mathcal{F}})\to(M,F)$ in the category of the foliations
$\mathfrak{F}\mathfrak{o}\mathfrak{l}$, then maps $\widehat{\tau}:B\to M/F$
and $s:W\to W/H\cong M/F$ are defined, and the following diagram

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is commutative.

Due to Proposition [15, Thm. 1] there are the Lie group isomorphisms

$\varepsilon:A_{B}(M,F)\to im(\varepsilon)\subset
A^{H}(W,\widetilde{\beta})\,\,and\,\,$
$\widehat{\varepsilon}:A_{B}(\widetilde{M},\widetilde{F})=A_{B}(\widehat{M},\widehat{F})\to
im(\widehat{\varepsilon})\subset A^{H}(\widehat{W},\widehat{\beta}).$

Let us define a map $\Theta:im(\varepsilon)\to N(\Phi)/\Phi$ by the following
a way. Take any $h\in im(\varepsilon)\subset A^{H}(W,\widetilde{\beta})$.
Denote the element $\varepsilon^{-1}(h)\in A_{B}(M,F)$ by $f\cdot
A_{L}(M,F)\in A_{B}(M,F)$, where $f\in A(M,F)$. Since
$\widetilde{\kappa}:\widetilde{M}\to M$ is the universal covering map there
exists $\widetilde{f}\in Diff(\widetilde{M})$ lying over $f$ relatively
$\widetilde{\kappa}$. It not difficult to see that $\widetilde{f}\in
A(\widetilde{M},\widetilde{F})$. Hence $\widetilde{f}\cdot
A_{L}(\widetilde{M},\widetilde{F})\in A_{B}(\widetilde{M},\widetilde{F})$.
Consider $\widehat{h}:=\widehat{\varepsilon}(\widetilde{f}\cdot
A_{L}(\widetilde{M},\widetilde{F}))\in im(\widehat{\varepsilon})\subset
A^{H}(\widehat{W},\widehat{\beta})$. The direct check shows that $\widehat{h}$
lies over $h$ respectively $\tau$. Remind that $\Phi$ is the deck
transformation group of the covering map $\tau:\widehat{W}\to W$. Applying the
Proposition 1, we get that $\widehat{h}\in N(\Phi)$, hence the set of all
automorphisms in $im(\widehat{\varepsilon})$ lying over $h$ is equal to the
set of transformations from the class $\widehat{h}\cdot\Phi.$ Let us put
$\Theta(h):=\widehat{h}\cdot\Phi\in N(\Phi)/\Phi.$ It is easy to check that
the map $\Theta:im(\varepsilon)\to N(\Phi)/\Phi$ is a group monomorphism.

The effectiveness of the Cartan geometry $\eta=(P(B,H),\omega)$ on $B,$ where
$P=\widehat{W},$ implies the existence of the Lie group isomorphism
$\sigma:A^{H}(\widehat{W},\widehat{\beta})\rightarrow Aut(B,\eta)$. Observe
that $\sigma({\Phi})=\Psi$ and $\sigma(N(\widetilde{\Phi}))=N(\Psi)$, hence
there exists the inducted Lie group isomorphism
$\widetilde{\sigma}:N({\Phi})/{\Phi}\to N(\Psi)/\Psi$. Thus, the composition
of the Lie group monomorphisms

$\displaystyle\delta:=\widetilde{\sigma}\circ{\Theta}\circ\varepsilon:A_{B}(M,F)\rightarrow
N(\Psi)/\Psi$

is the required Lie group monomorphism. Due to uniqueness of the Lie group
structure in $A_{B}(M,F)$, in conforming with [15, Thm. 1], the image
$im(\delta)$ is an open-closed subgroup of the Lie group $N({\Psi})/{\Psi}$. ∎

