# A cooperate-defect model for the spread of deviant behavior in social
networks

Sarah Rajtmajer∗, Christopher Griffin∗†, Derek Mikesell†, Anna Squicciarini+
Department of Mathematics∗, College of Information Sciences and Technology+
Applied Research Laboratory†
The Pennsylvania State University
University Park, PA, USA

###### Abstract

We present a game-theoretic model for the spread of deviant behavior in online
social networks. We utilize a two-strategy framework wherein each player’s
behavior is classified as normal or deviant and evolves according to the
cooperate-defect payoff scheme of the classic prisoner’s dilemma game. We
demonstrate convergence of individual behavior over time to a final strategy
vector and indicate counterexamples to this convergence outside the context of
prisoner’s dilemma. Theoretical results are validated on a real-world dataset
collected from a popular online forum.

##  I Introduction

Commenting systems on the Social Web have grown in popularity over the past
few years, from blogs and social media sites like YouTube and Flickr to major
news sites like NYTimes.com.

Following this trend, episodes of abuse online are proliferating [4, 7, 22,
19]. Generally, any behavior that is ”antisocial”, destructive, negative and
offensive is considered as abusive, although abusive behavior comes in many
forms ranging from minor to extremely harmful. Some of such forms include
grieving, trolling, flaming, harassment, threats trolling, multiple accounts,
shared accounts, advertising, plagiarism etc.

Users’ reactions to antisocial peers may range drastically, based on their
personality traits, predisposition toward certain behaviors, level of
influence, and other contextual factors. Users may choose to engage in the
discussion by mimicking the antisocial peer, confronting him and possibly
reporting the behavior to superusers, simply ignoring the negative influence,
or even leaving the network. Over time, many popular real-world communities
such as Reddit [23] have suffered damaging effects resulting from those few
participants that choose to behave in a manner that is counter to established
norms of behavior.

To date, despite a significant amount of work on the detection of social
spammers, deception [3], collective attention spam [19, 28, 18, 12, 25] and
online vandalism, the dynamics underlying online abusive behavior remain
uncertain [26].

Toward developing a better understanding of this phenomenon, in this paper, we
aim to build mathematical models to investigate the patterns of persuasion in
online social networks, with emphasis on antisocial behavior. We design an
evolutionary game over a social network graph to describe how users interact
with, influence and are influenced by both cooperative and antisocial
behavior. During the game, members of the community may behave cooperatively,
i.e. participating constructively in the network, or defectively, i.e.
antisocial behavior. Each pairwise interaction between users yields a distinct
payoff to each user, according to the extent of cooperation or defection
exhibited by both in that interaction. At each iteration of game, every user
observes his own total payoff as well as the actions and payoffs of his peers.
We allow users to change their behavior accordingly, and we posit that this
change in behavior will generally serve to mimic peers with observed higher
payoff.

To the best of our knowledge, this is the first effort investigating this
class of models to explain the spread of online abusive behavior.

We validate some of our findings on a real-world dataset collected from the
web. Precisely, we model the interactions amongst users who are part of a
threaded online community and discuss topics of interest with other peers.
Each user post is assessed and labelled on a scale from most cooperative to
most deviant. The trajectories of these scores are then tracked within the
context of the social network graph and compared against expected values of
our theoretical evolutionary game.

Following we present a summary of relevant, related work in the literature, as
it compares to our present study. We outline formally the problem we study and
the game theoretic framework we have established, and prove some important
theoretical properties of this game for both the special class of complete
graphs and in the general case. Finally, we compare our expectations with
measured deviance on a real-world online dataset.

##  II Related Work

Our work relates to the body of work on online deviance as it has been
investigated by the computer science and by mathematical modeling approaches.

Several recent works focus on detecting social spammers [28, 19, 18, 12, 25,
17]. Social spammers, according to this body of work, are users controlled
either by humans or bots, who use social networking sites and in particular
their social connections to promote products, advertise events, or simply post
useless and/or inappropriate comments. Lee et al. [19] studied social spammers
in online social networks. To do so, they deployed social honeypots for
harvesting deceptive spam profiles from social networking communities, and
created spam classifiers using machine learning methods (e.g., SVM) based on a
variety of features. Similarly, Kantchelian [18] developed an approach for
detecting comment spam by leveraging the informativeness level of a comment;
he showed that spammers’ comments have low information levels. Hu et al. [12]
also proposed a comprehensive framework for social spam detection, based on
social network content, attaining a high accuracy of detection. Our focus is
not on social spammers only but rather on activities related to vandalism and
misbehavior, which may or may not be generated by social spammers.

To deal with vandalism and bots, several tools exist (e.g. [2, 24]). Automated
bots (e.g., Cluebot), filters (e.g., abusefilter), and editing assistants
(e.g., Huggle and Twinkle) all aim to locate acts of vandalism. Such tools
work via regular expressions and manually-authored rule sets. Though useful,
these systems are unable to explain or predict where instances of abusive and
deviant behaviors will occur, and are often limited to language detection.

Finally, our work parallels the body of work on free-riding in peer-to-peer
systems [6, 29]. Peer-to-peer systems are designed to allow users to connect
with others and share resources. Similar to online communities, users are free
to access and contribute as much as desired, and few controls are in place. As
a result, in p2p systems peers may abuse their connections by exploiting other
peers’ resources, refusing to share owned resources, sharing broken or
corrupted resources, etc., draining the network without contributing it. In
online communities, the health of the community is heavily based on individual
peers’ reactions to selfish behavior, which they may choose to emulate or
disengage from. Punishment mechanisms can also be put in place, although these
are often considered not to be truly effective. To tackle these issues, the
most common solution is the implementation of incentive-based mechanisms.
Incentives are applied in certain online forums, whereby end users are given
special roles and privileges as a reward for good behavior.

From a theoretical perspective, our work can be placed in the context of the
DeGroot model [5] in which an individual changes her opinion dynamically and
in part through imitation. In that model, a discrete time Markov chain forms
the underlying behavior. From a game theoretic perspective, Morris [21]
studies behavioral contagion in coordination games. Our work in this paper
considers Prisoner’s dilemma, but our main result (Proposition 5.1) is
applicable to a broader class of games. In addition to this, our work is
inspired by work by the work of Jackson et al. [16, 13, 14, 15] when we
consider the proposed model on the presence of network changes [8].

In this work, we propose a model for the spread of deviance in online social
networks. We do not propose mechanisms for punishment and reward of deviant
behavior, but instead we model the inherent gain or loss for a user in the
network who exhibits antisocial, deviant behavior, and accordingly, how this
behavior as well as perceived gain or loss resulting from this behavior,
effects the strategies of other users in the system over time.

##  III Problem Statement

Consider a network of users represented as a graph, wherein users are
represented as nodes in the system and edges may represent stated friendships,
recorded interactions or generally any indication that those two users have
some influence on one another. Each player is assigned an initial probability
of choosing to cooperate during each interaction which we refer to as the
player’s initial _strategy_. In an online SN, the initial strategy represents
individual predisposition for cooperation, separate from the effects of peer
influence. 111 In practice, initial strategies may be assigned to each user
randomly, from a distribution of strategies over the entire population. Or, if
available, prior indicators of individual user behavior may be utilized to
provide a better estimate.

From this initial set of strategies, we introduce an evolutionary game wherein
pairwise interactions amongst users yield a distinct payoff to each player,
according to some payoff matrix. A player’s total payoff at each iteration is
given as the sum of his payoffs over all of his pairwise interactions. In our
study, we adopt a prisoner’s dilemma (PD)-type payoff matrix, wherein users
may choose to cooperate or defect at each interaction and payoffs follow
according to a canonical PD matrix. In the context of detecting deviant
behavior online, the defector like the deviant user in a social network aims
to influence or take advantage of a cooperative user and hence gains greatest
benefit by an interaction with one such user. The cooperative user, on the
other hand, suffers a negative payoff in this interaction, while the
cooperate-cooperate and defect-defect interactions are more neutral.

We assume that each player observes his own payoff and strategy at each
iteration of the game, as well as the total payoff and the strategy for each
of his neighbors. At each iteration of the game, a player may choose to change
his strategy by mimicking successful strategies in his neighborhood,
proportionally to their relative success. This play/strategy revision
procedure is repeated for some fixed number of iterations or until convergence
(guaranteed, see below). We assume a static graph in this work, and examine a
dynamic graph in a follow-up study [8].

##  IV Preliminaries

Consider our model of behavioral evolution on a graph $G=(V,E)$ on $n$ nodes.
Let $\mathbf{A}\in\mathbb{R}^{2\times 2}$ be a payoff matrix. Following the
prisoner’s dilemma framework, we let

$\mathbf{A}=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]$

where $c>a>d>0>b$.

If $i\in V$, let $x_{i}(t)\in[0,1]$ be the probability that player $i$ will
play her first strategy at time (iteration) $t$. Thus, the strategy vector for
Player $i$ is:

$\mathbf{x}_{i}=\begin{bmatrix}x_{i}\\\ 1-x_{i}\end{bmatrix}$

The payoff to Player $i$ is:

$P_{i}=\sum_{j\in N(i)}\mathbf{x}_{i}^{T}\mathbf{A}\mathbf{x}_{j}$

where $N(i)$ is the neighborhood of $i$ in $G$. In this paper, we will
consider only two-strategy games; therefore let $\mathbf{x}$ be the vector of
$x_{i}$ values. Define:

$\kappa_{ij}(\mathbf{x}):=\\\
\frac{\mathcal{H}\left(P_{j}(t)-P_{i}(t)\right)\left(P_{j}(t)-P_{i}(t)\right)}{\sum_{k\in
N(i)}\mathcal{H}\left(P_{k}(t)-P_{i}(t)\right)\left(P_{k}(t)-P_{i}(t)\right)}$
(1)

where $\mathcal{H}$ the Heaviside step function (defined as 0 at 0). In the
case when $P_{i}(t)=P_{j}(t)$ for all $j\in N(i)$, let
$\kappa_{ij}(\mathbf{x})=0$. Note that $\sum_{j}\kappa_{ij}(\mathbf{x})=1$,
just in case $\kappa_{ij}(\mathbf{x})\neq 0$ for all $j\in N(i)$. Let:

$f_{i}(\mathbf{x})=\sum_{j\in N(i)}\kappa_{ij}(t)(x_{j}(t)-x_{i}(t))$ (2)

The strategy update rule is given by

$x_{i}(t+\epsilon)=x_{i}(t)+\epsilon f_{i}(\mathbf{x})$ (3)

Here: $\epsilon>0$ and $\epsilon\ll 1$. This rule is a proportional success
mimicking rule in which players will drift toward (imitate) successful
behaviors.

