Giorgio Coricelli
email: coricelli@isc.cnrs.fr

Claude Montmarquette
email: claude.montmarquette@cirano.qc.ca

Marie Claire Villeval
email: villeval@gate.cnrs.fr

Cheating, Emotions, and Rationality: An Experiment on Tax Evasion

Keywords: Deception, tax evasion, emotions, physiological measures, experiment. JEL Classifications: C91, C92, D87, H26

The economics-of-crime approach usually ignores the emotional cost and benefit of cheating. In this paper, we investigate the relationships between emotions, deception, and rational decision-making by means of an experiment on tax evasion. Emotions are measured by skin conductance responses and self-reports. We show that the intensity of anticipated and anticipatory emotions before reporting positively correlates with both the decision to cheat and the proportion of evaded income. The experienced emotional arousal after an audit increases with the monetary sanctions and the arousal is even stronger when the evader's picture is publicly displayed. We also find that the risk of a public exposure of deception deters evasion whereas the amount of fines encourages evasion. These results suggest that an audit policy that strengthens the emotional dimension of cheating favors compliance.

INTRODUCTION

Many situations involve deception, like plagiarism, embellished résumés, or trickery in sports. Recent scandals at Enron, for example, remind us that cheating in markets is not unusual (Gerschlager, 2005). Tax evasion and social fraud provide other examples of substantial deception by economic agents (Slemrod, 2007). The standard economics-of-crime approach explains deception as resulting from a comparison between the expected benefit and cost of cheating. The cost of cheating is however restricted to the monetary consequences of detection, and the cost of the very act of cheating is neglected. Yet, recent studies have provided some evidence of lie or guilt aversion (Gneezy, 2005; Charness and Dufwenberg, 2006; Sanchez-Pages and Vorsatz, 2007). Regarding tax evasion, if empirical studies confirm the basic mechanisms of the deterrence models (Allingham and Sandmo, 1972, and Yitzhaki, 1974), they also reveal that some people comply although the benefits of cheating clearly outweigh its monetary cost (See Alm, 1991, Andreoni et al., 1998, Cowell, 1990, and Slemrod, 2007, for surveys). This has motivated the exploration of other dimensions of tax compliance, like tax morale, social norms (Slemrod, 1998; Torgler, 2007; Kirchler, 2007), ethics (Blumenthal et al., 2001), or conformity (Myles and Naylor, 1996; Fortin et al., 2007). We adopt a complementary approach by investigating the influence of emotions on cheating behavior with an application to tax evasion. Economists have not widely explored yet the role of emotions in cheating behavior. We aim at contributing to fill this gap. 1 We claim that both the cognitive deliberation preceding income reporting and the realization of an audit, possibly followed by sanctions, involves emotions. We 1 Although Bentham early incorporated emotions in his theory of utility, economists have only recently studied the role of emotions in decision-making (Elster, 1996, Loewenstein, 2000). They have identified their influence on reciprocity (Ben-Shakhar et al., 2007), punishment (de Quervain et al., 2004), or investment decisions (van Winden et al., 2008). assume that emotions associated with reporting, audit, and sanctions inform and influence the agents' deliberation and therefore, constitute a driving force of behavior.

Various emotions might be related to compliance or evasion. 2 Individuals might experience both anticipatory emotions when evaluating the risk associated with underreporting and anticipated emotions when anticipating how bad they will feel if audited and punished and how good and relieved if not audited. 3 We also expect that, ex post, the detection of cheating raises emotions which intensity is increased when deception is made public. In particular, detected evaders may feel regret and guilt even when the information is not spread out and shame if their cheating behavior is made public. 4 This is at least what the authorities expect when they publicly expose the offenders. 5 Since it is difficult to cope with a damaged reputation, shame avoidance might have a larger impact on future compliance than its private counterpart (on such amplification effect, see Kahneman and Miller, 1986).

Our paper is original in that our laboratory experiment links economic behavior in an income-reporting game to physiological measures of emotions and self-reports of emotions. In our benchmark treatment, each subject receives an income that is taxed at a proportional rate. He has to decide how much income he is willing to report. He faces a probability to be audited, in which case he must pay a penalty for underreporting. The picture treatment follows the same rules except that an 2 When the product of the tax is used for a charitable cause, Harbaugh et al. (2007) show that mandatory taxation entails neural activity in brain areas that are related to rewards. 3 As defined by Loewenstein et al. (2001): "Anticipatory emotions are immediate visceral reactions to risks and uncertainties. Anticipated emotions are typically not experienced in the immediate present but are expected to be experienced in the future…anticipated emotions are a component of the expected consequences of a decision..." (p. 268). 4 Interestingly, Allingham and Sandmo (1972) explicitly mention that tax evasion may be limited if the individuals fear losing their reputation, but do not include this consideration in their model. In psychology, see Seiter and Bruschke (2007) on the role of guilt and shame in deception. 5 While pillories have been used in many cultures, the Web provides, nowadays, many examples of such policies. For example, the Chicago police department displays the pictures and home addresses of arrested prostitute patrons; a listing of convicted pedophiles is available in every state in the U.S.A.; the government of Canada displays the names of citizens convicted of fiscal fraud. We have not found any empirical estimation of their efficiency.

individual's cheating behavior is publicly revealed through the public display of his picture. This allows us to differentiate the impact of monetary versus non-monetary sanctions by holding the expected monetary return of evasion constant.

To measure the emotional arousal associated with the reporting decisions and the feedback on audits, we record the subjects' Skin Conductance Responses (SCR) (Bradley, 2000). The SCR is a phasic component of the electrodermal activity primarily controlled by the sympathetic division of the autonomic nervous system (Dawson et al., 2000). It is associated to emotional arousal in a wide range of psychological states and processes (Boucsein, 1992). We also use affective selfreports to assess the emotional arousal (low-high intensity) and the hedonic valence (pleasant-unpleasant) dimensions of emotions as a robustness check and as a complement.6 Indeed, research in psychology has shown that emotional arousal and valence can explain most of the variance in emotions (Russel and Mehrabian, 1974; Lang et al., 1992; Anders et al., 2004). An additional originality of our experiment is that we elicit emotions in a group context.

Our experiment delivers several major findings. Subjects who express increased SCR, i.e. higher anticipated and anticipatory emotional arousal, are more likely to evade and evade more. Emotional arousal at the time of audit is increased by the prospect of additional earnings and the avoidance of certain losses, and also by monetary and non-monetary sanctions. These results show that, despite the sterility of the environment, it is possible to capture emotions and the fear of social stigma in the laboratory. Indeed, being in an environment where the picture of the detected evader is publicly displayed raises more emotions in case of an audit and deters cheating. In contrast, while being audited raises emotions and reduces the extent of evasion in the following period, the higher the fine the higher the proportion of evaded income. As Baldry (1986) has shown that tax evasion is not a gamble because it involves moral considerations, our findings strongly support the analyses on the importance of tax morale. People do not only feel anxiety because of the monetary prospects of a risky decision but also due to the moral implications associated with the risk of a public exposure of their behavior. This is a major finding because it suggests that a tax authority could better discourage evasion by increasing the emotional cost of evasion.