##  7 Basic automorphism groups of Cartan foliations
with an integrable Ehresmann connection

###  7.1 Proof of Theorem 3

1\. According to the conditions of the theorem being proved, $(M,F)$ has an
intagrable Ehresmann connection $\mathfrak{M}$. In this case, distribution
$\mathfrak{M}$ is integrable. In this case, there is $q$-dimensional foliation
$(M,F^{t})$ such that $TF^{t}=\mathfrak{M}.$

Let $\widetilde{\kappa}:\widetilde{M}\to M$ be the universal covering map.
According to the decomposition theorem belonging to S. Kashiwabara [10], the
universal covering manifold $\widetilde{M}$ is equal to the product of
manifolds $\widetilde{M}=\widetilde{Q}\times B$, where $\widetilde{Q}$ is the
universal covering manifold for any leaf of the foliation $(M,F)$, and $B$ is
the universal covering manifold for any leaf of the foliation $(M,F^{t})$. The
induced foliations
$\widetilde{F}=\widetilde{\kappa}^{*}F=\\{\widetilde{Q}\times\\{y\\}\,|\,y\in
B\\}$, $\widetilde{F}^{t}=\widetilde{\kappa}^{*}F^{t}=\\{\\{z\\}\times
B\,|\,z\in\widetilde{Q}\\}$ are defined. Therefore, (M, F) is covered by
fibration $\widetilde{s}:\widetilde{Q}\times B\to B$. In this case, by the
same way as in the proof of Theorem 1, the Cartan geometry $\eta$ is induced
on $B$ such that $(M,F)$ becomes an $(Aut(B,\eta),B)$-foliation.

2\. Let $\Psi$ be the global holonomy group of this foliation. Suppose now
that the normalizer $N(\Psi)$ is equal to the centralizer $Z(\Psi)$ of the
group $\Psi$ in the group $Aut(B,\eta)$.

Let us fix points $x_{0}\in M$ and
$(z_{0},y_{0})\in\widetilde{\kappa}^{-1}(x_{0})\in\widetilde{M}$. Then the
fundamental group $\pi_{1}(M,x_{0})$ acts on the universal covering space
$\widetilde{M}=\widetilde{Q}\times B$ as the deck transformation group
$\widetilde{G}\cong\pi_{1}(M,x_{0})$ of $\widetilde{\kappa}$. Since
$\widetilde{G}$ preserves both the inducted foliations
$(\widetilde{M},\widetilde{F})$ and $(\widetilde{M},\widetilde{F}^{t})$, then
every $\widetilde{g}\in\widetilde{G}$ may be written in the form
$\widetilde{g}=(\psi^{t},\psi)$, where $\psi^{t}$ generates a subgroup
$\Psi^{t}$ in $Diff(\widetilde{Q})$, $\psi\in\Psi$, and
$\widetilde{g}(z,y)=(\psi^{t}(z),\psi(y))$, $(z,y)\in\widetilde{Q}\times B.$
The maps $\widetilde{\chi}:\widetilde{G}\to\Psi:\widetilde{g}\to\psi$ and
$\widetilde{\chi}^{t}:\widetilde{G}\to\Psi^{t}:\widetilde{g}\to\psi^{t}$ are
the group epimorphisms.

Let $h$ be any element from $N(\Psi)/\Psi$. Since $N(\Psi)=Z(\Psi)$, we have
the following chain of equalities

$\widetilde{g}\circ(id_{\widetilde{Q}},h)=({\psi}^{t},\psi)\circ(id_{\widetilde{Q}},h)=({\psi}^{t}\circ
id_{\widetilde{Q}},\psi\circ h)=\\\ $
$(id_{\widetilde{Q}}\circ{\psi}^{t},h\circ\psi)=(id_{\widetilde{Q}},h)\circ({\psi}^{t},\psi)=(id_{\widetilde{Q}},h)\circ\widetilde{g}$

for any $\widetilde{g}=({\psi}^{t},\psi)\in\widetilde{G}$, i.e.
$\widetilde{G}\cdot(id_{\widetilde{Q}},h)=(id_{\widetilde{Q}},h)\cdot\widetilde{G}$.
Therefore, by the Proposition 1, for the deck transformation group
$\widetilde{G}$, there exists $\widetilde{h}\in Diff(M)$ such that
$(id_{\widetilde{Q}},h)$ lies over $\widetilde{h}$ respectively to
$\widetilde{\kappa}:\widetilde{M}\to M$.

