# Fundamental diagram of vibration-driven vehicles

German A. Patterson
Instituto Tecnológico de Buenos Aires (ITBA), CONICET
C.A. de Buenos Aires, Argentina
<EMAIL_ADDRESS>
&Daniel R. Parisi
Instituto Tecnológico de Buenos Aires (ITBA), CONICET
C.A. de Buenos Aires, Argentina
<EMAIL_ADDRESS>

###### Abstract

In this study, we conducted experimental investigations into the fundamental
diagram of vibration-driven vehicles (VDV) in a one-dimensional array. As
these mechanical agents interact solely through collisions, their mean speed
remains nearly constant at low and medium densities. However, there is a
reduction of between 25% and 40% when the lineal density approaches the
inverse of the contact distance. Remarkably, in this one-dimensional system,
the outcome is significantly influenced by the order in which agents, sorted
by their free speeds, are gradually introduced into the experiment. While a
significant speed difference is observed at low and medium densities based on
this ordering, both curves eventually converge to the same speed at maximum
density. Moreover, the attained speed in saturated systems is slower than the
speed of the slowest agent.

##  1 Introduction

The fundamental diagram for a set of self-propelled particles relates either
the flow rate or the mean velocity of these particles to their density. It has
been extensively used for characterizing vehicle traffic systems and as a tool
for their design [1, 2, 3, 4, 5, 6, 7, 8]. The relationship between speed and
density is monotonically decreasing: as density increases, speed decreases.

This same observable is frequently employed for characterizing and designing
pedestrian systems, which also exhibit the decreasing trend of mean speed with
density [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28].

While fundamental diagrams for vehicles and pedestrians are qualitatively
similar, there is a significant distinction. For vehicles, contact is strictly
prohibited. However, for pedestrians, contact between individuals naturally
occurs at high densities. In particularly dense situations, pushing or shoving
may be observed, especially in competitive behaviors [12, 16, 29].

In vehicular dynamics, the decrease in speed can be attributed to the driver’s
voluntary speed reduction, either by applying the brakes or easing off the
accelerator, to maintain a safe distance from the vehicle ahead. This
mechanism holds across the entirety of the vehicular fundamental diagram. For
pedestrians, a similar mechanism operates at low and medium densities:
individuals reduce their desired speed as available space becomes limited.
However, when physical contacts become prevalent due to increasing density,
the nature of speed reduction shifts, rooted in physical constraints.

Thus, we posit that there are at least two mechanisms shaping the speed-
density relation in pedestrian dynamics. One operates at low to medium
densities, where remote sensing (like vision) influences the desired speed,
which, in turn, affects the self-propulsion intensity, akin to vehicular
traffic. The other, at high densities, involves short-range limitations
imposed by physical constraints that curtail movement. The distinction between
these two mechanisms has not been previously emphasized in the literature.

To address this oversight, we aim to experimentally study a straightforward
system of physical agents that interact exclusively through contact, without
any capacity for remote sensing. This approach will help us understand and
isolate the section of the fundamental diagram related to contacts and
physical space limitations, at least within this minimal system. We used
commercially available vibration-driven vehicles (VDV) arranged in a one-
dimensional setup (a circular track).

Furthermore, another objective of this research is to establish a precedent
for utilizing the fundamental diagram as an insightful metric for a broad
spectrum of active matter systems. This applicability goes beyond the confines
of transport engineering, covering a spectrum of entities from animals and
cells to robots and VDVs. In fact, the particular VDVs we utilized have
already been employed to explore diverse facets of emergent behavior [30, 31,
32, 33, 34, 35].

##  2 Experimental setup

###  2.1 VDVs and circular track

The VDVs used in our experiments are the Hexbug Nano [36] (Fig. 1A). Their
body, with dimensions of $44\times 15\times 18\ \mathrm{mm}$, houses a
vibrating motor inside. These vehicles stand on an asymmetrical bristle that
rectifies the motor’s vibration, resulting in forward movement. The motor is
powered by a $1.5\ \mathrm{V}$, which, under normal usage conditions, provides
over 90 minutes of operation. In the absence of a container or obstacle, the
VDVs exhibit diffusive behavior in terms of their orientation of movement. In
this study, we restricted their movement to a one-dimensional track by
confining them within a circular channel with a width of $18\ \mathrm{mm}$. To
achieve this, we used two concentric circular rings: the first with an outer
radius $R_{1}=200\ \mathrm{mm}$, and the second with an inner radius
$R_{2}=218\ \mathrm{mm}$, as shown in Fig. 1B. The wooden ring walls were
covered with a PVC sheet to ensure a uniform contact surface with the VDVs.
The effective length of the channel, approximately $1313\ \mathrm{mm}$,
allowed us to introduce a maximum of 30 VDVs, as can be seen in Figs. 1B-D.

The experiments were recorded at a rate of 24 fps with a zenith camera. We
used a white label, with the front painted blue, placed on each agent for
tracking the VDVs (Fig. 1A). By employing a standard blob detection method, we
obtained the positions of the VDVs in each frame. Finally, velocities were
calculated using a fourth-order finite difference method.

Figure 1: Experimental setup: A) VDV photograph and dimensions. A blue-painted
label enables the tracking of trajectories. B) Snapshot of the system with
$N=10$ VDVs. Two wooden rings are used to construct the circular track. C)
Snapshot of the system with $N=20$ VDVs, and D) $N=30$.

###  2.2 Speed characterization

First, we carried out an individual speed characterization for each of the 37
available VDVs. For this purpose, we conducted 180-second experiments for each
VDV using fresh battery. We measured the average speed of each one and
discarded the fastest and slowest ones. In this way, we ended up with a set of
30 VDVs whose speeds were uniformly distributed in the range of $90-120\
\mathrm{mm/s}$.

##  3 Results

To investigate the velocity response as a function of density, experiments
were conducted over a duration of $180$ s for varying numbers $N$ of VDVs. The
trials spanned densities with $N\in\\{5,10,15,20,25,30\\}$. Specifically, the
initial run started with $N=5$ VDVs. After $180$ s, another $5$ VDVs were
introduced, and this increment continued until there were $N=30$ VDVs active
during the concluding $180$ s interval. Thus, the total duration for this
series of experiments, spanning these densities, was $T=6\times 180\
\mathrm{s}=1080\ \mathrm{s}$.

Three distinct sorting configurations were also examined:

1 - In the ’ascending’ configuration, the initial $N=5$ VDVs were those with
the lowest individual speeds, as detailed in Sec. 2.2. Subsequent
introductions added the next fastest set of $5$ VDVs until $N=30$ was
attained.

2 - Conversely, in the ’descending’ configuration, the process started with
the fastest $N=5$ VDVs. The next slower group of $5$ VDVs was then introduced,
and so on.

3 - A third configuration involved randomly selecting VDVs without considering
their individual velocities, to study the effects of increasing densities.

###  3.1 Global Velocities

As our first observable, we will calculate the average global velocities of
the VDVs as a function of $N$. The global velocity is defined as
$v^{g}_{i}=\frac{L_{i}}{180~{}\text{s}}$, where $L_{i}$ is the distance
traveled by particle $i$ during the $180$ s of the experiment. Figure 2
displays these quantities across the three scenarios: ’ascending’,
’descending’, and ’random’.

Figure 2: Mean speed as a function of $N$, for three sorting configurations.

As expected, for small values of $N$, the ’ascending’ scenario (where slower
VDVs are introduced first) exhibits a lower global velocity compared to the
’descending’ scenario (where the fastest VDVs are introduced first). In both
cases, a decreasing trend in global velocity is observed as the number of VDVs
increases. Furthermore, at maximum density ($N=30$), all scenarios converge to
a similar average global velocity. This convergence suggests that, when
reaching saturation and all VDVs are present, the order of introducing
particles based on their speeds no longer seems relevant.

The ’random’ scenario, in which VDVs are introduced without any specific
order, appears to fall between the two extreme scenarios of ’ascending’ and
’descending’. This observation supports the idea that random configurations
are bounded by these two ordered configurations.

In the individual characterization, as discussed in Sec.2.2, we found that the
slowest vehicle had a speed of 90 mm/s. It is noteworthy that, with an
increasing number of vehicles on the track, the overall speed decreases to
values lower than that of the slowest VDV. This illustrates that the system’s
speed is not constrained by the slowest vehicle but rather exhibits a
collapsing behavior

For a deeper understanding, the subsequent sub-sections will delve into the
microscopic states of the particles.

###  3.2 Distribution of Densities

We compute the individual density of each particle $i$ using the relation
$\rho_{i}=1/d_{ij}$, where $d_{ij}$ represents the distance along the circular
path to the nearest neighboring particle. In Fig. 3, the probability density
function (PDF) of densities, for all particles and all times, is displayed for
each experiment with varying numbers of VDVs, encompassing both the
’ascending’ and ’descending’ scenarios.

Figure 3: Distribution of individual densities. A) velocity ’ascending’ order.
B) velocity ’descending’.

For all cases, it can be observed that there is a peak in the distribution
around $\rho=0.023$ 1/mm. This value aligns with the minimum theoretical
distance between two contacting VDVs, given that each has a maximum length of
$44$ mm. However, tails of the distributions can be seen extending toward
higher density values. This can be attributed to the triangular geometry of
the VDV’s front and back. Slight misalignments, possibly due to the curvature
of the track, might allow for distances shorter than the theoretical minimum.
Other factors contributing to these increased densities could include
overlapping of VDVs (one mounting over another) and fluctuations in the
automatic tracking system’s detection of the label.

Another shared characteristic between the ’ascending’ and ’descending’
scenarios is that the probability in the low-density zone of the distribution
diminishes as $N$ increases.

In the peak zone, the ’ascending’ scenario (Fig. 3A) shows a clear separation
between the three greater values of $N={20,25,30}$ and the smaller ones
$N={5,10,15}$. The former set tends to exhibit a heightened probability
aligned with the maximum theoretical density. In the ’descending’ scenario
(Fig. 3B), this distinction is apparent only for $N=30$.

Overall, as the values of $N$ increase, the distributions tend to become more
pronounced, peaking at the maximum theoretical density dictated by the VDV’s
geometry.

###  3.3 Distribution of Velocities

Complementarily, we examine in detail the distribution of microscopic
velocities for each VDV and every time frame in Fig. 4.