##  V Convergence of Players’ Strategies to a Final Strategy Vector

We examine the trajectory of individual users’ strategies, assuming no changes
in network structure. The addition or deletion of network ties being more
infrequent than small shifts in individual behavior [11, 9], we here capture
those behavioral changes that precede a severing or creation of a structural
tie. We claim that in this case, assuming a static network structure, there
can be stable strategy equilibria. This holds for all network graphs, and we
show that in the special case of the complete graph this vector has the form

$\mathbf{x_{f}}=[x_{1_{f}},x_{2_{f}},\ldots,x_{n_{f}}]$

where $x_{1_{f}}=x_{2_{f}}=\ldots=x_{n_{f}}$.

###  V-A Equilibrium Points of Player Strategies on an Arbitrary Network

To study the fixed points of player strategies under the dynamics of
Expression 3 it is easier to pass to the continuous dynamics:

$\dot{x}_{i}=f_{i}(\mathbf{x})=\sum_{j\in
N(i)}\kappa_{ij}(\mathbf{x})(x_{j}-x_{i})$ (4)

There are three types of equilibria for the differential system defined by
Equation 4:

###### Definition 1

A Type 1 Equilibrium occurs when $x_{i}=x_{j}$ for all $i,j$. A Type 2
Equilibrium occurs when $P_{i}(\mathbf{x})=P_{j}(\mathbf{x})$ for all $i,j$
and there is at least one pair $i,j$ so that $x_{i}\neq x_{j}$. A Type 3
Equilibrium is any equilibrium not satisfying the conditions of Types 1 or 2.

Note, a Type 3 equilibria may contain groups (e.g., cliques) of vertices that
all share a strategy within group but have different strategies among the
groups. It is clear that every system of the type given by Equation 4 is
degenerate in the sense that it has an infinite number of equilibria. In this
case, it is similar to the SIR dynamics discussed in [10]. We show that under
certain conditions, Type 1 equilibria may be (at worst) neutrally stable. Note
first:

if $p=i$:

$\frac{\partial f_{i}}{\partial x_{p}}=\sum_{j\in
N(i)}\left\\{\frac{\partial\kappa_{ij}(\mathbf{x})}{\partial
x_{p}}(x_{j}-x_{i})-\kappa_{ij}(\mathbf{x})\right\\}$ (5)

if $p\in N(i)$:

$\frac{\partial f_{i}}{\partial x_{p}}=\sum_{j\in
N(i)}\left\\{\frac{\partial\kappa_{ij}(\mathbf{x})}{\partial
x_{p}}(x_{j}-x_{i})\right\\}+\kappa_{ij}(\mathbf{x})$ (6)

Otherwise:

$\frac{\partial f_{i}}{\partial x_{p}}=\sum_{j\in
N(i)}\frac{\partial\kappa_{ij}(\mathbf{x})}{\partial x_{p}}(x_{j}-x_{i})$ (7)

We are liberally abusing the derivative operator here, since
$\kappa_{ij}(\mathbf{x})$ is clearly not smooth. When we refer to its
derivative we refer to its derivative as a generalized function with
appropriate Dirac delta distributions used.

###### Proposition V.1

If $\mathbf{x}^{*}$ is an equilibrium point of Type 1 and $\sum_{j\in
N(i)}\kappa_{ij}(\mathbf{x}^{*})>0$ for at least one $i$, then
$\mathbf{x}^{*}$ is at worst neutrally stable.

###### Proof:

We make use of the face that $x\delta(x)=0$ in the generalized density
product, where $\delta(x)$ is the Dirac delta. From our observations and the
fact that $x_{i}=x_{j}$ for all $i,j$, we have that:

$\left.\frac{\partial f_{i}}{\partial
x_{p}}\right\rvert_{x=\mathbf{x}^{*}}=\begin{cases}-\sum_{j\in
N(i)}\kappa_{ij}(\mathbf{x}^{*})&\text{if $p=i$}\\\
\kappa_{ij}(\mathbf{x}^{*})&\text{if $p\in N(i)$}\\\
0&\text{otherwise}\end{cases}$ (8)

This implies:

$\left.\frac{\partial f_{i}}{\partial
x_{p}}\right\rvert_{x=\mathbf{x}^{*}}=\begin{cases}-1&\text{if $p=i$}\\\
\kappa_{ij}(\mathbf{x}^{*})&\text{if $p\in N(i)$}\\\
0&\text{otherwise}\end{cases}$ (9)

Applying the Gershgorin Disk Theorem, we know that every eigenvalue of the
Jacobian lies within a disk in the complex plane centered at $-1$ with radius
$1$. Thus, the real-parts of these eigenvalues are non-positive and the
resulting fixed point must be neutrally stable. ∎

###### Remark 1

It follows from Equation 1, that at least one row of the Jacobian matrix will
contain all zeros, since there is at least one vertex with greatest payoff.
Thus, at least one eigenvalue must be identically zero and the resulting Type
1 fixed points are never asymptotically stable; however they do exhibit a
basin of attraction as we illustrate in two examples.

###### Example 1

Consider the randomly generated graph with 10 vertices in Figure 1 (a). We use
the prisoner’s dilemma payoff matrix:

$\mathbf{A}=\begin{bmatrix}3&-7\\\ 5&2\end{bmatrix}$ (10)

Figure 1: (a)A random graph on 10 vertices used to illustrate the neutral
stability of Type 1 fixed points. (b) The weighted directed imitation graph at
the fixed point.

Any non-extreme (i.e., not all cooperating or defecting) Type 1 fixed point
will suffice for illustrate the stability properties of these fixed points.
Without loss of generality, will use the $x^{*}_{i}=0.795$ for $i=1,\dots,10$.
If we compute $\kappa_{ij}(\mathbf{x}^{*})$ for each pair, we can construct
the weighted directed imitation graph, shown in Figure 1 (b). Note, the
negative of the Laplacian for the imitation graph is the Jacobian matrix of
the dynamical system (when we extend the notion of the Laplacian to weighted
directed graphs). Vertices with $0$ out-degree have highest local payoff and
indicate directions in which we may perturb the equilibria and remain in the
basin of attraction. By way of example, consider the new point strategy set:

$\mathbf{x}^{\prime}=[0.79,0.795,0.79,0.8,0.795,0.795,\\\
0.792,0.795,0.795,0.7]$ (11)

Here we have perturbed players 1, 3, 4, 7, and 10 from their equilibrium
position, leaving players 2, 5, 6, 8 and 9 at the original equilibrium.
Analysis of the imitation graph suggests the critical players are 2 and 6,
both of whom have $0$ out-degree (and thus zero rows in the Jacobian matrix).
The return to equilibrium is illustrated in Figure 2 (a). This does not mean
that the basin of attraction for this point is a ball. From the the non-
equilibrium point:

$\mathbf{x}^{\prime}=[0.795,0.795,0.795,0.795,0.795,0.79,\\\
0.795,0.795,0.795,0.795]$ (12)

the system converges to a nearby Type 3 equilibrium, illustrating the neutral
stability of the original Type 1 point.

Figure 2: (a) The system is shown returning to equilibrium after being
perturbed. (b) Displacing Player 2 from equilibrium causes the system to move
to a new Type 3 equilibrium.

An interesting element of this example is Player 10, who becomes significantly
more defective is made more cooperative by peer imitation.

###### Remark 2

Notice the previous results do not rely on a Prisoner’s dilemma assumption and
the previous proof should generalize to payoff matrices in
$\mathbb{R}^{n\times n}$ with appropriate changes made.

###### Remark 3

Notice this analysis says nothing about Type 1 equilibria that also satisfy
the criteria of Type 2 equilibria. In this case, the Jacobian matrix consists
of all zeros and thus the imitation graph has no edges.

###### Remark 4

Analysis of Type 2 and 3 equilibria is even less satisfying. In the case of
Type 2 equilibria, we see we are at a point where all derivatives of
$f_{i}(\mathbf{x})$ fail to converge and consequently nothing can be said
about these points, however the stability is highly questionable. For a Type 3
equilibrium point satisfying the assumptions of Proposition V.1, the
Gershgorin disk theorem gives no information about the convergence other than
that these points may be stable or unstable depending on the nature of the
graph, the point and the payoff matrix. In practice, we see convergence to
Type 3 equilibria in which vertices can be partitioned into a few strategic
species; i.e., Type 3 equilibria, in practice, look like the amalgamations of
several Type 1 equilibria. Note, Type 3 equilibria satisfying the definition
of Type 2 equilibria may also have unusual stability properties.

###  V-B Convergence of Player Strategies in the Complete Case

In the case of the complete graph, we claim that all players will converge on
a single strategy (Type I), in particular the initial strategy of the most
deviant user in the graph. While the complete graph has a very strict tie
structure and online networks are generally not complete, this result provides
insight into local behavior in large networks since their typical small-world
structure is highly clustered [30], with densely-linked clusters also tending
to be more homogenous [20].

###### Theorem V.2

In the absence of network changes, the strategies of all players in a
completely connected network will converge to a unique value, in particular to
the minimal strategy (least cooperative) in the initial strategy vector.

###### Proof:

Without loss of generality, let $x_{l}=x_{1}(t_{0})\leq
x_{2}(t_{0})\leq\ldots\leq x_{n}(t_{0})$ be the initial strategies of $n$
completely connected nodes.