The remainder of this paper is organized as follows. Section 2 develops the experimental design and the predictions. Section 3 analyzes the results. Section 4 discusses the results and concludes.

THE EXPERIMENT

A. Experimental design

Two treatments. The benchmark treatment involves a group of eight players who receive an individual income I, with

. Each income level is randomly assigned to two subjects in each period, but this is not common information.

The players are requested to report their income that will be taxed at a uniform tax rate of 55%, by means of a scrollbar of which maximum graduation corresponds to the subject's actual income. To produce a stronger test of social emotions, we do not redistribute the tax funds to the subjects who are informed that these paybacks will go into research funds for other experiments. 7

7 We acknowledge that we do not know the subjects' valuation of the taxes and fines collected that may be considered as a specific form of a public good. The subjects do not, however, benefit from these funds directly. If this procedure has nevertheless influenced decisions, in one way or the others, there is no reason to believe that the subjects' reaction will be different between treatments since we use a within-subject design.

The players know that their reported income can be audited according to a certain cutoff audit scheme and that this audit will entail the payment of a fine if the reported income is less than the actual income. The realism of the design has been strengthened by replicating the structure of income reporting, by using non-neutral terms in the instructions so that it makes no doubt that underreporting means cheating, 8 and by introducing an endogenous audit rule. The probability of an audit is endogenous in that it depends on the median report in the group, and this is made common information. 9 If the reported income of a player is among the four highest reported incomes in the group, his audit probability is 35%. If his reported income is among the four lowest reported incomes, his audit probability is 65%. If all subjects report the same amount, the probability is uniform and equal to 50%. 10 The reason why low-reported-income players have a higher probability of being audited is because reporting low incomes signals to the tax authority, which knows the distribution of income but not each individual's income, that the individual might have underreported by a substantial amount. The tax authority has less to gain in auditing high reported incomes. Another reason for this endogenous scheme is that one's own evasion decreases the risk of audit of the other subjects (which is realistic since the auditing resources of the tax administration are constrained); therefore 8 Using loaded terms loses some control over the subjects' preferences; on the other hand, it introduces context that makes mental scripts less necessary (Alm, 1991). Alm, McClelland and Schulze (1992) compare two experiments using either loaded or neutral wording and find no difference in behavior. 9 In real settings, the probability of an audit depends on the taxpayer's decisions since reports convey information. While empirical analyses are plagued by such an endogeneity problem (Alm, 1991), experiments allow clearing it up by using random audit probabilities. Most experiments, however, do not analyze how endogenous probabilities influence decisions. Exceptions are Collins and Plumlee (1991), Alm et al. (1993), Cason and Gangadharan (2006), Cummings et al. (2009). 10 These probabilities are above the usual enforcement levels in order to collect enough detection observations. Note that one can also find high rates in reality, for example against taxpayers who have been caught in the past or in the presence of cutoff rules such that a report lower than some cutoff level will be audited with certainty (see Torgler, 2007). Again, we create an environment hostile to the phenomenon we want to study since we suspect that frequent detection erodes the emotions associated with cheating, due to a habituation effect (Dawson et al. 2000). And this is what we observe in this experiment (see below).

evading taxes does not hurt the group members. While creating some interdependencies between the subjects, this auditing rule is hostile to the emergence of a social stigma and provides a stronger test of social emotions (see below). The subjects are not informed on their individual audit probability (see Spicer and Thomas, 1982, and Alm, Jackson, and McKee, 1992, on the impact of uncertain audit probabilities on compliance). They do not receive any feedback on others' behavior. 11

If the player is not audited, his net income corresponds to his gross income minus the tax on his reported income. If an audit concludes to underreporting, both the tax on the actual income and a fine are charged. As in Yitzhaki (1974), the fine is proportional to the unpaid taxes; it is fixed at 20%, which is a realistic value (Andreoni et al., 1998).

The picture treatment follows the same rules as the benchmark. The only difference is that if an audit reveals that a player has underreported his income, his picture is displayed on his own screen and on the screens of other taxpayers. This treatment aims at identifying whether public information on one's own cheating involves additional emotions leading to more compliance. 12 In a period, only one picture can be displayed on the screen of any subject; if more than one cheater have been audited, there is a random display of each picture on the screen of the non-audited subjects and 11 An endogenous audit mechanism raises the possibility of potential coordination among participants. However, Alm and McKee (2004) have shown in their experiment that individuals find it difficult to coordinate on the zero-compliance equilibrium. Our endogenous audit rule depending on the median income report of the group of eight players makes this coordination even more difficult given that the game has no Nash equilibrium in pure strategies as discussed below.

12 If subjects perceive the environment as one about cheating the experimenter rather than the group members (because tax funds are used for further research or because one's own evasion reduces the audit probability of others), this treatment should not have a strong emotional or behavioral impact, especially since cheating the experimenter could be considered as acceptable from students who need money.

the honest audited taxpayers. 13 The amount evaded by a cheater is not displayed since it could be an additional source of influence on individual decisions.

We used a within-subject design to keep the subjects' characteristics constant and increase statistical power. The sessions consisted of 30 periods, divided into six blocks of five periods. The benchmark treatment was implemented in periods 1 to 5, 11 to 15, 21 to 25, and the picture treatment in the remaining periods.

Eliciting risk attitude. A parsimonious procedure (see Charness and Gneezy, 2003) has been used in the preliminary part of the experiment to elicit the individual degree of risk aversion. Each subject was endowed with 15 points (his show-up fee of €3)

and had to decide how much to invest in a risky asset and how much to keep. In case it fails, the amount invested is lost; in case of a success, the investment returns 2.5 times its amount. Each subject had also to choose one of two colors. If this color is randomly drawn at the end of the session, with a 50% chance, the investment is a success. A risk-neutral subject should invest his full endowment. The lower the amount invested, the higher the degree of risk aversion.

The "Physionomics lab" and the self-reports. We proceeded to the simultaneous and continuous electrophysiological (Skin Conductance Responses) recording of groups of eight interactive subjects until the end of the sessions. Indeed, the literature in psychology and neurophysiology has established that SCR capture emotional arousal (Bradley et al., 2001). 14 The method is exposed in details in Appendix 1. The SCR have been analyzed for the decision and the feedback on audit.