Figure 4: Distribution of individual velocities. A) VDVs introduced in an
’ascending’ sequence according to their free velocity. B) The same, but in a
’descending’ sequence. C) VDVs added without any specific order. D) PDF
combining all experiments with varying $N$, grouped solely by the type of
order of inclusion.

Similar to the analysis of densities, the ’ascending’ scenario depicted in
Fig. 4A shows a distinction between the velocity distributions for larger
($N={20,25,30}$) and smaller ($N={5,10,15}$) number of VDVs. This observation
suggests a marked shift toward lower velocities when there are more than 15
VDVs in the system. In the ’descending’ order scenario (Fig. 4B), only the
$N=30$ case distinctly separates from the others, indicating that when VDVs
are introduced from fastest to slowest, the system’s velocity substantially
decreases only at a very high density ($N=30$). Moreover, when comparing the
’ascending’ and ’descending’ configurations, one can observe that the peak
positions in the distributions shift toward higher speed values, consistent
with the average global velocity presented in Fig. 2. The random
configuration, as illustrated in Fig. 4C, does not exhibit a distinct
separation in the PDF, positioning itself between the two extreme ordered
configurations. This becomes more evident in Fig. 4D, which considers the
distribution across all $N$ values. In the ’ascending’ scenario, there is a
higher probability for low velocities and a reduced likelihood for high
velocities. Conversely, in the ’descending’ configuration, there is a
heightened probability for low velocities and a diminished chance for higher
velocities. Through this analysis, it is evident that the random order
configuration gives a PDF that falls between the two extremes.

Another noteworthy aspect is the distribution width. It can be observed that
in experiments with higher values of $N$ (Figs. 4)A-C, speed distributions
become narrower. This phenomenon occurs because, as the number of vehicles in
the channel increases, each particle experiences limited movement, resulting
in a more uniform rotation speed.

###  3.4 VDVs Fundamental Diagram

Now, after characterizing microscopic densities and velocities separately, we
delve into the speed-density diagram, as presented in Fig. 5. The first three
panels, Figs. 5A, B, and C, display the entire collection of microscopic data,
encompassing all values of $N$ and all time instances (with a total of
$F_{n}=1080\times 24=25920$ analyzed frames). As is clear, there is a wide
dispersion of data points, lacking a clear trend. However, when a moving
average of 1000 data points is plotted, a subtle decrease in speed becomes
discernible at higher density values. Still, overall, these averages remain
relatively flat. At the highest densities, a minimum, which corresponds to the
theoretical maximum density determined by the VDV’s geometry, is observed in
all three cases. The upward curve to the right of these minima can be
attributed to, misalignment, overlapping and tracking errors, as explained in
Sec. 3.2 in reference to Fig. 3.

By limiting the analysis to data up to the maximum theoretical density of
$\rho=0.023$ 1/mm and using a 20000-point moving average, we present the
speed-density plot for the three configurations in Fig. 5D. Three main
observations can be drawn from this analysis.

First, as expected, the results are in accordance with those regarding the
global velocity presented in Fig.2. In this context, the ’descending’
configuration, which involves introducing the fastest VDV first, exhibits
higher velocities when compared to the ’ascending’ configuration at lower and
moderate densities. However, as the density increases, both configurations
gradually converge to the same velocity value at the maximum achievable
density. Furthermore, the structure observed during the velocity PDF analysis
for all values of $N$ are confirmed: the random order scenario distinctly lies
between the two extreme ordered configurations, ’ascending’ and ’descending’.

Secondly, the velocity profile remains relatively stable for low to medium
densities until it approaches the density zone influenced by the geometric
constraints of the vehicles. The significant inflection point occurs at
approximately $\rho\sim 0.02$ 1/mm, which corresponds to an average distance
of about 50 mm, roughly 11% greater than the vehicle size. Since this value
represents an average, contact events start from this typical distance and
become more frequent as the mean distance between neighboring vehicles
decreases. This clarifies how speed-density curves behave without long-range
sensing, with speed reduction mainly due to agent collisions.

Finally, considering the reduction in speed looking at the speed-density
curves (Fig. 5) along with the global speed as a function of $N$ (Fig. 2), we
can establish that the maximum reduction in the mean speed ranges from 25% to
40%, depending on the arrangement of the vehicles.

Figure 5: Microscopic speed-density data. A) Vehicles introduced in an
’ascending’ order. B) ’Descending’ order. C) ’Random’ order. D) Moving average
of the three aforementioned data sets, calculated over 20,000 data points.

##  4 Conclusions

In summary, we experimentally studied a system of self-propelled particles
whose trajectories were, using a circular track, restricted to circular
motion. These particles, due to small differences in their construction, have
differences in the speed they can achieve. We found that as the number of VDVs
increased, the average speed of the system decreased regardless of the order
in which the vehicles were added. Moreover, the results showed that the
average speed of the system, when considering the highest number of VDVs, is
less than the lowest of the speeds of all vehicles.

We then analyzed the microscopic characteristics by means of the density and
speed experienced by each VDV. The density distributions showed a maximum
value that corresponds to the contact distance of two VDVs. In addition, as
$N$ increases, the distributions become more pronounced and the population in
the low-density region decreases significantly. Similarly, as the value of $N$
increased, the velocity distributions became narrower and more pronounced.
This occurred because the reduced available space resulted in a more uniform
system speed.

Compiling the microscopic data, we constructed the speed-density diagram for
VDVs. Given that these agents lack long-range sensing and interact solely
through contact, speed remains constant until the density reaches a point
roughly 10% beyond one agent’s length, at which stage it significantly
declines. This observation may provide insights into the behavior of other
systems in situations where long-range sensing is absent, and speed reduction
primarily arises from direct collisions between agents.

In a one-dimensional system, such as the one examined in this study, the
behavior in the low and medium density ranges is influenced by the order in
which agents are added to increase density. When fast VDVs are introduced
first, they result in higher speed values on the fundamental diagram compared
to the scenario where slower VDVs are added first. However, as the density
increases, both curves gradually converge to a common speed value at maximum
density. Additionally, we observed that both the speed-density curve and
velocity distributions for any arbitrary order configuration fall between
these two extreme ordered configurations.

From this observation, we can derive at least two significant insights: (a) If
this behavior remains consistent across all random configurations, then
organizing the particles based on their free speed could provide an efficient
method for capturing the full spectrum of potential speed-density curves,
rather than repetitively conducting the same experiment with randomly ordered
configurations. (b) Given the variability of speeds, the arrangement order of
agents significantly influences the specific shape of the fundamental diagram,
at least in one dimension. This factor may have been previously overlooked in
attempts to explain the variation observed in different pedestrian fundamental
diagrams.

## ACKNOWLEDGMENTS

This work was funded by Research Grant from HFSP (Ref.-No: RGP0053/2020) and
project PICT 2019-00511 (Agencia Nacional de Promoción Científica y
Tecnológica, Argentina).

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# Morse-Novikov cohomology on foliated manifolds

Md. Shariful Islam  Department of Mathematics
University of Dhaka
Dhaka 1000, Bangladesh<EMAIL_ADDRESS>
###### Abstract.

The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has
been utilized by many researchers to study important properties and invariants
of a manifold. Morse-Novikov cohomology is defined using the differential
$d_{\omega}=d+\omega\wedge$, where $\omega$ is a closed $1$-form. We study
Morse-Novikov cohomology relative to a foliation on a manifold and its
homotopy invariance and then extend it to more general type of forms on a
Riemannian foliation. We study the Laplacian and Hodge decompositions for the
corresponding differential operators on reduced leafwise Morse-Novikov
complexes. In the case of Riemannian foliations, we prove that the reduced
leafwise Morse-Novikov cohomology groups satisfy the Hodge theorem and
Poincaré duality. The resulting isomorphisms yield a Hodge diamond structure
for leafwise Morse-Novikov cohomology.

###### Key words and phrases:

foliation, cohomology, homotopy invariance, Hodge theory, Poincare duality

###### 2010 Mathematics Subject Classification:

57R30; 53C12; 58A14

*Corresponding author<EMAIL_ADDRESS>
##  1\. Introduction

Consider an $n$-dimensional smooth manifold $M$; denote by $\Omega^{k}(M)$ the
collection of all degree $k$ differential forms on $M$ and by $H^{k}(M)$ the
corresponding de Rham cohomology group. Let $\omega$ be a closed $1$-form that
is not necessarily exact. We consider the twisted operator
$d_{\omega}:\Omega^{k}(M)\rightarrow\Omega^{k+1}(M)$ defined by
$d_{\omega}=d+\omega\wedge$, where $d$ is the usual exterior derivative, so
that $(d_{\omega})^{2}=0$. The differential cochain complex
$(\Omega^{*}(M),d_{\omega})$ is called the Morse-Novikov complex of the
manifold $M$. The cohomology groups $H_{\omega}^{k}(M)$ of this cochain
complex are called the Morse-Novikov or Lichnerowicz cohomology groups of $M$
and have been utilized by many researchers. Morse-Novikov cohomology was first
studied by A. Lichnerowicz in [13], and used in the context of Poisson
geometry. The idea of Lichnerowicz has been exploited to study many properties
of manifolds. In [15] and [16] S.P.Novikov proved a generalization of the
Morse inequalities by comparing the ranks of these cohomology groups with
combinatorial invariants derived from the zeros of the form $\omega$.
Pazhintov [21] gave an analytic proof of the real part of Novikov’s
inequalities. E. Witten used the Morse-Novikov cohomology for exact $\omega$
in his famous discovery [27] of what is now known as Witten deformation. In
this case Morse-Novikov cohomology is isomorphic to de Rham cohomology. M.
Shubin and S. P. Novikov applied the deformation method to a rigorous
treatment of eigenvalue limits of Witten Laplacians for more general $1$-forms
and vector fields in [17] and [23]. Many other researchers have extended and
generalized this work, such as Braverman and Farber [4] in cases of
nonisolated zeros of $1$-forms and vector fields. See [8] for a good reference
on these related topics. Alexandra Otiman studied Morse-Novikov cohomology for
particular classes of closed $1$-forms in [20]. I. Vaisman studied locally
conformal symplectic manifolds in [24], and L. Ornea, and M. Verbitsky studied
Morse-Novikov cohomology of locally conformally Kähler manifolds in [19]. In
[5], X. Chen showed that if a Riemannian manifold $M$ has almost non-negative
sectional curvature and nonzero first de Rham cohomology group, then all the
Morse-Novikov cohomology groups of $M$ vanish irrespective of the choice of
the closed non-exact $1$-form $\omega$. In [14], L. Meng established an
analogue of the Leray-Hirsch Theorem for de Rham cohomology and a blowup
formula fo Dolbeault-Morse-Novikov cohomology on complex manifolds. Morse-
Novikov cohomology theory has also been used to study locally conformal
symplectic manifolds (see [25], [24], and [26]).