Taking into account that $(id_{\widehat{Q}},h)\in
A(\widetilde{M},\widetilde{F})$, it is not difficult to check that
$\widetilde{h}\leavevmode\nobreak\ \in\leavevmode\nobreak\ A(M,F)$. Hence,
$\varepsilon(\widetilde{h}\cdot A_{L}(M,F))=h.$ This means that
$\varepsilon:A_{B}(M,F)\to N(\Psi)/\Psi$ is surjective. Thus, $\varepsilon$ is
a the group isomorphism.

Since $N(\Psi)$ is a closed Lie subgroup of the automorphism Lie group
$Aut(B,\eta)$, and $\Psi$ is a discrete subgroup of $N(\Psi)$, the quotient
group $N(\Psi)/\Psi$ is a Lie group. Therefore, the group isomorphism
$\varepsilon$ induces a Lie group structure in $A_{B}(M,F)$ such that
$\varepsilon:A_{B}(M,F)\to N(\Psi)/\Psi$ becomes a Lie group isomorphism.
According to [15, Thm. 1], the Lie group structure in $A_{B}(M,F)$ is unique.
∎

###  7.2 Example of finding a basic automorphism group

Let $\mathbb{S}^{q}$ be a $q$-dimensional standard sphere, where $q\geq 3$. We
identify $\mathbb{S}^{q}$ with $\mathbb{R}^{q}\cup\\{\infty\\}$, where
$\\{\infty\\}$ is the point at infinity. Define the transformation
$\psi:\mathbb{S}^{q}\cong\mathbb{R}^{q}\cup\\{\infty\\}\to\mathbb{S}^{q}$ by
equality $\psi(z)=\lambda z$ for any
$z\in\mathbb{S}^{q}\cong\mathbb{R}^{q}\cup\\{\infty\\}$, where $\lambda$ is a
real number, and $0<\lambda<1$. We denote by $Conf(\mathbb{S}^{q})$ the Lie
group of all conformal transformations of the sphere $\mathbb{S}^{q}$.

Let $\Psi=<\psi>$ be the subgroup of the group $Conf(\mathbb{S}^{q})$
generated by $\psi$, and $\Psi$ is isomorphic to the group of integers
$\mathbb{Z}$. Define the action of the group $\mathbb{Z}$ on the product of
manifolds $\mathbb{R}^{1}\times\mathbb{S}^{q}$ by the equality
$n(t,z)=(t-n,\psi^{n}(z))$ for any $n\in\mathbb{Z}$,
$(t,z)\leavevmode\nobreak\ \in\leavevmode\nobreak\
\mathbb{R}^{1}\times\mathbb{Z}$. This action is free and properly
discontinuous. Therefore, the manifold of orbits $M\leavevmode\nobreak\
=\leavevmode\nobreak\ \mathbb{R}^{1}\times_{\mathbb{Z}}\mathbb{S}^{q}$ is
defined. Denote by $f:\mathbb{R}^{1}\times\mathbb{S}^{q}\to M$ the quotient
map. Fix a point $(t_{0},z_{0})\in\mathbb{R}^{1}\times\mathbb{S}^{q}$, put
$x_{0}=f(t_{0},z_{0})\in M$. Then the fundamental group $\pi_{1}(M,x_{0})$
acts on the universal covering space $\mathbb{R}^{1}\times\mathbb{S}^{q}$ as
the deck transformation group $\widetilde{G}\cong\pi_{1}(M,x_{0})$ of $f$.
Since the action $\widetilde{G}$ preserves the structure of the product
$\mathbb{R}^{1}\times\mathbb{S}^{q}$, then two foliations $(M,F)$ and
$(M,F^{t})$, covered by trivial fibrations
$pr_{2}:\mathbb{R}^{1}\times\mathbb{S}^{q}\to\mathbb{S}^{q}$ and
$pr_{1}:\mathbb{R}^{1}\times\mathbb{S}^{q}\to\mathbb{R}^{1}$ respectively, are
defined. Let us denote by
$\chi:\mathbb{R}^{1}\to\mathbb{S}^{1}=\mathbb{R}^{1}/\mathbb{Z}$ and
$\nu:\mathbb{S}^{q}\to\mathbb{S}^{q}/\Psi$ the quotient maps onto the orbit
spaces. Let $r:M\to M/F$ be the quotient map onto the leaf space. Observations
show that the topological spaces $M/F$ and $\mathbb{S}^{q}/\Psi$ are
homeomorphic and satisfy the commutative diagram