13 Displaying only one picture at a time does not allow the subjects to identify the number of evaders in the session. A subject who is caught cheating does not know if there were other cheaters and how many subjects can see his picture. 14 A connection between SCR generation and the brain areas involved on higher stages of information processing (e.g. information uptake, decision and storage processes) has also been evidenced. However, the role of SCR as an indicator of cognitive effort is not clearly defined, probably due to the wide variety of theoretical and experimental approaches used (Boucsein, 1992).

In addition, at the end of each period, the subjects had to report their feelings at the moment of the audit by means of two 7-point Likert-type scales of emotional arousal (from extremely weak to extremely strong) and hedonic valence (from extremely negative to extremely positive). Indeed, an adequate assessment of emotions requires an integration of behavior, verbal report and physiology (Bradley and Lang, 2001).

While the physiological measures provide an autonomic measure of the subject's emotional arousal, the self-reports unveil, backwardly, the subjective intensity and direction of emotions experienced by the subjects. Last, at the end of the session, we administered an attitudinal questionnaire on fraud. 15 The subjects' payment consisted of the average earnings in two randomly selected periods in order to avoid possible wealth effects. The instructions specified that a person who is not aware of the content of the experiment would pay the subjects in private in a different room; this was done to limit accountability.

B. Predictions

The predictions of this game in the vein of Allingham and Sandmo (1972) and Yitzhaki (1974) are the same for both treatments if players are rational. We assume that the utility function of the participants is linear in income. The expected gross income, I, is 125. For the symmetry of the game, we assume that each player believes that the others receive the same income with equal probability. The probability to be audited, assuming that each player uses the same strategy, is p = 0.5*A Low + 0.5*A High, with A Low (A High ) being the audit rate for those reporting income below (above) the median. Let t be the tax rate and f the penalty rate. If an individual reports an amount R of his gross income I, his expected net income, ENI, is:

I -[ p( tI + ft(I -R)) + t(1-p)R ] = ENI (1)
If an individual reports his full income, then his net income, NI, is:

(2)

The expected rate of return from reporting no income relatively to full income is:

(3)

With the following parameters, t = 55%, f = 20%, A Low = 65% and A High = 35% (p = 50%), the expected rate of return is 45%. Full reporting is therefore a dominated strategy. In addition, a subject may realize that it is easier to hide evasion when one receives a high income if he believes that other subjects have drawn a lower income.

The game has no Nash equilibrium in pure strategies and the solution is a mixed strategy equilibrium with a positive compliance rate because a subject has always an incentive to deviate by reporting one unit more than the others to reduce his own audit probability to 35%. The equilibrium cannot be easily characterized, but it should be similar in both treatments. 16 However, the high expected return from evasion let us suppose that theoretical compliance should be low.

In addition to risk aversion, anticipatory and anticipated emotions may lead players to comply more if the positive emotions raised by evasion, such as excitement due to the prospect of higher payoff, are outweighed by the expected psychic cost of being caught cheating. The only reason why we could observe different decisions and patterns of SCR in the picture treatment than in the benchmark is that additional negative emotions may arouse from the anticipation of seeing one's own picture or others' pictures disseminated in the group although cheating generates positive externalities on others.

C. Experimental procedures

The experiment consisted of six sessions. These sessions were conducted at the Groupe d'Analyse et de Théorie Economique (GATE), Lyon, France. 48 subjects (of which 63% were males), eight for each session, were recruited from undergraduate courses in the local business and engineering schools using the ORSEE software (Greiner, 2004). All of the subjects were inexperienced in this type of experiment.

The experiment was computerized using the REGATE software (Zeiliger (2000)).

Upon arrival, the subjects drew a tag indicating their randomly designated computer.

They were asked whether they allowed us to take a picture of their face and to do physiological recording (all have accepted). It was made explicit that this picture would be used at some point during the experiment, that they would be informed before the possible use of the picture, and that they could quit immediately against the payment of the show-up fee. They were guaranteed that their picture would be immediately destroyed at the end of the session and that they could attend the destruction. The volunteers were required to keep a neutral face when we took their picture since studies in psychiatry show that emotions expressed on pictures may be contagious (Wild, 2003). Next, after washing their hands with a neutral soap for a better skin conductance, the subjects sat down in front of their computer. The electrodemal electrodes filled with neutral-base paste were placed on the fingers of their non-dominant hand. The electrodermal signal was then monitored and we proceeded to any adjustment required. Concomitantly, the instructions for the preliminary part of the experiment were read aloud (Appendix 2). At the end of the questions answering (in private), more than five minutes from the placement of the last electrodes had elapsed, allowing time for the paste to be absorbed by the skin and the electrodes to make good contact. After the completion of the test of risk attitude, the instructions for the other parts were distributed and read aloud. For the sake of realism, the instructions used tax-specific language but they avoided loaded terms such as 'cheating' or 'lying'. We checked the understanding of the rules by means of a questionnaire. During each period, to avoid any overlapping between different sources of emotions, we imposed a fixed duration to some events (12 seconds after the feedback on audit, 6.5 seconds after the picture display, 3 seconds after the feedback on payoffs), while decisions and affective self-reports were self-paced.

At the end of the session, the subjects answered to the attitudinal questionnaire on fraud and to demographic questions, and they drew the color that determined whether their initial investment was a success or not. On average, a session lasted 75 minutes, including preparation of the subjects for the physiological recording and cash payment. The subjects received a show-up fee of €3 and earned €19.33 on average.

RESULTS

Overall, we find substantial treatment effects, as an environment where evading behavior is made public reduces the proportion of evaders and the amount evaded.

Emotional arousal correlates with cheating behavior and punishment, especially when made public. We provide descriptive statistics before considering the determinants of cheating behavior and the reactions to being audited.

A. Summary statistics and non-parametric analysis

Table 1 displays some descriptive statistics.

(Table 1 about here)

Table 1 indicates first that the proportion of evaders increases with income (all pairwise comparisons indicate that the differences are significant at the 5% level). 17

Second, the level of income exerts a non-linear influence on the proportion of evaded income in evaders. 18 Third, the proportion of evaders is marginally lower in the picture treatment than in the benchmark (p = 0.078), especially when incomes are either 50 (with a borderline p-value of p = 0.108) or 100 (p = 0.089). In contrast, the proportion of evaded income is higher in the picture treatment than in the benchmark for incomes of 50 or 100 and lower for higher incomes (but the Mann-Whitney tests fail concluding on significant differences). This suggests the existence of a selection effect: the subjects who care most about social sanctions do not evade when it is more risky to do so and those who take this higher risk do it for a higher proportion of income.