Two types of Morse-Novikov cohomology associated to foliations are basic (see
[22]) and leafwise. Liviu Ornea and Vladimir Slesar studied basic Morse-
Novikov cohomology in [18]. K. Richardson and G. Habib used basic Morse-
Novikov cohomology to prove that the basic signature and the Álvarez class of
a Riemannian foliation are homotopy invariants [10]. They have also used a
modified differential as in Morse-Novikov cohomology to define a twisted basic
cohomology for Riemannian foliations that satisfies Poincaré duality [9]. J.
A. Álvarez Lopez, Y. Kordyukov, and E. Leichtnam studied leafwise Hodge
decomposition on Riemannian foliations with bounded geometry and extended the
Morse-Novikov differential complex [2].

This paper is constructed as follows. Using $d+\omega\wedge$ as the
differential for a leafwise closed $1$-form $\omega$, we study leafwise Morse-
Novikov cohomology groups whose isomorphism classes turn out to be smooth
invariants of the foliation. In the cases where $\omega$ is truly a closed
$1$-form on the manifold, one further obtains Morse-Novikov cohomology groups
from the foliation. In Section 3 we study the basic properties of leafwise
Morse-Novikov cohomology groups, including the homotopy axiom in Proposition
3.7, and foliated homotopy invariance in Corollary 3.8. With the additional
assumption that the foliation is Riemannian, we give a proof of the Hodge
decomposition in Corollary 4.6 and Poincaré duality in Corollary 4.7. We
extend these results to more general settings of forms of $p,q$ type: homotopy
axioms for general leafwise Morse-Novikov cohomology in Proposition 5.3, Hodge
decomposition for general leafwise Morse-Novikov cohomology in Theorem 6.1,
Poincaré duality for general leafwise Morse-Novikov cohomology in Proposition
6.3. The assumption that the foliation is Riemannian is required to obtain
Hodge theory and Poincaré duality; for general smooth foliations, those
results are false, even for the case when $\omega=0$. In the remainder of
Section 6, the isomorphisms between the Morse-Novikov cohomology groups are
discussed, leading to Corollary 6.8, that shows the Hodge diamond structure.

Much but not all of the results in this paper were part of the authors Ph.D
thesis [12].

##  2\. Leafwise de Rham cohomology

In this section we review notations and known results. Let $M$ be a closed
compact oriented Riemannian manifold. Suppose we are given a smooth foliation
$\left(M,\mathcal{F}\right)$ on $M$. Let $T\mathcal{F}$ denote the leafwise
tangent bundle and $T^{*}\mathcal{F}$ denotes its dual bundle. Let the
conormal bundle $N^{*}_{x}\mathcal{F}$ be defined at each point $x\in M$ as
the set of all linear functionals that map each vector in $T_{x}\mathcal{F}$
to zero, and this bundle can be canonically identified with a subbundle of
$T^{*}_{x}M$ independent of the metric. Using the metric, the normal bundle
$Q=TM/T\mathcal{F}$ may be uniquely identified with a subbundle
$N\mathcal{F}=T\mathcal{F}^{\perp}$ of $TM$. Also, we can identify the dual
bundle $T^{*}\mathcal{F}$ with the set of covectors that kill $N\mathcal{F}$.
Now we can decompose all differential forms using

(2.1)
$\Lambda^{u,v}(M,\mathcal{F})=\Lambda^{u}N^{*}\mathcal{F}\wedge\Lambda^{v}T^{*}\mathcal{F}$

Let $\Omega^{u,v}(M,\mathcal{F})=\Gamma\Lambda^{u,v}(M,\mathcal{F})$. The
exterior derivative can then be decomposed as $d=d_{0,1}+d_{1,0}+d_{2,-1}$
with

$d_{i,j}\omega\in\Omega^{u+i,v+j}(M,\mathcal{F})$

for all $\omega\in\Omega^{u,v}(M,\mathcal{F})$. Then it is easy to see that
since $d^{2}=0$, we also have $d_{0,1}^{2}=0$.

The elements of $\Omega^{0,k}\left(M,\mathcal{F}\right)$ are called leafwise
$k$-forms. Let $\Gamma\left(T\mathcal{F}\right)$ be the set of smooth sections
of $T\mathcal{F}$. If $X,Y\in\Gamma\left(T\mathcal{F}\right)$, then by the
Frobenius theorem $[X,Y]\in\Gamma\left(T\mathcal{F}\right)$. The leafwise
exterior differential operator
$d_{0,1}^{k}=d_{\mathcal{F}}^{k}:\Omega^{0,k}\left(M,\mathcal{F}\right)\rightarrow\Omega^{0,k+1}\left(M,\mathcal{F}\right)$
may also be defined by

$\displaystyle
d_{\mathcal{F}}^{k}\omega\left(X_{0},\cdots,X_{k+1}\right)=\sum_{0\leq i\leq
k}(-1)^{i}[X_{i}\omega\left(X_{0},\cdots,\hat{X_{i}},\cdots,X_{k}\right)]$
$\displaystyle+\sum_{0\leq i<j\leq
k}(-1)^{i+j}\omega\left([X_{i},X_{j}],X_{0},\cdots,\hat{X_{i}},\cdots,\hat{X_{j}},\cdots,X_{k}\right)$

for $X_{0},\cdots,X_{k+1}\in\Gamma\left(T\mathcal{F}\right)$. The differential
operator $d_{\mathcal{F}}$ is the restriction of the usual differential on
differential forms on the leaves of $\mathcal{F}$. Similar to the usual
exterior differential, the leafwise differential satisfies
$d_{\mathcal{F}}^{k+1}\circ d_{\mathcal{F}}^{k}=0$. For $k\geq 0$, the
$k^{\mathrm{th}}$ leafwise cohomology group
$H^{k}\left(M,d_{\mathcal{F}}\right)$ is the $k^{\mathrm{th}}$ cohomology
group

$H^{k}\left(M,d_{\mathcal{F}}\right)=\frac{\ker
d_{\mathcal{F}}^{k}}{\mathrm{im}\,d_{\mathcal{F}}^{k-1}}$

of the cochain complex
$\left(\Omega^{0,*}\left(M,\mathcal{F}\right),d_{\mathcal{F}}\right)$.

For the purpose of having Laplacian and Hodge decompositions, we need to
consider reduced leafwise cohomology

$\bar{H}^{k}\left(M,d_{\mathcal{F}}\right)=\frac{\ker
d_{\mathcal{F}}^{k}}{\overline{\mathrm{im}\,d_{\mathcal{F}}^{k-1}}}.$

Here the closure $\overline{\mathrm{im}\,d_{\mathcal{F}}^{k-1}}$ is taken with
respect to the Frechét topology on $\Omega^{0,k}\left(M,\mathcal{F}\right)$.
The cup product induced from exterior product of forms makes
$\bar{H}^{*}\left(M,d_{\mathcal{F}}\right)$ into a graded commutative algebra
over $C^{\infty}(M)$.

Let $f:M\rightarrow N$ be a smooth map of the foliated manifold which maps
leaves into leaves. Then the pullback maps

$f^{*}:\Gamma\left(\Lambda^{k}T^{*}\mathcal{F}_{N}\right)\rightarrow\Gamma\left(\Lambda^{k}T^{*}\mathcal{F}_{M}\right)$

are defined for all $k$. They commute with $d_{\mathcal{F}}$ and respect the
exterior product; therefore they induce a continuous map of the reduced
cohomology ring.

$f^{*}:\bar{H}^{k}\left(N,d_{\mathcal{F}_{0}}\right)\rightarrow\bar{H}^{k}\left(M,d_{\mathcal{F}_{1}}\right).$

Such maps are called foliated maps. Two smooth foliated maps
$f,g:\left(N,\mathcal{F}_{0}\right)\rightarrow\left(M,\mathcal{F}_{1}\right)$
between two foliated manifolds are leafwise homotopic if there is a map
$F:N\times[0,1]\rightarrow M$ such that if $F_{t}$ denotes the restriction
$F_{t}=F|_{N\times{t}}:N\rightarrow M$ for $t\in[0,1]$, then
$F|_{0}=f,F|_{1}=g$, $F_{t}\left(\mathcal{F}_{0}\right)\in\mathcal{F}_{1}$ for
all $t\in[0,1]$, and for every $x\in N$ the points $F_{t_{1}}(x),F_{t_{2}}(x)$
lie in the same leaf of $\mathcal{F}_{1}$ for all $t_{1},t_{2}\in[0,1]$. Thus,
a leafwise homotopy consists of leaf-preserving maps. Denote the identity maps
of $\left(N,\mathcal{F}_{0}\right)$, $\left(M,\mathcal{F}_{1}\right)$ by
$l_{\mathcal{F}_{0}}$, $l_{\mathcal{F}_{1}}$ respectively. A leafwise map
$f:\left(N,\mathcal{F}_{0}\right)\rightarrow\left(M,\mathcal{F}_{1}\right)$ is
a leafwise homotopy equivalence if there exists a leafwise map
$g:\left(M,\mathcal{F}_{1}\right)\rightarrow\left(N,\mathcal{F}_{0}\right)$
with $f\circ g$ is leafwise homotopic to $l_{\mathcal{F}_{0}}$ and $g\circ f$
is leafwise homotopic to $l_{\mathcal{F}_{1}}$.

######  Proposition 2.1.

[11, Theorem I, 3.2]) If the maps
$f,g:\left(N,\mathcal{F}_{0}\right)\rightarrow\left(M,\mathcal{F}_{1}\right)$
between two foliated manifolds are leafwise homotopic, then
$f^{*}=g^{*}:H^{k}\left(M,d_{\mathcal{F}_{1}}\right)\rightarrow
H^{k}\left(N,d_{\mathcal{F}_{0}}\right)$. That is, leafwise homotopic maps
induce the same map on leafwise cohomology groups.