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where $p:M\to\mathbb{S}^{1}$ is the projection of the locally trivial
fibration transforming the orbit $\mathbb{Z}.(t,z)$ of a point
$(t,z)\in\mathbb{R}^{1}\times\mathbb{S}^{q}$, considered as a point from $M$,
into the orbit $\mathbb{Z}.{t}$ of a point $t\in\mathbb{R}^{1}$, considered as
a point of the circle $\mathbb{S}^{1}$. Since the manifold $M$ is the space of
a locally trivial fibration $p:M\to\mathbb{S}^{1}$ over the circle
$\mathbb{S}^{1}$ with a compact standard fiber $\mathbb{S}^{q}$, then $M$ is
compact.

The distribution $\mathfrak{M}$ tangent to $(M,F^{t})$, is an integrable
Ehresmann connection for the foliation $(M,F)$. The foliation $(M,F)$ has two
compact leaves $L_{1}$ and $L_{2}$ which are diffeomorphic to the circle
$\mathbb{S}^{1}$. Every other leaf $L$ of $(M,F)$ is diffeomorphic to
$\mathbb{R}^{1}$, and its closure $\overline{L}$ is equal to the union $L\cup
L_{1}\cup L_{2}$. We emphasize that $(M,F)$ is a proper conformal foliation,
which can be regarded as a Cartan foliation of type $(G,H)$, where
$G=Conf(\mathbb{S}^{q})$ and $H$ is a stationary subgroup of the group
$Conf(\mathbb{S}^{q})$ at some point in $\mathbb{S}^{q}$.

As is known, $H\cong CO(q)\ltimes\mathbb{R}^{q}$ is a semidirect product of a
conformal group $CO(q)\cong\mathbb{R}^{+}\times O(q)$ and a normal abelian
subgroup $\mathbb{R}^{q}$. Note that $\Psi$ is the global holonomy group of
the foliation $(M,F)$, and $\Psi$ is a discrete subgroup of the Lie group
$Conf(\mathbb{S}^{q})$.

The direct check shows that the normalizer of the group $\Psi$ is equal to
$N(\Psi)=\mathbb{R}^{+}\times O(q)$, and $N(\Psi)$ coincides with the
centralizer $Z(\Psi)$. Applying Theorem 3, we obtain that the group of basic
automorphisms $A_{B}(M,F)$ is a Lie group isomorphic to the quotient group
$N(\Psi)/\Psi\cong U(1)\times O(q)$, where $U(1)\cong\mathbb{S}^{1}$. Thus,
the Lie group of basic automorphisms $A_{B}(M,F)$ is isomorphic to the product
of Lie groups $U(1)\times O(q)$.

#### Acknowledgements

The author would like to thank N.I. Zhukova for helpful discussions and
comments. The work was supported by Laboratory of Dynamical Systems and
Applications NRU HSE, of the Ministry of science and higher education of the
RF grant ag. no. 075-15-2019-1931

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