The initial payoffs to each are node are ordered

$P_{1}(t_{0})\geq P_{2}(t_{0})\geq\ldots\geq P_{n}(t_{0})$

since according to our model, in each pairwise interaction the node less
likely to cooperate has a higher payoff.

Since each player $i$ observes all other players in the complete graph, he
will update his strategy according to:

$x_{i}(t+1)=x_{i}(t)+\\\ \epsilon\cdot\sum_{j\in
J}{\frac{(P_{j}(t)-P_{i}(t))(x_{j}(t)-x_{i}(t))}{\sum_{j\in
J}{P_{j}(t)-P_{i}(t)}}}$ (13)

where $J$ is the set of nodes such that $x_{j}(t)<x_{i}(t)$. This is a natural
modification of Equation 3. Thus we see that, $x_{i}(t+1)<x_{i}(t)$ for all
$x_{i}$ with the exception of $x_{l}$ which remains unchanged. Thus we have

$\lim_{t\to\infty}x_{i}(t)=x_{l}\quad\forall v_{i}\in V.$ (14)

∎

##  VI The Spread of Deviance on BodyBuilding.com

In this section, we describe our empirical measurements of the models against
a real-world dataset collected from the web.

###  VI-A Data Collection and Measures for Deviance

We collect data from a real-world online bulletin system, known for the
controversial and active nature of its discussions (BodyBuilding.com, a
popular and longstanding site for body builders and fans). Bodybuilding.com is
powered on a Vbulletin portal, wherein discussions are grouped by topic, and
each discussion is further given by a collection of threads. Logged users can
create a new thread and respond to any contributed comment within a thread.
For the purpose of our validation, we specifically concentrate on a network of
users active on the “Religion/Politics” sub-forum, which collects several
heated discussions on controversial topics. We build the users’ network by
joining users who have commented on the same thread with an undirected edge,
under the assumption that users who post to a thread have some degree of
familiarity with the other comments posted. This assumption is supported by
the longstanding presence of many of the community members, who contribute
regularly to the same topics.

We collected a total of $62,060$ posts, over $787$ threads, obtaining a
network of $2586$ distinct nodes (users) joined by $62,931$ unweighted,
undirected edges.

Figure 3: Interactions amongst users posting to the ”Religion/Politics” forum
on www.bodybuilding.com.

In order to explore the behavior of users in this network within the
theoretical game framework we have established, we split the network into two
pieces by time. Particularly, we create an early network and a late network of
user interactions obtained from the first $31,030$ posts and the second
$31,030$ posts, respectively. This allows us to fit our models in two distinct
temporal frames, comparing the evolution of individual deviant behavior.

Each post is assigned a deviance score in $[0,1]$, taken as a weighted average
of three measures: language, sentiment and content relevance. Here by
language, we mean the use of abusive, vulgar or inflammatory words. To obtain
a measure for each post with respect to this dimension, we extract distinctive
words from the text and compared them against dictionary of known abusive
words, to determine the ratio of abusive words used. If the result is above a
predefined threshold (e.g. $20\%$ in our context), we label the comment
negative otherwise it is neutral. To determine the post’s sentiment, we rely
on the AlchemyAPI tool [1]. Alchemy is a text mining platform with advanced
natural language processing capabilities for semantic analysis. It is widely
used both in academia and industry. In this regard, it is reasonable to assume
a reliable accuracy in terms of sentiment classification. Through Alchemy, the
sentiment of each post is marked as ”positive”, ”negative” or ”neutral”
according To confirm this hypothesis, we additionally validated sentiment
scores using the SentiStrength algorithm [27], which returned the same
sentiment results for 87% of the posts.

Content relevance measures the extent to which a particular comment is
degenerated relative to the post originating the discussion. It is measured by
considering the mutual information (MI) of the comment, with respect to the
category or topic of the thread. For a given thread $T$ and a comment $W$, it
is computed by measuring the the amount of information each term $w$ in $W$
relates with the thread $T$, wherein the relationship strength (i.e. amount of
information) is determined using the Wordnet dictionary as a reference
thesaurus. The less cohesive the comment is with respect to the whole thread,
the more degenerated or out of context it is likely to be.

The deviance scores for the subset of users present in both the early and late
networks ($586$ total) were compared against expected results using the
prisoners dilemma model for the spread of deviant behavior in social networks
presented above. The early deviance score for each user was set as the initial
strategy, while the late score was compared with the final strategy, obtained
after running the PD simulation until convergence ($\epsilon
f_{i}(\mathbf{x})\leq$ for all $i$). This is consistent with sociological
evidence that individuals will settle into behaviors over time and we have
ensured convergence within our model.

###  VI-B Empirical Analysis and Observations

Initial work with the data showed that fitting precise deviance values was
impossible (and ill-conceived, since we were using a computerized method to
determine deviance, which could add substantial noise to the measured values).
Let $Y_{0}$ be the early deviance scores and $Y$ be the late deviance scores
and let $S$ be the standard deviation of the $Y_{0}$. We binned the elements
of $Y$ using $S$ and $Y_{0}$ so that if $Y_{i}\in Y_{0_{i}}\pm S$, then it was
assigned Bin $0$. In general if:

$Y_{i}\in Y_{0_{i}}\pm kS\text{ and }Y_{i}\not\in Y_{0_{i}}\pm(k-1)S,$ (15)

then $Y_{i}$ is in bin $\pm k$. For simplicity, we used five bins, labeled
$-2$ through $2$. Letting $\hat{Y}$ be the estimated deviance values, we were
also able to compute bin values using the same technique. The objective was to
determine the general ability of the model to measure an individual’s
propensity to become more or less deviant as a function of time. The values of
the payoff matrix $\mathbf{A}$ were fit to minimize the sum of square bin-
error. The best-fit matrix is given by:

$\hat{\mathbf{A}}=\begin{bmatrix}0.1985&-0.6989\\\ 0.4927&0.0001\end{bmatrix}$
(16)

The confusion matrix that results from this model is given by:

$\mathbf{C}=\begin{bmatrix}28&11&6&0&0\\\ 6&17&47&0&0\\\ 7&14&319&42&0\\\
0&1&36&22&0\\\ 0&0&16&17&0\end{bmatrix}$

Rows are true bins, while columns are expected bins. From this we can compute
the confusion matrix in probabilities:

$\mathbf{P}=\begin{bmatrix}0.62222&0.24444&0.13333&0&0\\\
0.085714&0.24286&0.67143&0&0\\\ 0.018325&0.036649&0.83508&0.10995&0\\\
0&0.016949&0.61017&0.37288&0\\\ 0&0&0.48485&0.51515&0\end{bmatrix}$

Figure 4 also summarizes our fit using a histogram.

Figure 4: A histogram showing the true vs. expected distribution of users into
deviance bins.

This matrix is instructive on the capabilities and limitations of the modeling
approach. We make the following observations:

The model is very good at estimating when individuals will remain at
approximately the same deviance level. There is an 83% chance of correct
classification in this case.

The model is reasonably good at estimating when individuals will become much
more deviant (most likely because of the attractiveness of the deviant
strategy in the prisoner’s dilemma model).

The model tends to under-classify minor deviance and significantly under-
classify individuals who move toward extreme non-deviance (i.e., who move to a
+2 bin). What is surprising is the model’s tendency to under-represent minor
shifts toward more deviant behavior, since the prisoner’s dilemma has the
strict Nash equilibrium defect. This suggests that this approach is promising
as a way of modeling the emergence of more cooperative behaviors within
networks. It also suggests (given the under-representation of minor deviance)
that this game is not the ideal model for handling deviance. Further research
in both these areas is required.

##  VII Conclusion

In presenting a general evolutionary game model for the spread of deviant
behavior in social network graphs, we lay the foundation for an extension of
the general contagion model for influence online to include the notion of
individual payoff. We utilize a two-strategy prisoner’s dilemma model here,
but alternate models may be more deeply investigated in future work.
Additionally, we assume a single payoff matrix across all users. Further
research may incorporate multiple payoff matrices where this information is
available or detect users whose anomalous actions indicate that they are
operating with a different payoff scheme. Validation of this theoretical
framework with a large, online dataset gives preliminary indication that the
model may indeed work well when tuned within a particular domain.

## Acknowledgements

Portions of Dr. Griffin’s work were supported by the Army Research Office
under Grant W911NF-11-1-0487. Portions of Dr. Griffin’s, Dr. Squicciarini’s
and Dr. Ratjmajer’s work were supported by the Army Research Office under
grant W911NF-13-1-0271.

## References

  * [1] Alchemyapi.  http://www.alchemyapi.com.
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# Invariant volume forms and first integrals for geodesically equivalent
Finsler metrics

Ioan Bucataru  Faculty of Mathematics
Alexandru Ioan Cuza University
Iaşi, Romania<EMAIL_ADDRESS>http://www.math.uaic.ro/~bucataru/

###### Abstract.

Two geodesically (projectively) equivalent Finsler metrics determine a set of
invariant volume forms on the projective sphere bundle. Their proportionality
factors are geodesically invariant functions and hence they are first
integrals. Being $0$-homogeneous functions, the first integrals are common for
the entire projective class. In Theorem 1.1 we provide a practical and easy
way of computing these first integrals as the coefficients of a characteristic
polynomial.

###### Key words and phrases:

geodesically equivalent Finsler metrics, sphere bundle, Reeb vector field,
volume forms, first integral

###### 2000 Mathematics Subject Classification:

53C60, 53B40, 53D25, 53A20

##  1\. Introduction

Finsler geometry has been characterised by Chern as being “just Riemannian
geometry without the quadratic restriction” [6]. However, many important
results from Riemannian geometry cannot be easily extended to the Finslerian
framework without using techniques and tools that are specific to Finsler
geometry, [3].

In this work we will extend the results of Matveev and Topalov from [14],
regarding the existence of first integrals for geodesically equivalent
metrics, from the Riemannian to the Finslerian settings.