######  Corollary 2.2.

If a map
$f:\left(N,\mathcal{F}_{0}\right)\rightarrow\left(M,\mathcal{F}_{1}\right)$ is
a smooth foliated homotopy equivalence, then $f^{*}$ induces an isomorphism
between $H^{k}\left(N,d_{\mathcal{F}_{0}}\right)$ and
$H^{k}\left(M,d_{\mathcal{F}_{1}}\right)$.

##  3\. Leafwise Morse-Novikov Cohomology and Hodge Theory

From now we assume that our foliation is Riemannian, characterized by the
existence of a bundle-like metric $g$ such that a geodesic of the metric $g$
is orthogonal to all leaves that it meets whenever it is orthogonal to one of
them. We assume that a bundle like metric has been chosen.

Let $M$ be an oriented manifold endowed with a foliation $\mathcal{F}$ of
dimension $p$. The graded Frechét space
$\Omega^{0,*}\left(M,\mathcal{F}\right)$ can be endowed with the natural
metric

$\left(\alpha,\beta\right)=\int_{M}\left\langle\alpha,\beta\right\rangle_{\mathcal{F}}\mathrm{vol}.$

In this formula $\left\langle,\right\rangle_{\mathcal{F}}$ is the Riemannian
metric on $\Lambda T^{*}\mathcal{F}$ induced from the Riemannian metric $g$ on
$M$, and $\mathrm{vol}$ is the volume form associated to the metric $g$. We
denote the formal adjoint of the leafwise differential $d_{\mathcal{F}}$ with
respect to this inner product by $\delta_{\mathcal{F}}$; then the
corresponding Laplacian is

$\Delta_{\mathcal{F}}=d_{\mathcal{F}}\delta_{\mathcal{F}}+\delta_{\mathcal{F}}d_{\mathcal{F}}.$

Since $\mathcal{F}$ is Riemannian, the restriction of $\delta_{\mathcal{F}}$
to any leaf is the codifferential of the leaf with respect to the induced
metric [1, Lemma 3.2], i.e.

$\left(\delta_{\mathcal{F}}\alpha\right)|_{F}=\delta_{\mathcal{F}}\left(\alpha\right)|_{F}\text{
for all }\alpha\in\Omega^{0,k}\left(M,\mathcal{F}\right),$

where $F$ denotes a leaf of the foliation. Now we assume that the tangent
bundle $T\mathcal{F}$ is orientable. The choice of an orientation determines a
volume form $\chi_{\mathcal{F}}\in\Omega^{0,p}\left(M,\mathcal{F}\right)$. Now
we can define leafwise Hodge star-operator

$*_{\mathcal{F}}:\Lambda^{0,k}T^{*}\mathcal{F}\rightarrow\Lambda^{0,p-k}T^{*}\mathcal{F}\text{
for each }k\text{ and }x\in M,$

and it is determined by the relation

$\alpha\wedge*_{\mathcal{F}}\beta=\left\langle\alpha,\beta\right\rangle\chi_{\mathcal{F}},\text{
for }\alpha,\beta\in\Lambda^{0,k}T^{*}_{x}\mathcal{F}.$

This fibrewise star-operator determines the leafwise star-operator

$*_{\mathcal{F}}:\Omega^{0,k}\left(M,\mathcal{F}\right)\rightarrow\Omega^{0,p-k}\left(M,\mathcal{F}\right)\text{
for each }k.$

Now we state some important properties of leafwise cohomology. Suppose $M$ is
compact, and $\mathcal{F}$ is a $p$-dimensional oriented Riemannian foliation
of $M$ with a bundle-like metric $g$.

######  Proposition 3.1.

[7, Theorem 0.2] The map
$\phi:\ker\Delta^{k}_{\mathcal{F}}\rightarrow\bar{H}^{k}\left(M,d_{\mathcal{F}}\right)$
defined by
$\phi\left(\omega\right)=\omega\,\mathrm{mod}\,\overline{\mathrm{im}\,d^{k}_{\mathcal{F}}}$
is a topological isomorphism of Frechét spaces. This isomorphism, in general,
does not hold for non-Riemannian foliations.

Under the same assumptions, the next deep result is due to Álvarez López and
Kordyukov.

######  Theorem 3.2.

[1, Corollary C] The Hodge star-operator induces an isomorphism

$*_{\mathcal{F}}:\ker\Delta^{k}_{\mathcal{F}}\rightarrow\ker\Delta^{p-k}_{\mathcal{F}}.$

Moreover $*_{\mathcal{F}}$ commutes with $\Delta^{k}_{\mathcal{F}}$ up to a
sign. From the previous proposition we have the following isomorphism

$*_{\mathcal{F}}:\bar{H}^{k}\left(M,d_{\mathcal{F}}\right)\rightarrow\bar{H}^{p-k}\left(M,d_{\mathcal{F}}\right).$

Let $\omega$ be a leafwise closed $1$-form, which is not necessarily leafwise
exact. We consider the twisted operator
$d^{\omega}_{\mathcal{F}}:\Omega^{0,k}\left(M,\mathcal{F}\right)\rightarrow\Omega^{0,k+1}\left(M,\mathcal{F}\right)$
defined by $d^{\omega}_{\mathcal{F}}=d_{\mathcal{F}}+\omega\wedge$, where
$d_{\mathcal{F}}$ is the exterior derivative along the leaf. Since
$d_{\mathcal{F}}\circ d_{\mathcal{F}}=d^{2}_{\mathcal{F}}=0$,
$\omega\wedge\omega=0$, and $d_{\mathcal{F}}(\omega\wedge\alpha)=-\omega\wedge
d_{\mathcal{F}}\alpha$ for any $k$-form $\alpha$, it follows that
$\left(d^{\omega}_{\mathcal{F}}\right)^{2}=0$. The differential cochain
complex
$\left(\Omega^{0,*}\left(M,\mathcal{F}\right),d^{\omega}_{\mathcal{F}}\right)$
is called the leafwise Morse-Novikov complex of the foliated manifold
$\left(M,\mathcal{F}\right)$. Let $d_{\mathcal{F}}^{\omega,k}$ be the
restriction of $d^{\omega}_{\mathcal{F}}$ to
$\Omega^{0,k}\left(M,\mathcal{F}\right)$. The cohomology groups

$H^{k}_{\omega}\left(M,d_{\mathcal{F}}\right)=\frac{\ker\left(d_{\mathcal{F}}^{\omega,k}\right)}{\mathrm{im}\left(d_{\mathcal{F}}^{\omega,k-1}\right)}$

of this cochain complex are called the leafwise Morse-Novikov cohomology
groups of $\left(M,\mathcal{F}\right)$. For the purpose of obtaining Hodge
decomposition, we need to consider the reduced leafwise Morse-Novikov
cohomology

$\bar{H}^{k}_{\omega}\left(M,d_{\mathcal{F}}\right)=\frac{\ker\left(d_{\mathcal{F}}^{\omega,k}\right)}{\overline{\mathrm{im}\left(d_{\mathcal{F}}^{\omega,k-1}\right)}}.$

Here the closure
$\overline{\mathrm{im}\left(d_{\mathcal{F}}^{\omega,k-1}\right)}$ is taken
with respect to the Frechét topology on
$\Omega^{0,k}\left(M,\mathcal{F}\right)$. The cup product induced from
exterior product of forms makes
$\bar{H}^{*}_{\omega}\left(M,d_{\mathcal{F}}\right)$ into a graded commutative
algebra over $C^{\infty}(M)$.

######  Proposition 3.3.

If $\omega$ and $\theta=\omega+d_{\mathcal{F}}g$ are cohomologous in
$H^{1}\left(M,d_{\mathcal{F}}\right)$, then for each $k$, the leafwise Morse-
Novikov cohomology groups $H_{\omega}^{k}\left(M,d_{\mathcal{F}}\right)$ and
$H_{\theta}^{k}\left(M,d_{\mathcal{F}}\right)$ are isomorphic. That is, the
map $\Phi:H_{\omega}^{k}\left(M,d_{\mathcal{F}}\right)\rightarrow
H_{\theta}^{k}\left(M,d_{\mathcal{F}}\right)$ given by
$\Phi\left([\alpha]\right)=[e^{-g}\alpha]$ is an isomorphism.

###### Proof.

If $\omega$ and $\theta$ are cohomologous, then there exists
$g\in\Omega^{0}(M)$, such that $\theta-\omega=d_{\mathcal{F}}g$. Define the
mapping $\phi:H_{\omega}^{k}(M,d_{\mathcal{F}})\rightarrow
H_{\theta}^{k}(M,d_{\mathcal{F}})$ by
$\phi(\left[\alpha\right])=\left[e^{-g}\alpha\right]$. One can check that
$\phi$ is well-defined and is a group homomorphism, since

$\displaystyle\left(d_{\mathcal{F}}+\theta\wedge\right)\left(e^{-g}\alpha\right)$
$\displaystyle=$
$\displaystyle\left(d_{\mathcal{F}}+\omega\wedge+d_{\mathcal{F}}g\wedge\right)\left(e^{-g}\alpha\right),$
$\displaystyle=$ $\displaystyle
e^{-g}\left(d_{\mathcal{F}}+\omega\wedge\right)\left(\alpha\right),$

for all $\alpha\in\Omega^{k}\left(M,d_{\mathcal{F}}\right)$.

Suppose $\alpha,\beta\in\Omega^{k}\left(M,d_{\mathcal{F}}\right)$ are
cohomologous, then there exists
$\nu\in\Omega^{k-1}\left(M,d_{\mathcal{F}}\right)$ such that
$\alpha-\beta=\left(d_{\mathcal{F}}+\omega\wedge\right)\nu$. We have

$\displaystyle\phi\left([\alpha-\beta]\right)$ $\displaystyle=$
$\displaystyle[e^{-g}\left(d_{\mathcal{F}}+\omega\wedge\right)\nu],$
$\displaystyle\Rightarrow\left[\left(d_{\mathcal{F}}+\theta\wedge\right)\left(e^{-g}\nu\right)\right]$
$\displaystyle=$ $\displaystyle\left[0\right].$

Similarly if, $\phi(\left[\alpha\right])=0$, Then $\left[\alpha\right]=0\in
H_{\omega}^{k}(M,d_{\mathcal{F}})$, and $\phi$ is injective.
If $\left[\alpha\right]\in H_{\theta}^{k}(M,d_{\mathcal{F}})$ then we find
similarly that $\left[e^{g}\alpha\right]\in
H_{\omega}^{k}(M,d_{\mathcal{F}})$, so that $\phi$ is surjective. ∎

######  Corollary 3.4.

If $\omega$ is a $d_{\mathcal{F}}$ exact $1$-form, then for each $k$ the
leafwise Morse-Novikov cohomology group
$H_{\omega}^{k}\left(M,d_{\mathcal{F}}\right)$ and the leafwise de Rham
cohomology group $H^{k}\left(M,d_{\mathcal{F}}\right)$ are isomorphic.