Regarding emotions, Table 1 shows that the average SCR amplitude is higher when people underreport than when they comply (Wilcoxon test: p = 0.028). It is twice as high when related to the reporting decision and three times higher when related to the feedback on audit. Higher SCR are associated with more evasion. In the benchmark, the average SCR amplitude at the time of decision is 0.11 for the subjects who evade less than 25% of their income; it rises to 0.17 when they evade between 25% and 49%

of their income and to 0.22 when they evade between 50% and 74%, and 75% or more. In the picture treatment, the corresponding values are 0.10, 0.20, 0.26, and 17 In all the reported non-parametric statistics, a session gives one independent observation and all tests are two-sided. Mann-Whitney tests indicate that the proportion of evaders when the income is 200 is significantly higher than when the income is 150 (p = 0.034), 100 (p = 0.028) and 50 (p = 0.028). This proportion when the income is 150 is significantly higher than when the income is 100 (p = 0.046) and 50 (p = 0.028). It is also significantly higher when the income is 100 than when it is 50 (p = 0.034). 18 According to Wilcoxon signed-rank tests, the evaders dissimulate a higher proportion of their income when they earn 50 than when they earn 100 (p = 0.028), 150 (p = 0.027), or 200 (p = 0.028). On the other hand, the average proportion of evaded income is significantly higher when evaders receive an income of 200 than when their income is 150 (p = 0.046).

0.19, respectively. 19 For the subjects evading between 25% and 74% of their income, the picture treatment is significantly more arousing than the other treatment (Mann-Whitney tests: p = 0.037 for the SCR measured at the time of decision, and p = 0.004 for the SCR measured at the time of audit). In this treatment, when the subjects evade 75% of their income or more, the SCR is, however, not higher than when they evade between 50% and 74% (Mann-Whitney test, p > 0.10). The people who report nothing in this environment are more likely to experience less psychic cost in cheating; or perhaps knowing they will feel bad if detected, they are better evading totally than a fraction of income. This may explain why the average SCRs, both at the time of decision and audit, do not differ much by treatment in Table 1 (Mann-Whitney tests, p > 0.10).

To summarize, we find that there are marginally less evaders in the picture treatment than in the benchmark and that the average SCR amplitude is higher when people evade taxes than when they comply.

B. Determinants of the evasion decision

It is expected that the decision to evade and the amount of taxes unreported can be explained by different variables and by differential effects of the same variables affecting both decisions. In a panel setting, we will deal with this situation by referring to a two-step generalized Tobit model.

Table 2 displays the results of this estimation procedure. First we estimate the determinants of the decision to evade taxes by means of a random-effects Probit model (column 1). We include individual random effects to control for the lack of independence between each trial since each individual is observed 30 times. This is 19 At the time of audit, the corresponding values in the benchmark treatment are 0.08, 0.13, 0.17, and 0.21; in the picture treatment, they are 0.07, 0.18, 0.20, and 0.18, respectively.

confirmed by the significant value of the coefficient. The independent variables include each income value (the minimum income being the omitted reference category), the picture treatment, and a time trend. They also include individual characteristics such as the subject's SCR amplitude before the decision is entered, this variable interacted with the picture treatment, the subject's gender, his/her age, the decision time and its squared value (that may indicate the depth of reasoning), and the amount invested in the lottery.

We consider SCR amplitude as an exogenous variable and not as a consequence of the cheating decision. Indeed, we assume that emotions consist of anticipatory and anticipated responses and they are used as a source of information in the decisionmaking process, as stated by Loewenstein et al. (2001) -see also Zajonc (1984) on the primacy of affect. Furthermore, emotional arousal is measured before the subject submits his report. 20 In the second step, we estimate two models of the proportion of evaded income by means of random-effects GLS models with robust standard errors. In column 2 of Table 2, we add to the regressors the inverse of the Mill's ratio extracted from the first-step estimation (à la Heckman) to control for a potential correlation among the error terms of the two equations. 21 In column 3, we include individual time-invariant characteristics such as opinion on various types of fraud, mother's education (equal to 1 if the mother went to University), and existence of a loan to pay tuition fees as a 20 Studies in psychology on decision-making support a three-stage model: (1) stimulus detection, (2) central processing and (3) motor execution. Miller and Low (2001) showed that the duration of motor execution stage is invariant across different reaction time tasks and its mean duration is around 200 ms.

The stimulus detection's duration is also recognized as invariant within the same task, leading to the conclusion that duration variability across trials is ultimately due to differences at the central processing stage. In our experiment, both stimulus detection (income display) and motor execution (income reporting) procedures were kept constant across trials. Thus, we interpret the whole interval between stimulus detection and motor execution as part of the decision making process itself. 21 The variables used in the regressions were dictated by the literature and to achieve identification of the parameters to be estimated.

proxy for the individual's wealth. 22 We also control for the first underreporting, for an audit in the previous period and the amount of the fine, if any. 23 (Table 2 about here)

The decision to evade equation shows first that the likelihood of evading taxes increases directly in the level of income. A second major finding is that an environment where deception is made public favors tax compliance relative to an environment where fraud is only punished by monetary sanctions. Ceteris paribus, the risk of non-monetary sanctions diminishes by 8.2% the probability to evade taxes.

This effect would have been probably even larger if we had directly redistributed the product of taxes among the subjects or in the absence of positive audit externalities.

A third major result is that the SCR amplitude measured before the decision is made has a marginal positive influence on evasion (with no additional treatment effect). The more subjects are emotionally aroused, the more they choose to evade. The marginal effect of one SCR point is 13.9% (note that SCR values are distributed between 0 and 2.24). In contrast, a subject who rejects the idea of cheating decides straight away to comply without experiencing a significant level of emotions.

The decision to evade taxes is also associated, although non linearly, with a longer decision time (the compliers decide in 9.25 seconds on average while the evaders' decision takes 12.42 seconds). This could be associated with both the higher cognitive requirement involved in the decision to evade taxes if the subject elaborates mixed strategies and a possible tension between the expected payoff of underreporting 22 Asking direct questions on individual resources is not a better alternative since almost no subject is working while studying. Asking about the parents' income is not better because some students are reluctant to report this information or are simply not well informed. 23 These lagged variables are not included in model 1 because this equation is better estimated on the full sample of observations since emotional arousal is stronger at the beginning of the game. In addition, it could have introduced an endogeneity bias since being fined in the previous period is directly associated with the decision to evade. We have omitted the insignificant lagged audit variable.

and the emotional consequences of violating the social norm of non-cheating. Indeed, a higher decision time may be correlated with both a higher cognitive activity and a higher emotional arousal (contrary to what is suggested by Rubinstein, 2007).

Columns 2 and 3 in Table 2 show that conditional on the decision to evade, the picture treatment exerts a strong negative effect on the proportion evaded (-3.466 points) while the higher the SCR, the higher the proportion evaded (with no additional effect of the treatment). This behavior is probably emotionally driven since the dissemination of pictures in case of detection is not related to the importance of fraud.