It has been shown by Matveev and Topalov that two geodesically equivalent
Riemannian metrics, on an $n$-dimensional manifold, determine a set of $n$
first integrals, [14, Theorem 1]. In this work, we propose an extension of
this result to the Finslerian context. For two projectively related Finsler
metrics, their Hilbert $2$-forms are geodesically invariant. We use these
Hilbert forms to construct a set of $n$ volume forms on the projective sphere
bundle $SM$, which are invariant by the unitary geodesic vector field (the
Reeb vector field for an induced contact structure on $SM$). Therefore the
proportionality factors of these volume forms provide $n-1$ geodesically
invariant functions. These $n-1$ functions are $0$-homogeneous in the fibre
coordinates and hence they are common first integrals for all Finsler metrics
in the same projective class. The missing, nth first integral, is the energy
function, which is not $0$-homogeneous and cannot be obtained using this
technique.

We consider $M$ a smooth manifold, of dimension $n\geq 2$ and $TM$ its tangent
bundle. A continuous, positive, $1$-homogeneous (in the fiber coordinates)
function $F:TM\to[0,+\infty)$, and smooth on $T_{0}M=TM\setminus\\{0\\}$,
defines a Finsler structure if the metric tensor

$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^{2}F^{2}}{\partial y^{i}\partial
y^{j}}(x,y)$

is non-degenerate on $T_{0}M$. Here $(x,y)\in TM$, with $x\in M$ and $y\in
T_{x}M$.

In [14, Theorem 1], Matveev and Topalov use the characteristic polynomial of a
$(1,1)$-type tensor, constructed with two geodesically equivalent Riemannian
metrics, to generate $n$ first integrals that are quadratic in velocities. Two
alternative ways for obtaining quadratic first integrals for projectively
equivalent Riemannian metrics have been proposed by Crampin in [5]. In our
work we will use the angular metric to construct a characteristic polynomial,
(1.2), whose coefficients are first integrals.

The metric tensor $g_{ij}$ of the Finsler structure $F$ can be expressed in
terms of the angular metric $h_{ij}$ as follows:

(1.1) $\displaystyle g_{ij}=h_{ij}+\frac{\partial F}{\partial
y^{i}}\frac{\partial F}{\partial y^{j}}=h_{ij}+F_{y^{i}}F_{y^{j}},\quad
h_{ij}=F\frac{\partial^{2}F}{\partial y^{i}\partial y^{j}}=FF_{y^{i}y^{j}}.$

The metric tensor $g_{ij}$ has rank $n$ if and only if the angular metric
$h_{ij}$ has rank $n-1$, [13, Proposition 16.2]. The angular metric plays an
important role in projective Finsler geometry, [2].

Consider $F$ and $\widetilde{F}$ two Finsler metrics on the same manifold $M$
of dimension $n\geq 2$. The Finsler metrics $F$ and $\widetilde{F}$ are
geodesically equivalent, or projectively related, if they have the same
geodesics (considered as oriented unparameterised curves).

For two projectively related Finsler metrics $F$ and $\widetilde{F}$ consider
the characteristic polynomial:

(1.2) $\displaystyle
Q(\Lambda)=\det\left(\mathcal{H}^{i}_{j}+\Lambda\delta^{i}_{j}\right)=\sum_{\alpha=1}^{n}f_{\alpha}\Lambda^{\alpha}.$

The $(1,1)$-type tensor, of rank $n-1$, $\mathcal{H}^{i}_{j}$ is determined by
the angular metric $\widetilde{h}_{kj}$ of the Finsler structure
$\widetilde{F}$, the contravariant metric tensor $g^{ik}$ of the Finsler
structure $F$ and the two Finsler metrics $F$ and $\widetilde{F}$:

(1.3)
$\displaystyle\mathcal{H}^{i}_{j}=\dfrac{F}{\widetilde{F}}g^{ik}\widetilde{h}_{kj}=\dfrac{F}{\widetilde{F}}g^{ik}\left(\widetilde{g}_{kj}-\frac{\partial\widetilde{F}}{\partial
y^{k}}\frac{\partial\widetilde{F}}{\partial
y^{j}}\right)=\dfrac{F}{\widetilde{F}}g^{ik}\left(\widetilde{g}_{kj}-\dfrac{1}{\widetilde{F}^{2}}\widetilde{g}_{ks}y^{k}\widetilde{g}_{jl}y^{l}\right).$

We formulate now the main theorem of our work.

######  Theorem 1.1.

Consider $F$ and $\widetilde{F}$ two projectively related Finsler metrics. The
coefficients $f_{\alpha}$, $\alpha\in\\{1,...,n-1\\}$, of the characteristic
polynomial (1.2), are first integrals for the geodesic spray $S$ of the
Finsler metric $F$, which means that $S(f_{\alpha})=0$.

For $\alpha=n$ we have that $f_{n}=1$. Therefore, the coefficients of the
polynomial (1.2) give $n-1$ non-trivial first integrals. In order to prove
this, we will show first that the $0$-homogeneous functions $f_{\alpha}$
represent proportionality factors for some volume forms on the projective
sphere bundle $SM$. The key point for the proof of Theorem 1.1 is that these
volume forms are invariant by the unitary geodesic spray $S/F$, which is the
Reeb vector field of the contact manifold $(SM,F_{y^{i}}dx^{i})$. The idea of
using invariant differential $2$-forms to obtain invariant functions has been
used before by Tabachnikov in [17]. A modern reformulation of Tabachnikov’s
results has been used recently by Álvarez-Paiva to prove some rigidity results
for geodesically equivalent Finsler metrics, [1, Theorem VI].

For $\alpha=1$, the first integral $f_{1}$ can be expressed as:

(1.4) $\displaystyle
f_{1}=\frac{F^{n+1}}{\widetilde{F}^{n+1}}\frac{\det\widetilde{g}}{\det g}.$

An equivalent expression for the first integral $f_{1}$ has been obtained
recently in [4, Lemma 3.1] using different techniques. In formula (4.3), we
present an equivalent expression of this first integral, which in the
Riemannian case it corresponds to the Painlevé first integral $I_{0}$, [18,
Remark 1].

Another first integral can be obtained for $\alpha=n-1$:

(1.5) $\displaystyle
f_{n-1}=\operatorname{Tr}\left(\mathcal{H}^{i}_{j}\right)=\dfrac{F}{\widetilde{F}}g^{ij}\widetilde{h}_{ij}=\dfrac{F}{\widetilde{F}^{3}}g^{ij}\left(\widetilde{g}_{ij}\widetilde{F}^{2}-\widetilde{g}_{ik}y^{k}\widetilde{g}_{jl}y^{l}\right).$

An equivalent expression of this first integral is given by formula (4.4),
which in the Riemannian case corresponds to the first integral, $I_{1}$,
obtained by Topalov and Matveev, [18, Theorem 1].

In dimension $2$, Foulon and Ruggiero have shown the existence of a first
integral for the geodesic flow of a $k$-basic Finsler surface, [7]. This
result was extended to arbitrary dimension in [4], by providing a class of
Finsler manifolds that admit a first integral.

A different approach for obtaining first integrals in Finsler geometry is due
to Sarlet, who provides in [15] a recursive scheme of first integrals of the
geodesic flow of a Finsler manifold. Using Sarlet’s approach, for two
projectively related Finsler metrics, the tensor

$\displaystyle K^{i}_{j}=\left(\dfrac{\det\widetilde{g}}{\det
g}\right)^{\frac{1}{n+1}}\widetilde{g}^{ik}g_{kj}$

satisfies [15, (3)], which means that it is a special conformal Killing
tensor. The first integral generated by [15, Theorem 3], using this tensor
$K$, is an equivalent expression of the first integral $f_{1}$, see formula
(4.3).

##  2\. Projectively equivalent Finsler metrics

Consider $M$ a smooth manifold, of dimension $n\geq 2$, $TM$ its tangent
bundle and $T_{0}M=TM\setminus\\{0\\}$ the tangent bundle with the zero
section removed. We denote the local coordinates on the base manifold $M$ by
$(x^{i})$ and by $(x^{i},y^{i})$ the induced canonical coordinates on $TM$
(and $T_{0}M$).

The tangent bundle $TM$ carries two canonical structures: the Liouville
(dilation) vector field, ${\mathcal{C}}=y^{i}\frac{\partial}{\partial y^{i}}$,
and the tangent structure (vertical endomorphism), $J=\frac{\partial}{\partial
y^{i}}\otimes dx^{i}$.

A Finsler structure is defined by a continuous function $F:TM\to[0,+\infty)$,
smooth on $T_{0}M$, which satisfies the following two assumptions:

  * $F_{1}$)

$F$ is positively $1$-homogeneous ($1^{+}$-homogeneous): $F(x,\lambda
y)=\lambda F(x,y)$, $\forall\lambda>0,\forall(x,y)\in T_{0}M$;

  * $F_{2}$)

the Hessian of $F^{2}$

$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^{2}F^{2}}{\partial y^{i}\partial
y^{j}}(x,y)\ \textrm{is \ regular}.$

A pair $(M,F)$ is called a Finsler manifold. On a Finsler manifold we identify
the projective sphere bundle $SM=T_{0}M/{\mathbb{R}_{+}}$ with the indicatrix
bundle $IM=\\{(x,y)\in TM,F(x,y)=1\\}$. We note that functions defined on
$T_{0}M$ that are invariant under positive rescaling (are $0^{+}$-homogeneous)
can be restricted to $SM$. Some other geometric structures on $T_{0}M$ can
also be restricted to $SM$. For example, a $0^{+}$-homogeneous form
$\omega\in\Lambda^{k}(T_{0}M)$ can be restricted to $SM$ if and only if
$i_{\mathcal{C}}\omega=0$.

In this work we will use the Frölicher-Nijenhuis formalism to describe the
geometric setting on a Finsler manifold, [9, Chapter 2]. For a vector valued
$k$-form $K$, we will denote by $d_{K}$ the corresponding Lie derivation of
degree $k$. When $K$ is a vector field, the corresponding derivation of degree
$0$ is the standard Lie derivation $\mathcal{L}_{K}$.