$H_{\omega}^{k}\left(M,d_{\mathcal{F}}\right)\cong
H^{k}\left(M,d_{\mathcal{F}}\right).$

######  Corollary 3.5.

If the first leafwise de Rham cohomology group
$H^{1}\left(M,d_{\mathcal{F}}\right)$ equals $0$, then for every
$d_{\mathcal{F}}$ closed 1-form $\omega$ and for each $k$ the leafwise Morse-
Novikov cohomology groups satisfy
$H_{\omega}^{k}\left(M,d_{\mathcal{F}}\right)=H^{k}\left(M,d_{\mathcal{F}}\right)$.

######  Lemma 3.6.

For any smooth foliation $\left(M,\mathcal{F}\right)$ the leafwise Morse-
Novikov cohomology $H_{\omega}^{0}\left(M,d_{\mathcal{F}}\right)=\\{0\\}$ if
and only if $\omega$ is not $d_{\mathcal{F}}$ exact.

###### Proof.

Suppose first that $H_{\omega}^{0}\left(M,d_{\mathcal{F}}\right)\neq\\{0\\}$
for a $d_{\mathcal{F}}$ closed one form $\omega$ on
$\left(M,\mathcal{F}\right)$, then there is a nonzero function $f\in
C^{\infty}\left(M\right)$, such that

$\displaystyle\left(d_{\mathcal{F}}+\omega\right)f$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle d_{\mathcal{F}}f+f\omega$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle
d_{\mathcal{F}}\left(\log(\frac{1}{f})\right)$ $\displaystyle=$
$\displaystyle\omega,$

which implies $\omega$ is $d_{\mathcal{F}}$ exact. Conversely, suppose that
$\omega$ is $d_{\mathcal{F}}$ exact. There exists a function $g\in
C^{\infty}\left(M\right)$ such that $d_{\mathcal{F}}g=\omega$. Then

$\displaystyle\left(d_{\mathcal{F}}+\omega\right)\left(e^{-g}\right)$
$\displaystyle=$
$\displaystyle-e^{-g}d_{\mathcal{F}}g+e^{-g}d_{\mathcal{F}}g=0,$

which shows $H_{\omega}^{0}\left(M,d_{\mathcal{F}}\right)\neq\\{0\\}$. ∎

######  Proposition 3.7.

(Homotopy axiom for the leafwise Morse-Novikov cohomology). Let
$f:\left(M,\mathcal{F}_{0}\right)\rightarrow\left(N,\mathcal{F}_{1}\right)$
and
$g:\left(M,\mathcal{F}_{0}\right)\rightarrow\left(N,\mathcal{F}_{1}\right)$ be
foliated homotopic maps, and let $\omega$ be a leafwise closed $1-\text{form}$
on $\left(N,\mathcal{F}_{1}\right)$. Then there exists a positive function
$h:\left(M,\mathcal{F}_{0}\right)\rightarrow\mathbb{R}$ such that for all $k$

$f^{*}=hg^{*}:H^{k}_{\omega}\left(N,d_{\mathcal{F}_{1}}\right)\rightarrow
H^{k}_{f^{*}\omega}\left(M,d_{\mathcal{F}_{0}}\right).$

###### Proof.

Since $f$ and $g$ are foliated homotopic maps, by the homotopy axiom of
leafwise de Rham cohomology (Proposition 2.1), they induce the same map in
leafwise de Rham cohomology. Therefore, for any leafwise closed $1-$form
$\omega\in\Omega^{1}\left(N,\mathcal{F}_{1}\right)$, the pullback forms
$f^{*}\omega\text{ , }g^{*}\omega\in
H^{1}_{\omega}\left(M,d_{\mathcal{F}_{0}}\right)$ are cohomologous. There
exists a function $\nu:\left(M,\mathcal{F}_{0}\right)\rightarrow\mathbb{R}$
such that $g^{*}\omega-f^{*}\omega=d_{\mathcal{F}}\nu$. We define $h=e^{\nu}$.
Then from the proof of Proposition 3.3, for any $d^{\omega}_{\mathcal{F}}$
closed form $\alpha$ on $\left(N,\mathcal{F}_{1}\right)$,
$\left[hg^{*}\alpha\right]=\left[f^{*}\alpha\right]\in
H_{f^{*}\omega}^{k}\left(M,d_{\mathcal{F}_{0}}\right)$. ∎

######  Corollary 3.8.

If $f:\left(M,\mathcal{F}_{0}\right)\rightarrow\left(N,\mathcal{F}_{1}\right)$
is a foliated homotopy equivalence and $\omega$ is a leafwise closed
$1-\text{form}$, then the leafwise Morse-Novikov cohomology groups
$H_{\omega}^{*}\left(M,d_{\mathcal{F}_{0}}\right)$ and
$H_{f^{*}\omega}^{*}\left(N,d_{\mathcal{F}_{1}}\right)$ are isomorphic; i.e.

$H_{\omega}^{k}\left(M,d_{\mathcal{F}_{0}}\right)\cong
H_{f^{*}\omega}^{k}\left(N,d_{\mathcal{F}_{1}}\right),\text{ for all }k.$

###### Proof.

There exists a map
$g:\left(N,\mathcal{F}_{1}\right)\rightarrow\left(M,\mathcal{F}_{0}\right)$
such that $f\circ g$ is homotopic to the identity map $\mathbb{I}_{N}$ of $N$
and $g\circ f$ is homotopic to the identity map $\mathbb{I}_{M}$ of $M$ . We
have linear maps

$H^{*}_{\omega}\left(N,\mathcal{F}_{1}\right)\stackrel{{\scriptstyle
f^{*}}}{{\rightarrow}}H^{*}_{f^{*}\omega}\left(M,\mathcal{F}_{0}\right)\stackrel{{\scriptstyle
g^{*}}}{{\rightarrow}}H^{*}_{g^{*}f^{*}\omega}\left(N,\mathcal{F}_{1}\right).$

By the homotopy axiom of leafwise Morse-Novikov cohomology (Proposition 3.7),
there exists a positive function $h:N\rightarrow\mathbb{R}$ such that
$g^{*}f^{*}\omega=\omega+d_{\mathcal{F}_{1}}(\log(h))$ then we have

$\mathbb{I}_{N}=hg^{*}f^{*}=h\left(f\circ
g\right)^{*}:H^{*}_{\omega}\left(N,\mathcal{F}_{1}\right)\rightarrow
H^{*}_{\omega}\left(N,\mathcal{F}_{1}\right).$

And similarly, for some positive function $\bar{h}:M\rightarrow\mathbb{R}$
such that $f^{*}g^{*}\omega=\omega+d_{\mathcal{F}_{0}}(\log(\bar{h}))$ then we
have

$\mathbb{I}_{M}=\bar{h}f^{*}g^{*}=h\left(g\circ
f\right)^{*}:H^{*}_{\omega}\left(M,\mathcal{F}_{0}\right)\rightarrow
H^{*}_{\omega}\left(M,\mathcal{F}_{0}\right).$

Since multiplication by a positive function is an isomorphism of Leafwise
Morse-Novikov cohomology, $f^{*}$ and $g^{*}$ are isomorphisms. ∎

##  4\. Laplacian and Hodge decomposition on leafwise Morse-Novikov cohomology

In the following, assume $\dim(M)=n$, $\dim(\mathcal{F})=p$, and
$\mathrm{codim}(\mathcal{F})=q$. As in (Equation (2.1)), we have the bigrading

$\Omega^{u,v}\left(M,\mathcal{F}\right)=\Gamma(M,\Lambda^{v}T\mathcal{F}^{*}\otimes\Lambda^{u}T\mathcal{F}^{\perp*}).$

We choose a tangential and a transversal orientation for $\mathcal{F}$ on any
open subset $\mathcal{U}\subset M$. We obtain the Hodge star operator
$*_{\mathcal{F}}$ on $T\mathcal{F}^{*}$ and $*_{\bot}$ on
$T\mathcal{F}^{\bot*}$ to $\mathcal{U}$ such that
$*_{\bot}(1)\wedge*_{\mathcal{F}}(1)$ is a positive volume form on
$\mathcal{U}\subset M$.

######  Lemma 4.1.

(Lemma 3.2 in [1]) The Hodge star operator on
$\wedge\left(T^{*}M\right)=\wedge\left(T\mathcal{F}^{\bot*}\right)\otimes\wedge\left(T\mathcal{F}^{*}\right)$
on $\mathcal{U}$ satisfies

$*=(-1)^{(q-u)v}*_{\bot}\otimes*_{\mathcal{F}}:\Lambda^{u}T\mathcal{F}^{\bot*}\otimes\Lambda^{v}T\mathcal{F}^{*}\rightarrow\Lambda^{q-u}T\mathcal{F}^{\bot*}\otimes\Lambda^{p-v}T\mathcal{F}^{*}.$

######  Lemma 4.2.

$*^{2}=(-1)^{(u+v)(p+q+1)}$ on
$\Lambda^{u}T\mathcal{F}^{\bot*}\otimes\Lambda^{v}T\mathcal{F}^{*}$.

###### Proof.

Restricted to $\Lambda^{u}T\mathcal{F}^{\bot*}$ and
$\Lambda^{v}T\mathcal{F}^{*}$, we have $*_{\bot}^{2}=(-1)^{u(q+1)}$ and
$*_{\mathcal{F}}^{2}=(-1)^{v(p+1)}$.