The proportion of evaded income is significantly lower when income is higher than 50 although the relationship is not linear. This higher evasion when income is low could result from the subjects having a reference level of income below which they are willing to take a higher risk of audit for avoiding the pain associated to a certain loss in case of compliance (see Bernasconi and Zanardi, 2004, and Dhami and Al-Nowaihi, 2007, for an application of cumulative prospect theory to tax evasion). In contrast with the decision to evade taxes, the decision time has a negative effect; indeed, those who decide to evade their income in full decide more quickly (8.69 seconds instead of 13.14 for those who do not fully evade).

In addition to a marginal reduction of the proportion of evaded income over time, the model in column 3 indicates that whereas being audited in the previous period reduces evasion weakly, the amount of the fine exerts the opposite effect. Individuals probably attempt to regain the previous loss by evading more in the current period, which is also consistent with the reference income dependence. People evade less when they underreport for the first time but this is significant at the 5% level only if this variable is included in model 2 (non reported regression) because this usually happens in the first period of the game while this period is eliminated from the third regression. A higher amount invested in the risky asset in the preliminary part of the experiment is associated with a higher proportion of evaded income, but significantly so only when we control for auditing in the previous period. Males are not more likely to evade taxes than females but when they do, they dissimulate a higher proportion of income (which is consistent with the literature, see Slemrod, 2007).

Having a mother with a university education reduces the proportion evaded, perhaps through the transmission of norms. Having to reimburse a loan has a positive impact on evasion. Lastly, a more negative opinion regarding drivers contriving the Highway

Code is significantly correlated with less evasion, while a negative opinion on fare dodgers came short of significance (p = 0.116). Variables such as the number of times a subject can see his own picture or the picture of other group members are not significant and are omitted. Last, the inverse of the Mill's ratios are insignificant.

To summarize, an environment where deception is made public increases compliance and reduces the proportion of evaded taxes relative to a policy using only monetary sanctions. We also find that a higher emotional arousal predicts more evasion and that if being audited in the previous period slightly reduces evasion, the amount of the fine increases it.

C. Reactions to an audit

We now focus on the determinants of the physiological SCR amplitude and selfreported feelings when people receive a feedback on audits. Indeed, the self-reports that have been collected at the end of each period regarding audit offer a robustness check and a complement to our analysis of physiological measures since the SCR measures cannot indicate the direction of emotions. Table 1 has shown that the average self-reported emotional arousal, measured on a scale from 1 (extremely weak) to 7 (extremely strong), was higher for tax evaders (4.20) than for non-evaders (3.11).

This corroborates the SCR measures (0.16 and 0.05, respectively). Similarly, evaders report less positive hedonic valence regarding the audit (3.84) than non-evaders (4.27), on a scale from 1 (extremely negative) to 7 (extremely positive). In addition, the correlation coefficient between the SCR amplitude at the time of the audit and the self-reported arousal is significant and equal to 0.54.

Three econometric models have been estimated to identify the determinants of emotional reactions to a feedback on audit. Model 1 is a random-effects Tobit model in which the dependent variable is the SCR amplitude at the time of the audit. Models 2 and 3 are ordered Probit models with robust standard errors and clustering at the individual level in which the dependent variables are the self-reported emotional arousal and the hedonic valence of feelings, respectively. The independent variables comprise a time trend, the proportion of evaded income, and a dummy variable for the first evasion. We also include a dummy variable for the first audit, the amount of the fine, the display of the subject's own picture, the first time the subject's picture is displayed, the current number of times the subject's picture has been displayed since the beginning of the game. Overall we expect that being sanctioned should increase emotional arousal and should exert a negative impact on hedonic valence while not being caught should raise positive emotions. The results are reported in Table 3.

(Table 3 about here)

The SCR amplitude indicates that the emotional arousal increases marginally in the degree of tax evasion when the subject receives a feedback on audit; this feeling is however not self-reported. Other things equal, this could capture the emotional effect of a higher expected payoff. The SCR amplitude is also higher when a subject evades for the first time although the self-reported intensity of feelings indicates the opposite.

Reported feelings are more negative when subjects evade for the first time, possibly due to guilt. They become more intense and more and more positive over time, possibly because of a dissipation of guilt.

The first time a subject is audited also raises significantly more emotions. An evader who is not audited is also emotionally aroused. Since this is associated with a higher hedonic valence, this could be due to relief and the pleasure due to the avoidance of a loss. Indeed, the average hedonic valence is 5.23 (3.94) for the evaders (compliers)

who have not been audited and 2.49 (4.59) when they have been audited.

Accordingly, a regression conducted on the sole compliers (not reported here) shows that being audited generates significant positive feelings.

A major finding is related to the consequences of both monetary and non-monetary sanctions on emotions, with almost all of these effects being significant at the 1% level. The higher the fine, the more the subject is emotionally aroused, and the more he expresses negative feelings, probably related to regret and anger. Moreover, when the evader learns that his picture is going to be disseminated, his emotional arousal further increases (with an additional first-time effect on the SCR amplitude), although this effect tends to diminish with repetition. 24 This is associated with more negative feelings, possibly related to shame; this interpretation is reinforced by the significant negative coefficient associated with the current number of times a subject has seen his picture disseminated, with no stronger effect of the first time his picture is displayed.

These self-reports should be taken with caution however since the statistical power of the regressions is relatively low, especially in model 2. In addition, due to their 24 We have estimated the effect of seeing the pictures of other group members only on the selfreported intensity and valence of feelings since self-reports are made at the end of each period (not reported here). The display of another's picture has no impact on the emotional arousal but it is associated with positive feelings. These feelings may capture a subject's satisfaction of not having been audited himself if he evaded taxes, or his pleasure of seeing a cheater being caught if he complied.

We cannot test this effect on SCR since at the time of the audit, the subjects do not know yet whether another group member has been caught.

subjective nature, they may be biased. Physiological measures, that the subjects cannot easily control, are potentially more reliable. We also acknowledge that we only capture here the direction of emotions but not their nature. Overall, the results

show nevertheless a large consistency between the SCR measures, the self-reported intensity and valence of feelings, and our behavioral predictions.

To summarize, the higher the evasion, the higher the SCR magnitude when receiving a feedback on audit. The higher the fine, the more subjects are emotionally aroused, and the more negative are their feelings. This emotional arousal further increases when the evaders learn that their behavior will be exposed publicly. 25 4. DISCUSSION AND CONCLUSION Deception may generate anticipatory emotions related to the risk of being detected and anticipated emotions related to both moral considerations and the prospect of higher monetary payoffs associated with cheating. Tax evasion constitutes an archetypical example of deception by the economic agents. This paper explores a complementary avenue to the standard deterrence models of tax evasion by testing the hypothesis that the prospect of evading taxes generates emotions in the individuals, which affect their actual behavior, especially when deception can be made public.