For a Finsler metric $F$, its Hilbert $1$ and $2$-forms:

$\displaystyle d_{J}F=\frac{\partial F}{\partial y^{i}}dx^{i},\quad
dd_{J}F=\frac{1}{2}\left(\frac{\partial^{2}F}{\partial y^{i}\partial
x^{j}}-\frac{\partial^{2}F}{\partial y^{j}\partial x^{i}}\right)dx^{i}\wedge
dx^{j}+\frac{1}{F}h_{ij}dy^{i}\wedge dx^{j}$

are $0^{+}$-homogeneous. Moreover, $i_{\mathcal{C}}d_{J}F=0$ and
$i_{\mathcal{C}}dd_{J}F=0$ and therefore we can restrict these Hilbert forms
to $SM$, $d_{J}F\in\Lambda^{1}(SM)$, $dd_{J}F\in\Lambda^{2}(SM)$.

The regularity condition $F_{2}$, for the metric tensor of a Finsler
structure, ensures that there is a unique vector field
$S\in\mathfrak{X}(T_{0}M)$ that satisfies the Euler-Lagrange equations:

$\displaystyle{\mathcal{L}}_{S}d_{J}F^{2}-dF^{2}=0.$

The vector field $S$ is $2^{+}$-homogeneous and it is called the geodesic
vector field (spray) of the Finsler metric $F$. In local coordinates, the
geodesic spray is given by:

$\displaystyle S=y^{i}\frac{\partial}{\partial
x^{i}}-2G^{i}\frac{\partial}{\partial y^{i}},$

where $G^{i}$ are $2^{+}$-homogeneous functions, locally defined, on $T_{0}M$.

For a Finsler metric $F$, we use the geometric framework induced by its
geodesic spray $S$. We consider the horizontal and the vertical distributions
on $T_{0}M$ determined by the horizontal and vertical projectors, [8]:

$\displaystyle h=\frac{1}{2}\left(\operatorname{Id}-[S,J]\right),\quad
v=\frac{1}{2}\left(\operatorname{Id}+[S,J]\right).$

Locally, the two projectors $h$ and $v$ have the following expressions:

$\displaystyle h=\frac{\delta}{\delta x^{i}}\otimes dx^{i},\quad
v=\frac{\partial}{\partial y^{i}}\otimes\delta y^{i},\quad\textrm{ where: \
}\frac{\delta}{\delta x^{i}}=\frac{\partial}{\partial x^{i}}-\frac{\partial
G^{j}}{\partial y^{i}}\frac{\partial}{\partial y^{j}},\quad\delta
y^{i}=dy^{i}+\frac{\partial G^{i}}{\partial y^{j}}dx^{j}.$

######  Definition 2.1.

Two Finsler metrics $F$ and $\widetilde{F}$ are geodesically equivalent
(projectively related) if their geodesics coincide as unparameterised oriented
curves.

Two Finsler structures $F$ and $\widetilde{F}$ are projectively related if and
only if either one of the following equivalent Rapcsák equations are
satisfied:

  * ($R_{1}$)

${\mathcal{L}}_{S}d_{J}\widetilde{F}=d\widetilde{F}$;

  * ($R_{2}$)

$i_{S}dd_{J}\widetilde{F}=0$;

  * ($R_{3}$)

${\mathcal{L}}_{S}dd_{J}\widetilde{F}=0$;

  * ($R_{4}$)

$d_{h}d_{J}\widetilde{F}=0$.

For an extensive set of Rapcsák equations, we refer to [16, Theorem 9.2.22].
The third Rapcsák equation $(R_{3})$ ensures that the Hilbert $2$-form
$dd_{J}\widetilde{F}$ (of the Finsler metric $\widetilde{F}$) is geodesically
invariant (with respect to the Finsler metric $F$).

The fourth Rapcsák equation $(R_{4})$ ensures that the Hilbert $2$-forms
$dd_{J}F$ and $dd_{J}\widetilde{F}$ have a special form with respect to the
horizontal and vertical distributions induced by the Finsler metric $F$:

(2.1) $\displaystyle dd_{J}F=\frac{1}{F}h_{ij}\delta y^{i}\wedge dx^{j},\quad
dd_{J}\widetilde{F}=\frac{1}{\widetilde{F}}\widetilde{h}_{ij}\delta
y^{i}\wedge dx^{j}.$

According to this formula, the Hilbert $2$-forms $dd_{J}F$ and
$dd_{J}\widetilde{F}$ vanish whenever their arguments are either both
horizontal vector fields or both vertical vector fields.

##  3\. Invariant volume forms on the projective sphere bundle $SM$

We have seen already that for a Finsler metric $F$, its Hilbert $1$-form
$d_{J}F$ is $0^{+}$-homogeneous and hence it can be restricted to the
projective sphere bundle $SM$. This $1$-form is a contact structure on the
$(2n-1)$-dimensional manifold $SM$, which means that
$d_{J}F\wedge\left(dd_{J}F\right)^{(n-1)}\neq 0$. Therefore, the contact
manifold $(SM,d_{J}F)$ has a canonical volume form:

(3.1)
$\displaystyle\Omega_{F}=\frac{(-1)^{(n-1)(n-2)/2}}{(n-1)!}d_{J}F\wedge\left(dd_{J}F\right)^{(n-1)}\in\Lambda^{2n-1}(SM).$

For an introduction to contact structures and induced volume forms we refer to
[11, §10.1]. For the contact manifold $(SM,d_{J}F)$ one can easily see that
the corresponding Reeb vector field is the normalized geodesic spray $S/F$,
since it satisfies:

$\displaystyle d_{J}F\left(\dfrac{S}{F}\right)=1,\
i_{S/F}dd_{J}F=\dfrac{1}{F}i_{S}dd_{J}F=\dfrac{1}{F}\left(\mathcal{L}_{S}d_{J}F-di_{S}d_{J}F\right)=0.$

Consider now $F$ and $\widetilde{F}$ two projectively related Finsler metrics
and their Hilbert forms: $d_{J}F\in\Lambda^{1}(SM)$,
$dd_{J}F\in\Lambda^{2}(SM)$ and $dd_{J}\widetilde{F}\in\Lambda^{2}(SM)$. Using
these ingredients, we can define the following $(2n-1)-$forms, for each
$\alpha\in\\{1,...,n\\}$:

(3.2)
$\displaystyle\Omega_{\alpha}=\frac{(-1)^{(n-1)(n-2)/2}}{(\alpha-1)!(n-\alpha)!}d_{J}F\wedge\left(dd_{J}F\right)^{(\alpha-1)}\wedge\left(dd_{J}\widetilde{F}\right)^{(n-\alpha)}\in\Lambda^{2n-1}(SM).$

In the next section, we will prove that the volume forms (3.2) are invariant
by the Reeb vector field $S/F$.

Now, we express the proportionality factors between $\Omega_{\alpha}$ and
$\Omega_{F}=\Omega_{n}$ using two characteristic polynomials, which we
construct with the help of the angular metrics $h_{ij}$ and
$\widetilde{h}_{ij}$ of the two Finsler metrics $F$ and $\widetilde{F}$:

$\displaystyle P(\Lambda)$ $\displaystyle=$
$\displaystyle\det\left(\dfrac{F}{\widetilde{F}}\widetilde{h}_{ij}+\Lambda
g_{ij}\right)=\sum_{\alpha=1}^{n}\delta_{\alpha}\Lambda^{\alpha},$
$\displaystyle Q(\Lambda)$ $\displaystyle=$
$\displaystyle\det\left(\mathcal{H}^{i}_{j}+\Lambda\delta^{i}_{j}\right)=\frac{1}{\det
g}P\left(\Lambda\right)=\sum_{\alpha=1}^{n}f_{\alpha}\Lambda^{\alpha}.$

The $(1,1)$-type tensor, of rank $n-1$, $\mathcal{H}^{i}_{j}$ is given by

$\displaystyle\mathcal{H}^{i}_{j}=\dfrac{F}{\widetilde{F}}g^{ik}\widetilde{h}_{kj}=\dfrac{F}{\widetilde{F}}g^{ik}\left(\widetilde{g}_{kj}-\frac{\partial\widetilde{F}}{\partial
y^{k}}\frac{\partial\widetilde{F}}{\partial y^{j}}\right).$

We note that the coefficients $\delta_{\alpha}$ of the polynomial $P$ are not
globally defined functions on $T_{0}M$ (or $SM$), under a change of
coordinates they obey the same transformation law as $\det g$. However, the
quotient $f_{\alpha}=\delta_{\alpha}/\det g$, the coefficients of the
polynomial $Q$, are globally defined functions on $T_{0}M$, and being
$0^{+}$-homogeneous functions, they are globally defined on $SM$, for each
$\alpha\in\\{1,2,...,n\\}$.

The two polynomials $P$ and $Q$ have no free terms,
$P(0)=\det(\frac{F}{\widetilde{F}}\widetilde{h}_{ij})=0$ and
$Q(0)=\det(\mathcal{H}^{i}_{j})=0$. In the next lemma, we provide an explicit
formula for the coefficients $\delta_{\alpha}$ of the polynomial $P$.