$\displaystyle*^{2}$ $\displaystyle=$
$\displaystyle(-1)^{(q-u)v}(-1)^{u(p-v)}*_{\bot}^{2}\otimes*_{\mathcal{F}}^{2}$
$\displaystyle=$
$\displaystyle(-1)^{(q-u)v}(-1)^{u(p-v)}(-1)^{u(q+1)}(-1)^{v(p+1)}id_{\bot}\otimes
id_{\mathcal{F}}$ $\displaystyle=$
$\displaystyle(-1)^{(u+v)(p+q+1)}id_{\bot}\otimes id_{\mathcal{F}}.$

∎

######  Lemma 4.3.

(Formula 17 in [1]) The adjoint $\delta_{\mathcal{F}}=d_{\mathcal{F}}^{*}$ of
$d_{\mathcal{F}}$ is given by

$\delta_{\mathcal{F}}\beta=d_{\mathcal{F}}^{*}\beta=(-1)^{pk+p+1}*_{\mathcal{F}}d_{\mathcal{F}}*_{\mathcal{F}}\beta,$

for any $\beta\in\Omega^{0,k}\left(M,\mathcal{F}\right)$.

###### Proof.

The standard proof that $d^{*}=\left(-1\right)^{nk+n+1}*d*$ on $n$-manifolds
applies on a foliated manifold in a local neighborhood. ∎

######  Lemma 4.4.

$\omega\lrcorner=(-1)^{pk+p}*_{\mathcal{F}}\left(\omega\wedge\right)*_{\mathcal{F}}$
for all $\omega\in\Omega^{0,k}\left(M,\mathcal{F}\right)$.

###### Proof.

Suppose $\tau$ denotes the tangential volume form. For any
$\beta\in\Omega^{0,k}$, we have

$\displaystyle\omega\lrcorner\beta$ $\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(k+1\right)}*\left(\omega\wedge\right)*\beta$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(k+1\right)}*\left(\omega\wedge\right)\left(*_{\bot}\otimes*_{\mathcal{F}}\right)\left(1\otimes\beta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(k+1\right)}\left(-1\right)^{qk}\left(-1\right)^{q}*\left(\tau\wedge\omega\wedge*_{\mathcal{F}}\beta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(k+1\right)}\left(-1\right)^{qk}\left(-1\right)^{q}\left(*_{\bot}\otimes*_{\mathcal{F}}\right)\left(\tau\wedge\omega\wedge*_{\mathcal{F}}\beta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{p\left(k+1\right)}*_{\mathcal{F}}\left(\omega\wedge\right)*_{\mathcal{F}}\beta.$

∎

By Lemmas above and the identity
$*_{\mathcal{F}}^{2}=\left(-1\right)^{k\left(p-k\right)}$, we have on
$\Omega^{0,k}\left(M,\mathcal{F}\right)$

$\displaystyle\left(\omega\lrcorner\right)*_{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{k}*_{\mathcal{F}}\left(\omega\wedge\right)$
${}_{\mathcal{F}}\left(\omega\lrcorner\right)$ $\displaystyle=$
$\displaystyle\left(-1\right)^{k+1}\left(\omega\wedge\right)*_{\mathcal{F}}$
${}_{\mathcal{F}}d_{\mathcal{F}}^{*}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{k}d_{\mathcal{F}}*_{\mathcal{F}}$
$\displaystyle d_{\mathcal{F}}^{*}*_{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{k+1}*_{\mathcal{F}}d_{\mathcal{F}}.$

The adjoint of the leafwise differential
$\left(d_{\mathcal{F}}+\omega\wedge\right)$ is
$\left(d_{\mathcal{F}}^{*}+\omega\lrcorner\right)$. We denote the Laplacian
corresponding to the differential $d_{\mathcal{F}}+\omega\wedge$ by
$\Delta_{\mathcal{F},\omega}$. Then

$\Delta_{\mathcal{F},\omega}=\left(d_{\mathcal{F}}+\omega\wedge\right)\left(d_{\mathcal{F}}^{*}+\omega\lrcorner\right)+\left(d_{\mathcal{F}}^{*}+\omega\lrcorner\right)\left(d_{\mathcal{F}}+\omega\wedge\right).$

######  Proposition 4.5.

If $\omega\in\Omega^{0,1}\left(M,\mathcal{F}\right)$ is a leafwise closed
$1$-form, then the Hodge star operator $*_{\mathcal{F}}$ satisfies

$*_{\mathcal{F}}\Delta_{\mathcal{F},\omega}=\Delta_{\mathcal{F},-\omega}*_{\mathcal{F}}.$

###### Proof.

For all $\beta\in\Omega^{0,k}\left(M,\mathcal{F}\right)$, we have

$\displaystyle*_{\mathcal{F}}\Delta_{\mathcal{F},\omega}\beta=*_{\mathcal{F}}(d_{\mathcal{F}}+\omega\wedge)(d^{*}_{\mathcal{F}}+\omega\lrcorner)\beta+*_{\mathcal{F}}(d^{*}_{\mathcal{F}}+\omega\lrcorner)(d_{\mathcal{F}}+\omega\wedge)\beta$
$\displaystyle=\left(-1\right)^{k}(d^{*}_{\mathcal{F}}-\omega\lrcorner)*_{\mathcal{F}}(d^{*}_{\mathcal{F}}+\omega\lrcorner)\beta+\left(-1\right)^{k+1}(d_{\mathcal{F}}-\omega\wedge)*_{\mathcal{F}}(d_{\mathcal{F}}+\omega\wedge)\beta$
$\displaystyle=\left(-1\right)^{k}\left(-1\right)^{k}(d^{*}_{\mathcal{F}}-\omega\lrcorner)(d_{\mathcal{F}}-\omega\wedge)*_{\mathcal{F}}\beta+\left(-1\right)^{k+1}\left(-1\right)^{k+1}(d_{\mathcal{F}}-\omega\wedge)(d^{*}_{\mathcal{F}}-\omega\lrcorner)*_{\mathcal{F}}\beta$
$\displaystyle=\left((d^{*}_{\mathcal{F}}-\omega\lrcorner)(d_{\mathcal{F}}-\omega\wedge)+(d_{\mathcal{F}}-\omega\wedge)(d^{*}_{\mathcal{F}}-\omega\lrcorner)\right)*_{\mathcal{F}}\beta$
$\displaystyle=\Delta_{\mathcal{F},-\omega}*_{\mathcal{F}}\beta.$

Thus the operator $*_{\mathcal{F}}$ maps
$\Delta_{\mathcal{F},\omega}$-harmonic forms to
$\Delta_{\mathcal{F},-\omega}$-harmonic forms. ∎

######  Corollary 4.6.

If we restrict the Laplacian $\Delta_{\mathcal{F},\omega}$ on
$\Omega^{0,v}\left(M,\mathcal{F}\right)$, then
$\ker\Delta_{\mathcal{F},\omega}$ is finite dimensional, and every reduced
leafwise Morse-Novikov cohomology class has a $\Delta_{\mathcal{F},\omega}$
harmonic representative.

###### Proof.

Notice that the operator $\Delta_{\mathcal{F},\omega}$ is defined on all forms
in $\Omega^{u,v}\left(M,\mathcal{F}\right)$, but it is elliptic when
restricted on the forms $\Omega^{0,v}\left(M,\mathcal{F}\right)$ along the
leaves of the foliation (Section 1 in [1]). Using this ellipticity and the
arguments similar to [1], we can conclude that
$\mathcal{H}^{k}_{\omega}\left(M,d_{\mathcal{F}}\right)=\ker\Delta_{\mathcal{F},\omega}\subset\Omega^{0,k}\left(M,\mathcal{F}\right)$
is isomorphic to $\bar{H}_{\omega}^{k}\left(M,d_{\mathcal{F}}\right)$, and

$\mathcal{H}^{k}_{\omega}\left(M,d_{\mathcal{F}}\right)\cong\bar{H}_{\omega}^{k}\left(M,d_{\mathcal{F}}\right).$

∎

######  Corollary 4.7.

$\bar{H}^{k}_{\omega}\left(M,d_{\mathcal{F}}\right)\cong\bar{H}_{-\omega}^{p-k}\left(M,d_{\mathcal{F}}\right)$.

###### Proof.

Since the operator $*_{\mathcal{F}}$ maps
$\Delta_{\mathcal{F},\omega}$-harmonic forms to
$\Delta_{\mathcal{F},-\omega}$-harmonic forms, it induces the isomorphism

$\mathcal{H}^{k}_{\omega}\left(M,d_{\mathcal{F}}\right)\cong\mathcal{H}_{-\omega}^{p-k}\left(M,d_{\mathcal{F}}\right).$

∎

##  5\. Extension of leafwise Morse-Novikov cohomology to forms of general
$u,v$ type

We now extend leafwise Morse-Novikov cohomology to forms of general $u,v$
type. Let $\omega\in\Omega^{0,1}\left(M,\mathcal{F}\right)$ be a leafwise
closed $1$-form which is not necessarily exact. We consider the twisted
operator
$d^{\omega}_{\mathcal{F}}:\Omega^{u,v}\left(M,\mathcal{F}\right)\rightarrow\Omega^{u,v+1}\left(M,\mathcal{F}\right)$
defined by $d^{\omega}_{\mathcal{F}}=d_{\mathcal{F}}+\omega\wedge$, where
$d_{\mathcal{F}}$ is the exterior derivative along the leaf.

######  Proposition 5.1.

$\left(d_{\mathcal{F}}+\omega\wedge\right)^{2}=0.$ Therefore
$d_{\mathcal{F}}+\omega\wedge$ is a differential of the sections of
$\Omega^{u,v}\left(M,\mathcal{F}\right)$.