We link individual behavior both to Skin Conductance Responses that provide a physiological measure of the emotional arousal uncontrolled by the individuals and to affective self-reports that inform on the valence and intensity of emotions.

25 As suggest by a referee, exposing the amount not reported or in terms of the percentage of income not reported along with the picture of the participant caught underreporting his income will have added a nice feature to this research in terms of policy issue. Interestingly, Halla and Schneider (2009) based on a large micro data set of respondents from 29 OECD member countries suggest that moral values evolve endogenously and are determined by prices (i.e. the cost of acting morally). The picture treatment definitively affects the real price of cheating.

Our results are striking. We provide the first evidence that cheating is correlated with the emotional arousal of the subjects before submitting their decision. A higher emotional arousal is positively associated with both the likelihood of tax evasion and the amount evaded, indicating that this decision does not only result from the "cold" comparison between the monetary benefits and costs of evading taxes (without mentioning the social benefits that the subjects may derive from future research funding through taxes). The cognitive deliberation involves "hot" processes activated by the emotional brain. The subjects who comply are those who do not experience such emotions and they decide more quickly than evaders. Our results add to the literature in neurosciences showing that rational decision-making should not be opposed to emotions (see the theory of somatic markers Damasio, 1994).

Our findings also strongly support the importance of tax morale and justify incorporating moral dimensions in the standard models of tax compliance. Indeed, when the subjects receive a feedback on their report audit, controlling for the proportion of evaded income, we find a strong physiological impact of the public display of the evaders' pictures on the emotional arousal in addition to monetary sanctions. This may explain that when they are in an environment where nonmonetary sanctions coexist with monetary sanctions the subjects are less likely to cheat and when they do, they evade less. Furthermore, while monetary fines increase later evasion, showing the tax evaders' picture has the opposite effect. The selfreports reveal that both monetary and non-monetary sanctions raise negative emotions. In contrast, learning that their report has not been audited generates positive feelings in evaders, probably due to the conjunction of relief and the joy of getting higher earnings than if they had complied.

There are reasons for caution in extrapolating these results. In particular, students may not be representative of taxpayers and the size of our sample is relatively small.

We have also used audit probabilities that are above the average enforcement values and we cannot measure how various audit probabilities would have changed decisions and emotions (which would be an interesting extension of our paper). Moreover, cheating in our experiment may not be considered as bad as in a real setting since more cheating may lower the audit probability of other players. One can, however, be relatively confident in the external validity of our results because if one observes that emotions arise even in the sterile environment of the laboratory with small monetary stakes, such high audit probabilities leading to frequent audits, and positive immediate externalities in the laboratory, they should be even greater when a yearly income is at stake in actual reporting decisions, with less frequent audits and negative externalities.

Indeed, we have found that the first time a subject evades taxes, his emotions are stronger and more negative when he receives a feedback on being audited, while observing frequently one's own picture affects emotional arousal negatively. One can also suspect that when cheating hurts directly other members in a group, as if the product of taxes was immediately redistributed among the taxpayers, emotions would probably be stronger. Therefore, we hope to have contributed to identify some relationships between emotions, sanctions, and cheating.

A limitation is that our methodology does not allow us to identify the very nature of emotions. We can, however, suggest interpretations. Since higher SCR measured at the time of decision is associated with a higher probability to cheat and to a larger deception, this may be due both to the cheater's anxiety related to the risk of deterrence and to the anticipated excitement of earning more money if not detected, and anticipated negative emotions if detected. The emotions experienced after a positive audit in the benchmark treatment are probably associated with private emotions such as guilt, regret, or anger, whereas those experienced in the picture treatment may be associated with public emotions such as shame or embarrassment.

One can suggest two main directions for further research. First, regarding economic modeling, the emotional costs and benefits of cheating should be included in the individual's utility function in interaction with its expected monetary return. With the public exposure of deception, people evade less. Since we held the expected monetary return from evasion constant across treatments, this suggests that the net emotional cost of deception has been increased. An interpretation is that when the net cost of anticipated and anticipatory emotions increases, people evade less. Brain imaging could help in measuring these emotional costs and benefits with a distinction -if relevant-between the emotional costs associated with risk-taking and the emotional cost associated with norm violation.

A second direction for research is related to institutional design. We have found that, ceteris paribus, being in an environment where deception is made public reduces by 8.20% the probability to cheat; for the evaders, it reduces by 3.88 points the proportion of evaded income although by design the non-monetary sanction does not depend on the amount evaded. In contrast, while being audited reduces the proportion of evaded income by 2.92 points, we observe that the higher the fine, the higher the dissimulation of income in the following period. This suggests that a policy threatening to denounce cheaters publicly with a reduced amount of fines -i.e.

augmenting the emotional cost and reducing the monetary cost-might be more effective. Further research is however needed to explore the potential backfiring effects of a public exposure of deception. Indeed, once their reputation is lost, evaders may react by evading even more. We have found that frequent exposure raises more negative feelings but it also reduces the emotional arousal in case of an audit and overall we do not observe in the Picture treatment an increase of compliance over time. It would be useful to manipulate experimentally the duration of social sanctions to see whether reintegrative shaming -i.e. offering an opportunity to restore reputation-circumvents the potential backfiring effects of these sanctions. Appendix. The physionomics laboratory and the skin conductance data analysis Experimental sessions took place in a noiseless room with stable temperature set to 21 o C. Skin conductance activity was recorded with a BIOPAC MP150W system and two TEL100C telemetry modules (BIOPAC Systems, EU). Two Ag/AgCl electrodes filled with 0.5% saline in a neutral-base paste were placed on the subject's distal phalanges of the middle and the index fingers of the non-dominant hand, after the attachment site had been cleaned with a neutral soap (Dawson et al., 2000). A constant voltage of 0.5 V was applied between the electrodes. The skin conductance signal was amplified (x 2000) and low-pass filtered (30 Hz) before being sampled at 125 Hz. Skin conductance activity was continuously recorded until the end of the session.

Regarding the skin conductance analysis, the skin conductance signal was low-pass filtered at 0.5 Hz offline, using a fifth-order Butterworth low-pass digital filter. Skin Conductance Responses (SCR) onset and peak were automatically detected, when the first derivative of the filtered signal changed sign, by a routine written in Matlab (The MathWorks Inc., USA). Onsets were identified by a negative to positive zero crossing, while peaks were identified by a positive to negative zero crossing. The SCR amplitude was calculated as the difference between the signal amplitude at the peak and the onset times. The SCR amplitude was thresholded at 0.02 µS (Dawson et al., 2000). The whole signal was visually inspected prior to further analysis and false SCR detections were removed. The detection of an accelerative deflection during the interval between onset and peak times indicated SCRs overlapping during rise time. In that case, the two SCRs were i) separated, if they could be related to different events (according to the epochs defined below), or ii) summed together, if they were related to a single event (see Boucsein, 1992). In the case of responses overlapping during recovery time, Edelberg (1967) has shown that the amplitude scoring based on the difference between the signal amplitude at the peak and the onset times is sufficiently accurate and it is used as a standard procedure. In order to minimize SCR overlapping between events of interest, we imposed a minimum interval between events of six seconds.