######  Lemma 3.1.

The coefficients $\delta_{\alpha}$ of the polynomial $P$ are
$0^{+}$-homogeneous functions, given by

$\displaystyle\delta_{\alpha}$ $\displaystyle=$
$\displaystyle\dfrac{F^{n-\alpha}}{\widetilde{F}^{n-\alpha}}\frac{1}{(\alpha-1)!(n-\alpha)!}\sum_{\sigma_{1},\sigma_{2}\in
S_{n}}\varepsilon(\sigma_{1}\sigma_{2})h_{\sigma_{1}(1)\sigma_{2}(1)}\cdots
h_{\sigma_{1}(\alpha-1)\sigma_{2}(\alpha-1)}$
$\displaystyle\cdot\widetilde{h}_{\sigma_{1}(\alpha)\sigma_{2}(\alpha)}\cdots\widetilde{h}_{\sigma_{1}(n-1)\sigma_{2}(n-1)}\frac{\partial
F}{\partial y^{\sigma_{1}(n)}}\frac{\partial F}{\partial y^{\sigma_{2}(n)}}.$

###### Proof.

From the definition of the polynomial $P$ we have

$\displaystyle
P(\Lambda)=\det\left(\dfrac{F}{\widetilde{F}}\widetilde{h}_{ij}+\Lambda
g_{ij}\right)=\sum_{\sigma\in
S_{n}}\varepsilon(\sigma)\left(\dfrac{F}{\widetilde{F}}\widetilde{h}_{1\sigma(1)}+\Lambda
g_{1\sigma(1)}\right)\cdots\left(\dfrac{F}{\widetilde{F}}\widetilde{h}_{n\sigma(n)}+\Lambda
g_{n\sigma(n)}\right)=\sum_{\alpha=1}^{n}\delta_{\alpha}\Lambda^{\alpha}.$

Therefore, the coefficient $\delta_{\alpha}$ collects all terms from the above
sum that contain $\alpha$ factors of elements of $g_{ij}$ and $(n-\alpha)$
factors of elements of $\frac{F}{\widetilde{F}}\widetilde{h}_{ij}$:

$\displaystyle\delta_{\alpha}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$
$\displaystyle\dfrac{F^{n-\alpha}}{\widetilde{F}^{n-\alpha}}\sum_{\sigma\in
S_{n}}\varepsilon(\sigma)\sum_{\begin{subarray}{c}1\leq
i_{1}<\cdots<i_{\alpha}\leq n\\\ 1\leq j_{1}<\cdots<j_{n-\alpha}\leq n\\\
\\{i_{1},...,i_{\alpha}\\}\cap\\{j_{1},...,j_{n-\alpha}\\}=\emptyset\end{subarray}}g_{i_{1}\sigma(i_{1})}\cdots
g_{i_{\alpha}\sigma(i_{\alpha})}\widetilde{h}_{j_{1}\sigma(j_{1})}\cdots\widetilde{h}_{j_{n-{\alpha}}\sigma(j_{n-{\alpha}})}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}$
$\displaystyle\dfrac{F^{n-\alpha}}{\widetilde{F}^{n-\alpha}}\frac{1}{\alpha!(n-\alpha)!}\sum_{\sigma,\sigma_{1}\in
S_{n}}\varepsilon(\sigma)g_{\sigma_{1}(1)\sigma(\sigma_{1}(1))}\cdots
g_{\sigma_{1}(\alpha)\sigma(\sigma_{1}(\alpha))}\widetilde{h}_{\sigma_{1}(\alpha+1)\sigma(\sigma_{1}(\alpha+1))}\cdots\widetilde{h}_{\sigma_{1}(n)\sigma(\sigma_{1}(n))}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}$
$\displaystyle\dfrac{F^{n-\alpha}}{\widetilde{F}^{n-\alpha}}\frac{1}{\alpha!(n-\alpha)!}\sum_{\sigma_{1},\sigma_{2}\in
S_{n}}\varepsilon(\sigma_{1}\sigma_{2})g_{\sigma_{1}(1)\sigma_{2}(1)}\cdots
g_{\sigma_{1}(\alpha)\sigma_{2}(\alpha)}\widetilde{h}_{\sigma_{1}(\alpha+1)\sigma_{2}(\alpha+1)}\cdots\widetilde{h}_{\sigma_{1}(n)\sigma_{2}(n)}$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}$
$\displaystyle\dfrac{F^{n-\alpha}}{\widetilde{F}^{n-\alpha}}\frac{1}{(\alpha-1)!(n-\alpha)!}\sum_{\sigma_{1},\sigma_{2}\in
S_{n}}\varepsilon(\sigma_{1}\sigma_{2})h_{\sigma_{1}(1)\sigma_{2}(1)}\cdots
h_{\sigma_{1}(\alpha-1)\sigma_{2}(\alpha-1)}$
$\displaystyle\cdot\widetilde{h}_{\sigma_{1}(\alpha)\sigma_{2}(\alpha)}\cdots\widetilde{h}_{\sigma_{1}(n-1)\sigma_{2}(n-1)}\frac{\partial
F}{\partial y^{\sigma_{1}(n)}}\frac{\partial F}{\partial y^{\sigma_{2}(n)}},$

which gives formula (3.1).

The equality (a) can be obtained also using Marvin Marcus’ formula, [12, (1)],
for the sum of a determinant:

(3.4)
$\displaystyle\det(A+B)=\sum_{r}\sum_{a,b}(-1)^{s(a)}(-1)^{s(b)}\det\left(A[a|b]\right)\det\left(B[a|b]\right),$

where $r$ is an integer from $0$ to $n$, and $a$ an $b$ are increasing integer
sequences of length $r$ chosen from $1$ to $n$. Moreover, $A[a|b]$ is the
$r$-square matrix obtained by taking rows $a$ and columns $b$, while $B[a|b]$
is the $(n-r)$-square matrix taking the complementary rows to $a$ and $b$,
respectively. Also, $s(a)$ and $s(b)$ are the sums of the integers in $a$ and
$b$, respectively.

We can apply formula (3.4) to compute
$P(\Lambda)=\det\left(\frac{F}{\widetilde{F}}\widetilde{h}_{ij}+\Lambda
g_{ij}\right)$. As we are interested in computing the coefficient of
$\Lambda^{\alpha}$, we must consider only the terms with $r=n-\alpha$, for
$a=\left(i_{1}<\cdots<i_{\alpha}\right)$ and
$b=\left(j_{1}<\cdots<j_{n-\alpha}\right)$. This gives the right hand side of
equality $(a)$.

While proving the formula for $\delta_{\alpha}$, we made use of the following
arguments:

(b) The two sets $\\{i_{1},...,i_{\alpha}\\}$ and
$\\{j_{1},...,j_{n-\alpha}\\}$ partition the set $\\{1,2,...,n\\}$ and
therefore there is a permutation $\sigma_{1}\in S_{n}$ such that
$i_{1}=\sigma_{1}(1),...,i_{\alpha}=\sigma_{1}(\alpha),j_{1}=\sigma_{1}(\alpha+1),...,j_{n-\alpha}=\sigma_{1}(n)$.
While $i_{1}<\cdots<i_{\alpha}$ and $j_{1}<\cdots<j_{n-\alpha}$ are in a fixed
order, the values of the permutation $\sigma_{1}$ (grouped in two sets, one
with $\alpha$ elements and one with $n-\alpha$ elements) are not ordered.
Hence the factor $1/\alpha!(n-\alpha)!$ in front of the sum.

(c) If we denote $\sigma_{2}=\sigma\sigma_{1}$, then, for a fixed
$\sigma_{1}\in S_{n}$, $\sigma$ covers $S_{n}$ if and only if $\sigma_{2}$
covers $S_{n}$. Their signatures are related by
$\varepsilon(\sigma)=\varepsilon(\sigma_{1}\sigma\sigma_{1})=\varepsilon(\sigma_{1}\sigma_{2})$.

(d) We use formula (1.1) to replace the metric tensor $g_{ij}$ in terms of the
angular metric $h_{ij}$, of rank $n-1$, and the components
$F_{y^{i}}F_{y^{j}}$ of rank $1$. Therefore, in each product of $\alpha$
factors

(3.5) $\displaystyle\left(h_{\sigma_{1}(1)\sigma_{2}(1)}+\frac{\partial
F}{\partial y^{\sigma_{1}(1)}}\frac{\partial F}{\partial
y^{\sigma_{2}(1)}}\right)\cdots\left(h_{\sigma_{1}(\alpha)\sigma_{2}(\alpha)}+\frac{\partial
F}{\partial y^{\sigma_{1}(\alpha)}}\frac{\partial F}{\partial
y^{\sigma_{2}(\alpha)}}\right),$

the components $\frac{\partial F}{\partial y^{\sigma_{1}(j)}}\frac{\partial
F}{\partial y^{\sigma_{2}(j)}}$ will appear exactly once, for each
$j\in\\{1,...,\alpha\\}$.

The terms of formula (3.5), where the components of $F_{y^{i}}F_{y^{j}}$ do
not appear, are the products $h_{\sigma_{1}(1)\sigma_{2}(1)}\cdots
h_{\sigma_{1}(\alpha)\sigma_{2}(\alpha)}$. These products, in the right hand
side of equality $(c)$, will come together with the products
$\widetilde{h}_{\sigma_{1}(\alpha+1)\sigma_{2}(\alpha+1)}\cdots\widetilde{h}_{\sigma_{1}(n)\sigma_{2}(n)}$,
and their corresponding sum

$\displaystyle\sum_{\sigma_{1},\sigma_{2}\in
S_{n}}\varepsilon(\sigma_{1}\sigma_{2})h_{\sigma_{1}(1)\sigma_{2}(1)}\cdots
h_{\sigma_{1}(\alpha)\sigma_{2}(\alpha)}\widetilde{h}_{\sigma_{1}(\alpha+1)\sigma_{2}(\alpha+1)}\cdots\widetilde{h}_{\sigma_{1}(n)\sigma_{2}(n)}$

will vanish since it represents the coefficient of $\Lambda^{\alpha}$ of the
polynomial $\det\left(\widetilde{h}_{ij}+\Lambda h_{ij}\right)=0$. Here I used
that both angular metrics have the same kernel, which implies that
$\left(\widetilde{h}_{ij}+\Lambda h_{ij}\right)y^{j}=0$ and hence
$\det\left(\widetilde{h}_{ij}+\Lambda h_{ij}\right)=0$.

While expanding formula (3.5), there are $\alpha$ identical terms that contain
$\frac{\partial F}{\partial y^{\sigma_{1}(j)}}\frac{\partial F}{\partial
y^{\sigma_{2}(j)}}$, for each $j\in\\{1,...,\alpha\\}$. We group them and
hence the factor in front of the sum from the right hand side of equality
$(d)$ becomes $1/(\alpha-1)!(n-\alpha)!$. Then, we replace both permutations
$\sigma_{1}$ and $\sigma_{2}$ with $\sigma_{1}\circ\tau_{\alpha n}$ and
$\sigma_{2}\circ\tau_{\alpha n}$ ($\tau_{\alpha n}$ is the transposition that
interchanges $\alpha$ and $n$) and obtain formula (3.1). ∎

The following lemma shows the importance of the coefficients $\delta_{\alpha}$
and $f_{\alpha}$, of the two characteristic polynomials $P$ and $Q$, for
expressing the volume forms $\Omega_{\alpha}$ in a natural coordinate system
or with respect to the canonical volume form.