###### Proof.

Observe that for any section
$\alpha\wedge\beta\in\Omega^{u,v}\left(M,\mathcal{F}\right)$, we have

$\displaystyle\left(d_{\mathcal{F}}+\omega\wedge\right)^{2}\left(\alpha\wedge\beta\right)=\left(d_{\mathcal{F}}+\omega\wedge\right)\left((-1)^{u}\alpha\wedge
d_{\mathcal{F}}\beta+\omega\wedge\left(\alpha\wedge\beta\right)\right)$
$\displaystyle=d_{\mathcal{F}}\left((-1)^{u}\alpha\wedge
d_{\mathcal{F}}\beta\right)+(-1)^{u+1}\omega\wedge\left(\alpha\wedge
d_{\mathcal{F}}\beta\right)+(-1)^{u}\left(\omega\wedge\left(\alpha\wedge
d_{\mathcal{F}}\beta\right)\right)+\omega\wedge\left(\omega\wedge\left(\alpha\wedge\beta\right)\right)$
$\displaystyle=(-1)^{u+1}\left(\omega\wedge\left(\alpha\wedge
d_{\mathcal{F}}\beta\right)-\omega\wedge\left(\alpha\wedge
d_{\mathcal{F}}\beta\right)\right)=0.$

∎

We call the differential cochain complex
$\left(\Omega^{*,*}\left(M,\mathcal{F}\right),d^{\omega}_{\mathcal{F}}\right)$
the general leafwise Morse-Novikov complex of the foliated manifold
$\left(M,\mathcal{F}\right)$. The cohomology groups

$H^{*,*}_{\omega}\left(M,d_{\mathcal{F}}\right)=\frac{\ker\left(d_{\mathcal{F}}^{\omega}\right)}{\mathrm{im}\left(d_{\mathcal{F}}^{\omega}\right)}$

of this cochain complex are called the general leafwise Morse-Novikov
cohomology groups of $\left(M,\mathcal{F}\right)$. For the purpose of having
Laplacian and Hodge decomposition, we need to consider reduced general
leafwise Morse-Novikov cohomology

$\bar{H}^{*,*}_{\omega}\left(M,d_{\mathcal{F}}\right)=\frac{\ker\left(d_{\mathcal{F}}^{\omega}\right)}{\overline{\mathrm{im}\left(d_{\mathcal{F}}^{\omega}\right)}}.$

Again the closure
$\overline{\mathrm{im}\left(d_{\mathcal{F}}^{\omega}\right)}$ is taken with
respect to the Frechét topology on $\Omega^{*,*}\left(M,\mathcal{F}\right)$.

######  Proposition 5.2.

If $\omega$ and $\theta=\omega+d_{\mathcal{F}}g$ are cohomologous in
$H^{1}\left(M,d_{\mathcal{F}}\right)$, then for each $\ell,k$, the general
leafwise Morse-Novikov cohomology groups
$H_{\omega}^{\ell,k}\left(M,d_{\mathcal{F}}\right)$ and
$H_{\theta}^{\ell,k}\left(M,d_{\mathcal{F}}\right)$ are isomorphic via the
isomorphism $\left[\alpha\right]\mapsto\left[e^{-g}\alpha\right]$.

###### Proof.

Similar to the proof of Proposition 3.3. ∎

######  Proposition 5.3.

(Homotopy axiom for the general leafwise Morse-Novikov cohomology). Let
$f:\left(M,\mathcal{F}_{0}\right)\rightarrow\left(N,\mathcal{F}_{1}\right)$
and
$g:\left(M,\mathcal{F}_{0}\right)\rightarrow\left(N,\mathcal{F}_{1}\right)$ be
foliated homotopic maps, and
$\omega\in\Omega^{0,1}\left(N,\mathcal{F}_{1}\right)$ be a leafwise closed
$1-\text{form}$ on $\left(N,\mathcal{F}_{1}\right)$. Then there exists a
positive function $h:\left(M,\mathcal{F}_{0}\right)\rightarrow\mathbb{R}$ such
that, for all $\ell,k$

$f^{*}=hg^{*}:H^{\ell,k}_{\omega}\left(N,d_{\mathcal{F}_{1}}\right)\rightarrow
H^{\ell,k}_{f^{*}\omega}\left(M,d_{\mathcal{F}_{0}}\right).$

###### Proof.

Similar to the proof of Proposition 3.7. ∎

##  6\. Leafwise Morse-Novikov Hodge Theory of forms of general
$\left(u,v\right)$ type

Let $P=d_{\mathcal{F}}+\omega\wedge$, then its formal adjoint $P^{*}$ is
$\delta_{\mathcal{F}}-\omega\lrcorner$. Let $D=P+P^{*}$ be the corresponding
Dirac operator. Then the corresponding Laplacian
$\Delta_{\mathcal{F},\omega}^{u,v}=\left(P+P^{*}\right)^{2}=PP^{*}+P^{*}P$ is
a nonnegative, self-adjoint second order differential operator on the smooth
sections on $\Omega^{u,v}\left(M,\mathcal{F}\right)$. For each integer $k\geq
0$, let $H_{k}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)$ be the Hilbert
space completion of the space $\Omega^{u,v}\left(M,\mathcal{F}\right)$ with
respect to the scalar product

$\left\langle\alpha,\beta\right\rangle=\sum_{j=0}^{j=k}{\left\langle\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)^{j}\alpha,\beta\right\rangle}$

for $\alpha,\beta\in\Omega^{u,v}\left(M,\mathcal{F}\right)$. For the
corresponding norm $\left\|.\right\|_{k}$, we have

$k\leq
k^{\prime}\Rightarrow\left\|\alpha\right\|_{k}\leq\left\|\alpha\right\|_{k^{\prime}}\text{
for all }\alpha\in\Omega^{u,v}\left(M,\mathcal{F}\right).$

Thus we obtain the chain of continuous inclusions

$H=H_{0}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)\supset
H_{1}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)\supset
H_{2}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)\supset\cdots\supset
H_{\infty}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right),$

where

$H_{\infty}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)=\bigcap_{k\geq
0}H_{k}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)$

equipped with the Frechét topology.

######  Theorem 6.1.

Let $\left(M,\mathcal{F}\right)$ be a smooth foliation of a closed Riemannian
manifold with a bundle like metric and
$\omega\in\Omega^{0,1}\left(M,\mathcal{F}\right)$. The Laplacian
$\Delta_{\mathcal{F},\omega}^{u,v}$ on
$\Omega^{u,v}\left(M,\mathcal{F}\right)$ gives rise to an orthogonal direct
sum decomposition

$H_{\infty}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)\cong\ker\overline{\Delta_{\mathcal{F},\omega,\infty}^{u,v}}\oplus\overline{\mathrm{im}\overline{\Delta_{\mathcal{F},\omega,\infty}^{u,v}}}\cong\ker\overline{\Delta_{\mathcal{F},\omega,\infty}^{u,v}}\oplus\overline{\mathrm{im}\overline{P_{\infty}}}\oplus\overline{\mathrm{im}\overline{P^{*}_{\infty}}},$

where $\overline{\Delta_{\mathcal{F},\omega,\infty}^{u,v}}$,
$\overline{P_{\infty}}$, and $\overline{P^{*}_{\infty}}$ are canonical
continuous extensions of the corresponding differential operators.

###### Proof.

The complexification of the Dirac operator $D=P+P^{*}$ satisfies the
hypothesis of Chernoff’s Lemma 2.1 in [6]. This can be verified from Corollary
1.4 of [6]. Then with the ideas explained in Section 2 of [3], we have the
real Hilbert spaces $H_{k}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)$ and
$H_{\infty}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)$. We can extend the
operator $D$ to $\overline{D_{\infty}}$ and
$\Delta_{\mathcal{F},\infty}^{\omega}$ to

$\overline{\Delta_{\mathcal{F},\omega,\infty}^{u,v}}:H_{\infty}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)\rightarrow
H_{\infty}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right),$

yielding the orthogonal decompositions

$H_{\infty}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)\cong\ker\overline{\Delta_{\mathcal{F},\omega,\infty}^{u,v}}\oplus\overline{\mathrm{im}\overline{\Delta_{\mathcal{F},\omega,\infty}^{u,v}}}\cong\ker\overline{D_{\infty}}\oplus\overline{\mathrm{im}\overline{D_{\infty}}}.$

Notice the spaces $\Omega^{p,q}\left(M,\mathcal{F}\right)$ are orthogonal to
each other with respect to the inner product $\left\langle,\right\rangle_{k}$
defined above, for eack $k\geq 0$. Therefore, it follows that
$\overline{D_{\infty}}$ can be decomposed as the sum of the continuous
operators

$\overline{P_{\infty}},\overline{P_{\infty}^{*}}:H_{\infty}\rightarrow
H_{\infty},$

which are extensions of $P_{\infty}$ and $P_{\infty}^{*}$ respectively. Since
$\mathrm{im}P$ and $\mathrm{im}P^{*}$ are
$\left\langle,\right\rangle_{k}$-orthogonal for each $k\geq 0$, we obtain the
following orthogonal decomposition:

$H_{\infty}\left(\Delta_{\mathcal{F},\omega}^{u,v}\right)\cong\ker\overline{\Delta_{\mathcal{F},\omega,\infty}^{u,v}}\oplus\overline{\mathrm{im}\overline{P_{\infty}}}\oplus\overline{\mathrm{im}\overline{P_{\infty}^{*}}}.$

∎

######  Corollary 6.2.

Every reduced general leafwise Morse-Novikov cohomology class has a
$\Delta_{\mathcal{F},\omega}^{u,v}$-harmonic representative.

For any $\alpha\wedge\beta\in\Omega^{u,v}\left(M,\mathcal{F}\right)$, from
formula 17 in [3] we have
$\delta_{\mathcal{F}}=(-1)^{n(u+v)+n+1}*d_{\mathcal{F}}*$. By using the
identities

$\displaystyle\omega\lrcorner=(-1)^{n(u+v)+n}*\omega\wedge*$
$\displaystyle\text{ and }*^{2}=(-1)^{(u+v)(n+1)},$

it can be shown that

$\displaystyle\left(\omega\lrcorner\right)*=\left(-1\right)^{u+v}*\left(\omega\wedge\right).$
$\displaystyle\left(\omega\lrcorner\right)=\left(-1\right)^{u+v+1}\left(\omega\wedge\right)*.$
$\displaystyle\delta_{\mathcal{F}}=\left(-1\right)^{u+v}d_{\mathcal{F}}*.$
$\displaystyle\delta_{\mathcal{F}}*=\left(-1\right)^{u+v+1}*d_{\mathcal{F}}.$

######  Proposition 6.3.

Let $\left(M,\mathcal{F}\right)$, and $\omega$ be as in Theorem 6.1. The Hodge
operator
$*:\Omega^{u,v}\left(M,\mathcal{F}\right)\rightarrow\Omega^{q-u,p-v}\left(M,\mathcal{F}\right)$
satisfies

$*\Delta_{\mathcal{F},\omega}^{u,v}=\Delta_{\mathcal{F},-\omega}^{q-u,p-v}*.$

###### Proof.

Similar to Proposition 4.5, using the formulas above. Thus the operator $*$
maps $\Delta_{\mathcal{F},\omega}^{u,v}$-harmonic forms to
$\Delta_{\mathcal{F},-\omega}^{q-u,p-v}$-harmonic forms. ∎

######  Lemma 6.4.

$\omega\lrcorner=(-1)^{p\left(v+1\right)}*_{\mathcal{F}}\left(\omega\wedge\right)*_{\mathcal{F}}$
for all $\omega\in\Omega^{0,1}\left(M,\mathcal{F}\right)$ on
$\Omega^{u,v}\left(M,\mathcal{F}\right)$.