In long-term exosomatic continous current recording, skin conductance signal may show a downward drift that is usually associated to a progressive polarization of electrodermal electrodes (Edelberg, 1967). An adequate correction of drift is only possible when calibration values are recorded during the signal acquisition and it is not an imperative pre-processing procedure for short-term recordings (Boucsein, 1992). For that reason, SCR amplitudes were not corrected for electrodermal activity drift in our analysis. SCR habituation effect was balanced across treatments since every treatment had the same number of periods. The habituation effect was further accounted within the regression model by the introduction of time as confound. SCRs were analyzed for the decision, the feedback on audit and the feedback on payoff events. SCRs with onset between 1.0 and 3.0 s after the beginning of an audit or payoff were assumed specific to the event. With respect to the decision event, given its self-paced duration, SCRs were allowed onset time from 1.0 s after the subject is informed on his income until his decision. Indeed, we are especially interested in measuring the intensity of emotions before the subject makes his reporting decision, because it should capture the emotional content of the subject's deliberation.

Table 1 .

 1 Summary statistics The Table indicates average values; the standard deviations are in parentheses.

	Benchmark	Picture	All treatments
	treatment	treatment

Self-reports on the emotional valence and arousal dimensions are correlated with autonomic and somatic responses to emotional stimuli, while SCR positively correlate with the emotional arousal independently of valence(Bradley and Lang, 2001)

Note that the self-reports do not inform more than SCR measures on the very nature of emotions (anger, guilt, …). An alternative would have been to ask the subjects to report the intensity of each of various possible feelings at the end of each period. But in addition to be time-consuming and boring, this could have focused too artificially the subjects' attention on emotions. In the attitudinal questionnaire, the subjects had to report on a 7-point Likert-type scale their feelings (from extremely negative to extremely positive) regarding tax evasion by shopkeepers, fare-dodging, and offences against the Highway Code.

We cannot characterize the mixed strategy equilibrium because we use endogenous audit probabilities. In the picture treatment players get to know other players through the dissemination of pictures. We do not believe however that the game should be considered as a repeated game since only one picture can be disseminated in one period and not all group members evade.

California at Santa Barbara, at the Universities of Aarhus, Copenhagen, and Liege, and at the WZB in Berlin for their comments and suggestions. We thank R. Zeiliger for programming the experiment. This research was supported by a grant from the Rhône-Alpes Region (CIBLE program).

significance at the 0.10 and 0.01 level, respectively. The SCR values lie between 0 and 2.129. The self-reported emotional arousal variable can take any integer value between 1 (extremely low) and 7 (extremely high). The hedonic valence variable is coded from 1 (extremely negative) to 7 (extremely positive).


Vladimir S Matveev

Pierre Mounoud

Gallot-Tanno Theorem for closed incomplete pseudo-Riemannian manifolds and applications

Gallot-Tanno Theorem for closed incomplete pseudo-Riemannian manifolds and applications.

1 Introduction.

Main result

Let (M, g) be a pseudo-Riemannian manifold and let D be the corresponding covariant derivative. Along this paper we will consider Riemannian metrics as pseudo-Riemannian. We are interested in the existence of non-constant functions α on M such that for all vector fields X, Y , Z on M we have:

DDDα(X, Y, Z) + c • 2(Dα ⊗ g)(X, Y, Z) + (Dα ⊗ g)(Y, X, Z) + (Dα ⊗ g)(Z, X, Y ) = 0, [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF] where c is a real constant. This is a linear system of PDE on α, in the "tensor" notation it reads α ,ijk = c • (2α ,k g ij + α ,j g ki + α ,i g kj ) .

The main result of this article is the following Theorem 1. Let (M, g) be a closed (i.e., compact without boundary) connected pseudo-Riemannian manifold and c be a real number. If α : M → R is a non-constant function satisfying [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF], then c = 0 and the metric c • g is a Riemannian (i.e., positively definite) metric of constant curvature 1.

History, motivation, and applications

The equation (1) has already been studied, mostly in the Riemannian setting. The motivation of Gallot and Tanno to study this equation came from the spectral geometry: it is well-known (see for example [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF]) that, on the standard sphere S n ⊂ R n+1 of dimension n > 1, all eigenfunctions corresponding to the second biggest eigenvalue -n of the laplacian satisfy the equation

DDα(X, Y ) + α • g(X, Y ) = 0, (2)
(for all vector fields X, Y , Z on M ). The eigenfunctions corresponding to the third biggest eigenvalue -2(n + 1) satisfy [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF] with c = 1. Moreover, Obata has shown [15, Theorem A] that, on closed Riemannian manifolds, the existence of a nonconstant solution of [START_REF] Carrière | Complétude des métriques lorentziennes de T 2 et difféormorphismes du cercle[END_REF] implies that the metric has constant curvature 1. Later, he [START_REF] Obata | Riemannian manifolds admitting a solution of a certain system of differential equations[END_REF], and, according to Gallot [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF], Lichnerowicz, asked the question whether the same holds for the equation (1) (assuming c = 1). The affirmative answer was given in [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF][START_REF] Tanno | Some differential equations on Riemannian manifolds[END_REF]. Theorem 1, which is the main result of our paper, generalizes the results of [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF][START_REF] Tanno | Some differential equations on Riemannian manifolds[END_REF] to pseudo-Riemannian metrics.

This equation also appears in the context of geodesic equivalence. Recall that two metrics g and g ′ are geodesically equivalent (affinely equivalent, resp.) if every g-geodesic is a reparametrized g ′ -geodesic (if their Levi-Civita connections coincide, resp.) A vector field V on (M, g) is called projective (affine, resp.), if its local flow acts by projective transformations (affine transformations, resp.), i.e., takes unparameterized geodesics to geodesics (preserves the Levi-Civita connection, resp.).

In particular, Solodovnikov [START_REF] Solodovnikov | Projective transformations of Riemannian spaces[END_REF] has showed that if a Riemannian metrics g admits "lots" (we formalize this notion in section 5) of geodesically equivalent, but not affinely equivalent metrics, then there exists a real number c and a non-constant solution of (1), see section 5 for definitions and more precise statements. This result has been recently extended to the pseudo-Riemannian setting in Kiosak et al [START_REF] Matveev | Proof of the Lichnerowicz Conjecture[END_REF]. Hiramatu [START_REF] Hiramatu | Riemannian manifolds admitting a projective vector field[END_REF] has shown that if a Riemannian metric of constant scalar curvature on a closed manifold admits a nonaffine projective vector field, then there also exists a nonconstant solution of the equation ( 1) for a certain constant c. Kiosak et al [START_REF] Kiosak | Complete Einstein metrics are geodesically rigid[END_REF] has shown that if an Einstein metric g admits a geodesically equivalent, but not affinely equivalent metric g ′ , then there exists a non-constant solution of (1) for a certain constant c.