######  Lemma 3.2.

Consider $F$ and $\widetilde{F}$ two projectively related Finsler metrics. In
terms of a natural coordinate system, the $(2n-1)$-forms $\Omega_{\alpha}$
given by (3.2) can be expressed as follows:

(3.6)
$\displaystyle\Omega_{\alpha}=(-1)^{n-1}\dfrac{\delta_{\alpha}}{F^{n}}dx\wedge
i_{\mathcal{C}}dy,\quad\forall\alpha\in\\{1,...,n\\},$

where $\delta_{\alpha}$ are the coefficients of the polynomial $P$ and

$i_{\mathcal{C}}dy=\sum_{i=1}^{n}(-1)^{i-1}y^{i}dy^{1}\wedge\cdots
dy^{i-1}\wedge dy^{i+1}\wedge\cdots\wedge dy^{n}.$

With respect to the volume form $\Omega_{F}$, we have:

(3.7)
$\displaystyle\Omega_{\alpha}=f_{\alpha}\Omega_{F},\quad\forall\alpha\in\\{1,...,n\\},$

where $f_{\alpha}$ are the coefficients of the polynomial $Q$.

###### Proof.

Since $S$ is the geodesic spray of the Finsler metric $F$, it follows that
$d_{h}F=0$, and therefore we have:

$\displaystyle\dfrac{1}{F}dF=\dfrac{1}{F}d_{v}F=\dfrac{1}{F}\dfrac{\partial
F}{\partial y^{i}}\delta y^{i}.$

Together with the $(2n-1)-$forms (3.2), we will consider the following
$2n-$forms, which are volume forms on $T_{0}M$ for each
$\alpha\in\\{1,..,n\\}$:

$\displaystyle\Omega^{\prime}_{\alpha}$ $\displaystyle=$
$\displaystyle\dfrac{1}{F}dF\wedge\Omega_{\alpha}=\dfrac{(-1)^{(n-1)(n-2)/2}}{(\alpha-1)!(n-\alpha)!}\dfrac{1}{F}d_{v}F\wedge
d_{J}F\wedge\left(dd_{J}F\right)^{(\alpha-1)}\wedge\left(dd_{J}\widetilde{F}\right)^{(n-\alpha)}$
$\displaystyle=$
$\displaystyle\dfrac{(-1)^{(n-1)(n-2)/2}}{(\alpha-1)!(n-\alpha)!}\dfrac{1}{F}\dfrac{\partial
F}{\partial y^{i}}\delta y^{i}\wedge\dfrac{\partial F}{\partial
y^{j}}dx^{j}\wedge\left(\dfrac{1}{F}h_{ij}\delta y^{i}\wedge
dx^{j}\right)^{(\alpha-1)}\wedge\left(\dfrac{1}{\widetilde{F}}\widetilde{h}_{ij}\delta
y^{i}\wedge dx^{j}\right)^{(n-\alpha)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{deltak}}}{{=}}$
$\displaystyle\dfrac{(-1)^{(n-1)(n-2)/2}}{F^{n}}\delta_{\alpha}dy^{1}\wedge
dx^{1}\wedge\cdots\wedge dy^{n}\wedge
dx^{n}=\dfrac{-\delta_{\alpha}}{F^{n}}dx\wedge dy.$

Now, we can prove formula (3.6):

$\displaystyle\Omega_{\alpha}=i_{\mathcal{C}}\Omega^{\prime}_{\alpha}=\dfrac{(-1)^{n-1}\delta_{\alpha}}{F^{n}}dx\wedge
i_{\mathcal{C}}dy.$

In order to prove formula (3.7), we pay attention now to the volume form
$\Omega_{n}=\Omega_{F}$. The coefficient $\delta_{n}$ of the polynomial $P$ is
given by $\delta_{n}=\det g$. Therefore, for $\alpha=n$, formula (3.6) reads

(3.8) $\displaystyle\Omega_{n}=(-1)^{n-1}\frac{\delta_{n}}{F^{n}}dx\wedge
i_{\mathcal{C}}dy=(-1)^{n-1}\frac{\det g}{F^{n}}dx\wedge i_{\mathcal{C}}dy.$

Now, the two formulae (3.6) and (3.8) provide the formula (3.7) that expresses
the $(2n-1)$-forms $\Omega_{\alpha}$ in terms of the volume form $\Omega_{n}$.
∎

The volume form $\Omega_{n}=\Omega_{F}$ has been introduced in [10] using an
orthonormal frame. In [10, (1.5)], the expression of this volume form is given
in terms of a natural coordinate system and does not contain the factor
$(-1)^{n-1}$, which appears in formula (3.8).

##  4\. Proof of Theorem 1.1 and some particular cases

In this section we provide the proof of the main result, Theorem 1.1, and use
it to obtain explicit formulae for the first integrals $f_{\alpha}$, for
$\alpha=1$ and $\alpha=n-1$. The expression for $f_{1}$ has been obtained
before in [4, Lemma 3.1], using different techniques. The expression for
$f_{n-1}$ is new. We show that in the Riemannian case, the two integrals (1.4)
and (1.5) reduce to the two first integrals $I_{0}$ and $I_{1}$ obtained by
Topalov and Matveev in [18, Theorem 1].

In the previous section, in Lemma 3.2, we have seen that all volume forms
$\Omega_{\alpha}$, given by (3.2), can be expressed in terms of the canonical
volume form $\Omega_{F}=\Omega_{n}$, the proportionality factors being given
by formula (3.7):

$\displaystyle\Omega_{\alpha}=f_{\alpha}\Omega_{F},\quad\forall\alpha\in\\{1,..,n\\},$

for $f_{\alpha}$ the coefficients of the characteristic polynomial (1.2). In
order to complete the proof of Theorem 1.1, we will show now that the volume
forms $\Omega_{\alpha}$ are invariant by the Reeb vector field.

######  Lemma 4.1.

For two projectively related Finsler metrics $F$ and $\widetilde{F}$, the
$(2n-1)$-forms (3.2) are invariant by the Reeb vector field $S/F$.

###### Proof.

According to formula (3.2), it suffices to prove that
$\mathcal{L}_{S/F}d_{J}F=0$, $\mathcal{L}_{S/F}dd_{J}F=0$ and
$\mathcal{L}_{S/F}dd_{J}\widetilde{F}=0$.

First, we rewrite the Rapcsack equation $(R_{1})$ in terms of the Reeb vector
field $S/F$:

(4.1)
$\displaystyle\mathcal{L}_{S/F}d_{J}\widetilde{F}=\dfrac{1}{F}\mathcal{L}_{S}d_{J}\widetilde{F}-\widetilde{F}\dfrac{dF}{F^{2}}=\dfrac{1}{F}\left(\mathcal{L}_{S}d_{J}\widetilde{F}-d\widetilde{F}\right)+d\left(\dfrac{\widetilde{F}}{F}\right)\stackrel{{\scriptstyle(R_{1})}}{{=}}d\left(\dfrac{\widetilde{F}}{F}\right).$

From the above formula (4.1), we obtain that the Hilbert forms $d_{J}F$,
$dd_{J}F$ and $dd_{J}\widetilde{F}$ are invariant by the Reeb vector field
$S/F$ and hence the volume forms (3.2) are invariant as well. ∎

Next, we use formulae (3.1) to obtain some explicit values for the first
integrals $f_{\alpha}$.

For $\alpha=1$, we have $f_{1}=\delta_{1}/\det g$. From formula (3.1), for
$\alpha=1$, we have

(4.2) $\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle\dfrac{F^{n-1}}{\widetilde{F}^{n-1}}\frac{1}{(n-1)!}\sum_{\sigma_{1},\sigma_{2}\in
S_{n}}\varepsilon(\sigma_{1}\sigma_{2})\prod_{i=1}^{n-1}\widetilde{h}_{\sigma_{1}(i)\sigma_{2}(i)}\frac{\partial
F}{\partial y^{\sigma_{1}(n)}}\frac{\partial F}{\partial y^{\sigma_{2}(n)}}$
$\displaystyle=$
$\displaystyle\dfrac{F^{n-1}}{\widetilde{F}^{n-1}}\frac{1}{(n-1)!}\sum_{j=1}^{n}\sum_{\sigma_{1},\sigma_{2}\in
S_{n},\sigma_{1}(n)=j}\varepsilon(\sigma_{1}\sigma_{2})\prod_{k=1,k\neq
j}^{n}\widetilde{h}_{k\sigma_{2}\sigma_{1}^{-1}(k)}\frac{\partial F}{\partial
y^{j}}\frac{\partial F}{\partial y^{\sigma_{2}\sigma_{1}^{-1}(j)}}$
$\displaystyle\stackrel{{\scriptstyle\sigma_{3}=\sigma_{2}\sigma_{1}^{-1}}}{{=}}$
$\displaystyle\dfrac{F^{n-1}}{\widetilde{F}^{n-1}}\frac{1}{(n-1)!}\sum_{j=1}^{n}\sum_{\sigma_{1},\sigma_{3}\in
S_{n},\sigma_{1}(n)=j}\varepsilon(\sigma_{3})\prod_{k=1,k\neq
j}^{n}h_{k\sigma_{3}(k)}\frac{\partial F}{\partial y^{j}}\frac{\partial
F}{\partial y^{\sigma_{3}(j)}}$ $\displaystyle=$
$\displaystyle\dfrac{F^{n-1}}{\widetilde{F}^{n-1}}\sum_{j=1}^{n}\sum_{\sigma_{3}\in
S_{n}}\varepsilon(\sigma_{3})\prod_{k=1,k\neq
j}^{n}\widetilde{h}_{k\sigma_{3}(k)}\frac{\partial F}{\partial
y^{j}}\frac{\partial F}{\partial y^{\sigma_{3}(j)}}$ $\displaystyle=$
$\displaystyle\dfrac{F^{n-1}}{\widetilde{F}^{n-1}}\det\left(\widetilde{h}_{ij}+\frac{\partial
F}{\partial y^{i}}\frac{\partial F}{\partial
y^{j}}\right)=\frac{F^{n+1}}{\widetilde{F}^{n+1}}\det\widetilde{g}.$

For the last equality above we made use of [4, Lemma 2.2]. If we replace the
value of $\delta_{1}$ in $f_{1}=\delta_{1}/\det g$, we obtain the first
integral (1.4).