###### Proof.

For any $\alpha\wedge\beta\in\Omega^{u,v}\left(M,\mathcal{F}\right)$, we have

$\displaystyle\omega\lrcorner\left(\alpha\wedge\beta\right)$ $\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(u+v+1\right)}*\left(\omega\wedge\right)*\left(\alpha\wedge\beta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(u+v+1\right)}\left(-1\right)^{\left(q-u\right)v}\left(-1\right)^{\left(q-u\right)}*\left(*_{\bot}\alpha\wedge\omega\wedge*_{\mathcal{F}}\beta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(u+v+1\right)}\left(-1\right)^{\left(q-u\right)v}\left(-1\right)^{\left(q-u\right)}\left(-1\right)^{\left(q-q+u\right)\left(p-v+1\right)}*_{\bot}^{2}\alpha\wedge*_{\mathcal{F}}\left(\omega\wedge*_{\mathcal{F}}\beta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(u+v+1\right)}\left(-1\right)^{\left(q-u\right)v}\left(-1\right)^{\left(q-u\right)}\left(-1\right)^{\left(u\right)\left(p-v+1\right)}\left(-1\right)^{u\left(q+1\right)}\alpha\wedge*_{\mathcal{F}}\left(\omega\wedge*_{\mathcal{F}}\beta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(u+v+1\right)}\left(-1\right)^{\left(q-u\right)v}\left(-1\right)^{\left(q-u\right)}\left(-1\right)^{\left(u\right)\left(p-v+1\right)}\left(-1\right)^{u\left(q+1\right)}\left(-1\right)^{u\left(v-1\right)}*_{\mathcal{F}}\left(\omega\wedge\right)*_{\mathcal{F}}\beta\wedge\alpha$
$\displaystyle=$
$\displaystyle\left(-1\right)^{p\left(v+1\right)+uv}\left(-1\right)^{uv}*_{\mathcal{F}}\left(\omega\wedge\right)*_{\mathcal{F}}\alpha\wedge\beta$
$\displaystyle=$
$\displaystyle\left(-1\right)^{p\left(v+1\right)}*_{\mathcal{F}}\left(\omega\wedge\right)*_{\mathcal{F}}\alpha\wedge\beta.$

∎

In a Riemannian foliation, any element of $\Omega^{u,v}(M,\mathcal{F})$, on a
local foliated chart, can be expressed as a linear combination of forms of the
type $\alpha\wedge\beta$, where $\alpha$ is a basic form in
$\Omega^{u,0}(M,\mathcal{F})$ and $\beta$ is a form in
$\Omega^{0,v}(M,\mathcal{F})$. Then in particular $d_{\mathcal{F}}\alpha=0$.
See [1, Lemma 3.4].

######  Lemma 6.5.

$d_{\mathcal{F}}^{*}=\left(-1\right)^{p\left(v+1\right)+1}*_{\mathcal{F}}d_{\mathcal{F}}*_{\mathcal{F}}$
for all on $\Omega^{u,v}\left(M,\mathcal{F}\right)$.

###### Proof.

For any $\alpha\wedge\beta\in\Omega^{u,v}\left(M,\mathcal{F}\right)$, we have

$\displaystyle d_{\mathcal{F}}^{*}\left(\alpha\wedge\beta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(u+v\right)+n+1}*d_{\mathcal{F}}*\left(\alpha\wedge\beta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(u+v\right)+n+1+\left(q-u\right)\left(v+1\right)}*\left(*_{\perp}\alpha\wedge
d_{\mathcal{F}}*_{\mathcal{F}}\beta\right)$ $\displaystyle=$
$\displaystyle\left(-1\right)^{n\left(u+v\right)+n+1+\left(q-u\right)\left(v+1\right)+u\left(p-v+1\right)+u(q+1)}\left(\alpha\wedge\left(*_{\mathcal{F}}d_{\mathcal{F}}*_{\mathcal{F}}\beta\right)\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{p\left(v+1\right)+u+1}\left(\alpha\wedge\left(*_{\mathcal{F}}d_{\mathcal{F}}*_{\mathcal{F}}\beta\right)\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{p\left(v+1\right)+u+1+u\left(v-1\right)}\left(*_{\mathcal{F}}d_{\mathcal{F}}*_{\mathcal{F}}\beta\right)\wedge\alpha$
$\displaystyle=$
$\displaystyle\left(-1\right)^{p\left(v+1\right)+1}\left(*_{\mathcal{F}}d_{\mathcal{F}}*_{\mathcal{F}}\right)\alpha\wedge\beta.$

∎

By Lemmas above and the identity
$*_{\mathcal{F}}^{2}=\left(-1\right)^{v\left(p-v\right)}$, we have on
$\Omega^{u,v}\left(M,\mathcal{F}\right)$

$\displaystyle\left(\omega\lrcorner\right)*_{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{p(p-v+1)}\left(\omega\lrcorner\right)*_{\mathcal{F}}^{2}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{P(p-v+1)+v(p-v)}*_{\mathcal{F}}\left(\omega\wedge\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{v}*_{\mathcal{F}}\left(\omega\wedge\right),$

and

$\displaystyle*_{\mathcal{F}}\left(\omega\lrcorner\right)$ $\displaystyle=$
$\displaystyle\left(-1\right)^{p\left(v+1\right)}*_{\mathcal{F}}^{2}\left(\omega\wedge\right)*_{\mathcal{F}}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{p\left(v+1\right)}\left(-1\right)^{\left(p-v\right)\left(v+1\right)}\left(\omega\wedge\right)*_{\mathcal{F}}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{\left(v+1\right)}\left(\omega\wedge\right)*_{\mathcal{F}}.$

Similarly on $\Omega^{u,v}\left(M,\mathcal{F}\right)$ we also have

$\displaystyle*_{\mathcal{F}}d_{\mathcal{F}}^{*}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{v}d_{\mathcal{F}}*_{\mathcal{F}}$
$\displaystyle d_{\mathcal{F}}^{*}*_{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{v+1}*_{\mathcal{F}}d_{\mathcal{F}}.$

######  Proposition 6.6.

Let $\left(M,\mathcal{F}\right)$, and $\omega$ be as in Theorem 6.1. The Hodge
operator
$*_{\mathcal{F}}:\Omega^{u,v}\left(M,\mathcal{F}\right)\rightarrow\Omega^{u,p-v}\left(M,\mathcal{F}\right)$
satisfies

$*_{\mathcal{F}}\Delta_{\mathcal{F},\omega}^{u,v}=\Delta_{\mathcal{F},-\omega}^{u,p-v}*_{\mathcal{F}}.$

###### Proof.

Similar to Proposition 4.5, using the formulas above. Thus the operator
$*_{\mathcal{F}}$ maps $\Delta_{\mathcal{F},\omega}^{u,v}$-harmonic forms to
$\Delta_{\mathcal{F},-\omega}^{u,p-v}$-harmonic forms. ∎

Notice on $\Omega^{u,v}\left(M,\mathcal{F}\right)$, we have

$\displaystyle**_{\mathcal{F}}=(-1)^{(q-u)(p-v)}*_{\perp}*_{\mathcal{F}}*_{\mathcal{F}}$
$\displaystyle*_{\mathcal{F}}=(-1)^{(q-u)(p-v)+v(p-v)}*_{\perp}$
${}_{\perp}=(-1)^{(q-u)(p-v)+v(p-v)}**_{\mathcal{F}}.$

From the Propositions 6.3and 6.5 we have,

######  Corollary 6.7.

Let $\left(M,\mathcal{F}\right)$, and $\omega$ be as in Theorem 6.1. The Hodge
operator
$*_{\perp}:\Omega^{u,v}\left(M,\mathcal{F}\right)\rightarrow\Omega^{q-u,v}\left(M,\mathcal{F}\right)$
satisfies

$*_{\perp}\Delta_{\mathcal{F},\omega}^{u,v}=\pm\Delta_{\mathcal{F},\omega}^{q-u,v}*_{\perp}.$

###### Proof.

$\displaystyle*_{\perp}\Delta_{\mathcal{F},\omega}^{u,v}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{v(n-u-v)}**_{\mathcal{F}}\Delta_{\mathcal{F},\omega}^{u,v}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{v(n-u-v)}*\Delta_{\mathcal{F},-\omega}^{u,p-v}*_{\mathcal{F}}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{v(n-u-v)}\Delta_{\mathcal{F},\omega}^{q-u,v}**_{\mathcal{F}}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{v(n-u-v)}\left(-1\right)^{(q-u)(p-v)+v(p-v)}\Delta_{\mathcal{F},\omega}^{q-u,v}*_{\perp}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{p(q-u)}\Delta_{\mathcal{F},\omega}^{q-u,v}*_{\perp}.$

∎

Let $h^{u,v}_{\omega}$, $h^{u,v}_{-\omega}$ be the dimensions of the
Cohomology groups $H_{\omega}^{u,v}\left(M,d_{\mathcal{F}}\right)$ , and
$H_{-\omega}^{u,v}\left(M,d_{\mathcal{F}}\right)$ respectively. We observe
that, Proposition 6.3 implies, $h^{u,v}_{\omega}$=$h^{q-u,p-v}_{-\omega}$,
Proposition 6.6 implies, $h^{u,v}_{\omega}$=$h^{u,p-v}_{-\omega}$, and
Corollary 6.7 implies, $h^{u,v}_{\omega}$=$h^{q-u,v}_{\omega}$.

######  Corollary 6.8.

( Hodge Diamond Structure )For a Riemannian foliation of a manifold

$h^{u,v}_{\omega}=h^{q-u,p-v}_{-\omega}=h^{u,p-v}_{-\omega}=h^{q-u,v}_{\omega}.$

In particular, we consider a diagram of the dimensions of the cohomology
classes for a manifold of dimension $n=5$ with $p=2$, and $q=3$.

The dimensions in the same color are equal.

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