If we denote:

$\displaystyle\mu=\left(\dfrac{\det
g}{\det\widetilde{g}}\right)^{\frac{1}{n+1}},$

then formula (1.4) for the first integral $f_{1}$ can be written as:

(4.3) $\displaystyle\dfrac{F^{2}}{f_{1}^{2/n+1}}=\mu^{2}\widetilde{F}^{2}.$

Therefore, the right hand side of formula (4.3) is a first integral. In the
Riemannian case it is quadratic in velocities and it reduces to the Painlevé
first integral $I_{0}$, [18, Remark 1]. An equivalent expression of this first
integral has been obtained by Crampin in [5] as a $0^{+}$-homogeneous
function, $\kappa=\frac{F}{\mu\widetilde{F}}$. The right hand side of formula
(4.3) can be obtained also as a first integral for two projectively related
Finsler metrics using [15, Theorem 3].

For $\alpha=n-1$, we obtain another first integral using directly the fact
that $f_{\alpha}$ are the coefficients of the polynomial (1.2):

$\displaystyle
f_{n-1}=\operatorname{Tr}\left(\mathcal{H}^{i}_{j}\right)=\dfrac{F}{\widetilde{F}}g^{ij}\left(\widetilde{g}_{ij}-\dfrac{\partial\widetilde{F}}{\partial
y^{i}}\dfrac{\partial\widetilde{F}}{\partial y^{j}}\right).$

The above expression for the first integral $f_{n-1}$ can be written as:

(4.4) $\displaystyle f_{n-1}\dfrac{\widetilde{F}^{3}\mu^{3}}{F}$
$\displaystyle=$
$\displaystyle\mu^{3}g^{ij}\left(\widetilde{g}_{ij}\widetilde{g}_{kl}-\widetilde{g}_{ik}\widetilde{g}_{jl}\right)y^{k}y^{l}.$

It follows that the right hand side of formula (4.4) is a first integral. In
the Riemannian case it is quadratic in velocities and corresponds to the the
first integral $I_{1}$ from [18, Theorem 1].

In dimension $2$, we can obtain directly the $0$-homogeneous integral $f_{1}$
and then show that it agrees with either one of the two formulae (1.4) or
(1.5). Consider $F$ and $\widetilde{F}$ two geodesically equivalent Finsler
metrics on a $2$-dimensional manifold. The forms (3.2) are now

$\displaystyle\Omega_{1}=d_{J}F\wedge
dd_{J}\widetilde{F},\quad\Omega_{2}=d_{J}F\wedge dd_{J}F.$

For the volume form $\Omega_{2}$ we have:

$\displaystyle\Omega_{2}$ $\displaystyle=$ $\displaystyle\frac{\partial
F}{\partial y^{i}}dx^{i}\wedge\frac{1}{F}h_{ij}\delta y^{i}\wedge
dx^{j}=\frac{1}{F}\frac{\partial F}{\partial y^{i}}dx^{i}\wedge
h_{ij}dy^{i}\wedge dx^{j}$ $\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$
$\displaystyle\frac{1}{F^{2}}\left(g_{2i}y^{i}h_{j1}dy^{j}-g_{1i}y^{i}h_{j2}dy^{j}\right)\wedge
dx^{1}\wedge dx^{2}$ $\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}$
$\displaystyle\frac{\det g}{F^{2}}\left(y^{2}dy^{1}-y^{1}dy^{2}\right)\wedge
dx^{1}\wedge dx^{2}=\frac{-\det g}{F^{2}}dx^{1}\wedge
dx^{2}\wedge\left(y^{1}dy^{2}-y^{2}dy^{1}\right),$

which is formula (3.8), for $n=2$. In the previous calculations we used:

(a) $\frac{\partial F}{\partial y^{i}}=\frac{1}{F}g_{ij}y^{j}$;

(b) the expression (1.1) of the angular metric $h_{ij}$ in terms of the metric
$g_{ij}$.

For the volume form $\Omega_{1}$ we have:

$\displaystyle\Omega_{1}$ $\displaystyle=$ $\displaystyle\frac{\partial
F}{\partial y^{i}}dx^{i}\wedge\frac{1}{\widetilde{F}}\widetilde{h}_{ij}\delta
y^{i}\wedge dx^{j}=\frac{1}{\widetilde{F}}\frac{\partial F}{\partial
y^{i}}dx^{i}\wedge\widetilde{h}_{ij}dy^{i}\wedge dx^{j}$ $\displaystyle=$
$\displaystyle\frac{1}{\widetilde{F}}\left(\frac{\partial F}{\partial
y^{2}}\widetilde{h}_{11}-\frac{\partial F}{\partial
y^{1}}\widetilde{h}_{12}\right)dx^{1}\wedge dx^{2}\wedge
dy^{1}+\frac{1}{\widetilde{F}}\left(\frac{\partial F}{\partial
y^{2}}\widetilde{h}_{21}-\frac{\partial F}{\partial
y^{1}}\widetilde{h}_{22}\right)dx^{1}\wedge dx^{2}\wedge dy^{2}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}$
$\displaystyle\dfrac{-\delta_{1}}{F^{2}}dx^{1}\wedge
dx^{2}\wedge\left(y^{1}dy^{2}-y^{2}dy^{1}\right).$

In the equality $(c)$ above, we introduce the function $\delta_{1}$ to show
that for $n=2$ we recover also formula (3.6). In order to obtain the function
$\delta_{1}$, we identify the terms in both sides of equality (c). We obtain:

$\displaystyle y^{2}\dfrac{\delta_{1}}{F^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\widetilde{F}}\left(\frac{\partial F}{\partial
y^{2}}\widetilde{h}_{11}-\frac{\partial F}{\partial
y^{1}}\widetilde{h}_{12}\right)$
$\displaystyle-y^{1}\dfrac{\delta_{1}}{F^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\widetilde{F}}\left(\frac{\partial F}{\partial
y^{2}}\widetilde{h}_{21}-\frac{\partial F}{\partial
y^{1}}\widetilde{h}_{22}\right).$

We multiply the first equation above by ${\partial F}/{\partial y^{2}}$, the
second one by $-{\partial F}/{\partial y^{1}}$, we sum these expressions and
obtain:

$\displaystyle\delta_{1}\dfrac{\widetilde{F}}{F}=-\begin{vmatrix}\widetilde{h}_{11}&\widetilde{h}_{12}&\dfrac{\partial
F}{\partial y^{1}}\\\ \widetilde{h}_{21}&\widetilde{h}_{22}&\dfrac{\partial
F}{\partial y^{2}}\\\ \dfrac{\partial F}{\partial y^{1}}&\dfrac{\partial
F}{\partial
y^{2}}&0\end{vmatrix}\stackrel{{\scriptstyle(d)}}{{=}}\dfrac{F^{2}}{\widetilde{F}^{2}}\det\widetilde{g},\quad\delta_{1}=\dfrac{F^{3}}{\widetilde{F}^{3}}\det\widetilde{g}.$

For the equality $(d)$ above we did use [4, Lemma 2.2]. The above expression
for $\delta_{1}$ agrees, for $n=2$, with the expression (4.2).

With the value of the function $\delta_{1}$, we obtain

$\displaystyle\Omega_{1}=\frac{-F}{\widetilde{F}^{3}}\det\widetilde{g}\
dx^{1}\wedge dx^{2}\wedge\left(y^{1}dy^{2}-y^{2}dy^{1}\right).$

By comparing the expressions for the volume forms $\Omega_{1}$ and
$\Omega_{2}$, we obtain

$\displaystyle\Omega_{1}=f_{1}\Omega_{2},\quad
f_{1}=\frac{F^{3}}{\widetilde{F}^{3}}\frac{\det\widetilde{g}}{\det g},$

which agrees with formula (1.4). We will show that in the $2$-dimensional
case, this formula agrees also with (1.5). From (1.5), for $n=2$, we have

$\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\dfrac{F}{\widetilde{F}}\left(g^{11}\widetilde{h}_{11}+g^{12}\widetilde{h}_{12}+g^{21}\widetilde{h}_{21}+g^{22}\widetilde{h}_{22}\right)$
$\displaystyle=$ $\displaystyle\dfrac{F}{\widetilde{F}}\dfrac{1}{\det
g}\left(g_{22}\widetilde{h}_{11}-g_{12}\widetilde{h}_{21}-g_{21}\widetilde{h}_{12}+g_{11}\widetilde{h}_{22}\right)=\dfrac{\delta_{1}}{\det
g}=\dfrac{F^{3}}{\widetilde{F}^{3}}\dfrac{\det\widetilde{g}}{\det g}.$

In the $2$-dimensional case, the proportionality factor and hence the first
integral $f_{1}$, can be obtained using a different argument. The angular
metrics $\widetilde{h}_{ij}$ and $h_{ij}$ have rank $1$ and they are
proportional,
$\frac{1}{\widetilde{F}}\widetilde{h}_{ij}=f_{1}\frac{1}{F}h_{ij}$. It follows
the proportionality of the Hilbert $2$-forms
$dd_{J}\widetilde{F}=f_{1}dd_{J}F$ and hence the proportionality of the
induced volume forms on $SM$, $\Omega_{1}=f_{1}\Omega_{2}$.

### Acknowledgements

I am grateful to Vladimir Matveev for very useful discussions regarding the
Riemannian correspondents of the first integrals (1.4) and (1.5). I express my
thanks to the referees for their comments and suggestions that improved the
proofs of the results.

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