The equation ( 1) naturally appears in the study of the geometry of the metric cones, see Gallot [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF] or Alekseevsky et al [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF]. We will explain the relation between cones with decomposable holonomy and the equation (1) in section 3. In fact, this relation is one of the main tools of our proof.

Combining theorem 1 with the results listed above, we obtain that on a closed manifold any pseudo-Riemannian metric admitting "lots" of geodesically equivalent, but not affinely equivalent metrics is, up to multiplication by a constant, the Riemannian metric of constant curvature 1 (cf. corollary 5.2).

We also obtain that Einstein pseudo-Riemannian metrics of nonconstant curvature on closed manifolds are geodesically rigid, in the sense that every metric geodesically equivalent to them are actually affinely equivalent to them (cf. corollary 5.1).

We also obtain that metric cones with decomposable holonomy group over closed pseudo-Riemannian metrics are Riemannian and flat (cf. proposition 4.1). We apply this result to closed nonzero constant curvature (non-Riemannian) manifolds i.e., to manifolds locally modeled on a pseudosphere S p,q . We obtain that, for such a manifold, the holonomy group of the associated (O(p + 1, q), S p,q )-structure does not preserve any non degenerate splitting of R p+1,q (cf. corollary 6.1). In [START_REF] Zeghib | On closed anti de Sitter spacetimes[END_REF], Zeghib proved this statement under the additional condition of completeness.

Previous results

Partial versions of theorem 1 were known before. In the Riemannian case, theorem 1 is due to Gallot [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF]Corollaire 3.3] and Tanno [18, Theorem A] under the assumption c > 0, and is due to Hiramatu [START_REF] Hiramatu | Riemannian manifolds admitting a projective vector field[END_REF]Lemma 2] under the assumption c ≤ 0.

Moreover, Gallot and Tanno assumed only completeness (instead of closedness). In the realm of Riemannian geometry, the unit tangent bundle of a compact manifold being compact, closeness implies completeness. This is no more true in the pseudo-Riemannian geometry, where incomplete metrics on compact manifolds are abundant. For example, by Carrière et al [START_REF] Carrière | Complétude des métriques lorentziennes de T 2 et difféormorphismes du cercle[END_REF] the set of incomplete Lorentzian 2 dimensional tori is dense in the set of Lorentzian tori. Completeness and closeness are quite independent properties in the pseudo-Riemannian geometry, and it is not an easy task to understand whether a given metric on a closed manifold is complete. Moreover, [1, Example 3.1] from Alekseevsky et al provides non-compact complete pseudo-Riemannian manifolds of non-constant curvature admitting non-constant solutions to [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF]. Moreover, under the additional assumption that the metric is complete theorem 1 is easy, see [START_REF] Matveev | Gallot-Tanno theorem for pseudo-Riemannian manifolds and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist[END_REF]Theorems 1,[START_REF] Carrière | Complétude des métriques lorentziennes de T 2 et difféormorphismes du cercle[END_REF].

1.4 Organisation of the paper and the converse statement.

The round sphere S n := {(x 1 , ..., x n+1 ) ∈ R n+1 | (x 1 ) 2 + ... + (x n+1 ) 2 = 1} with the standard metric admits a lot of nonconstant solutions of (1) (with c = 1): as we mentioned in §1.2, every eigenfunction of the Laplacian corresponding to the third biggest eigenvalue -2(n + 1) satisfies [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF]. By our theorem, any closed manifold admitting a non constant solution of (1) is, up to a constant, a quotient of S n , but certain of those quotients do not admit nonconstant solutions of (1).

Indeed, let M be the quotient of S n by a discrete subgroup Γ ⊂ O(n + 1). The cone over M is the quotient of the cone over S n (i.e., of R n+1 \ {0} endowed with the euclidean metric) by Γ. By proposition 3.4, M admits a non-trivial solution of (1), if and only if its cone is decomposable i.e., if and only if Γ preserves an orthogonal splitting of R n+1 , or equivalently if and only if there exists 0 < p < n + 1 such that Γ ⊂ O(p) × O(n + 1p). Thus, the only quotients of the sphere S 3 admitting nontrivial solutions of (1) are the lens spaces. It follows that the Poincaré homology sphere (which is the quotient of the standard 3-sphere by the lift of the group of direct isometries of the regular dodecahedron) admits no nonconstant solution of [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF].

The organization of the article is as follows. In section 2 we prove theorem 1 under the additional assumption c = 0. The rest of the paper is devoted to the case c = 0 -we will explain in remark 2.1, that if c = 0, then without loss of generality we can assume c = 1. In section 3 we establish a link between solutions of (1) (with c = 1) and parallel symmetric (0, 2)-tensors on the cone over (M, g): we show that the existence of a non-constant solution of (1) is equivalent to that the cone is decomposable. In section 4 decomposable cones are studied and theorem 1 is proved. Section 5 is devoted to the application of theorem 1 in the theory of geodesically equivalent metrics. Section 6 is devoted to the study of the holonomy of closed manifolds with constant nonzero curvature.

2 Proof of theorem 1 under the assumption c = 0.

Assume c = 0. Equation [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF] implies that the Hessian of α is parallel. Since the manifold is closed, α has a minimum and a maximum. At a minimum, the Hessian must be nonnegatively definite, and at a maximum it must be nonpositevely definite. Therefore the Hessian is null, and the gradient of α is parallel. But as it vanishes at the extremal points, it vanishes everywhere and α is constant. Theorem 1 is proved under the assumption c = 0. Remark 2.1. If c = 0, without loss of generality we can assume c = 1. Indeed, if a function α is a solution of (1) with c = 0, then it is also a solution of the equation

DDDα(X, Y, Z) + 2(Dα ⊗ g ′ )(X, Y, Z) + (Dα ⊗ g ′ )(Y, X, Z) + (Dα ⊗ g ′ )(Z, X, Y ) = 0 (3)
for g ′ := c • g. Since the Levi-Civita connections of g and of g ′ coincide, the equation (3) is the equation (1) with respect to the metric g ′ with c = 1.

3 Parallel symmetric (0, 2)-tensors on the cone over a manifold and nonconstant solutions of (1) for c = 1.

Let (M, g) be a pseudo-Riemannian manifold. The cone manifold over (M, g) is the manifold M = R >0 × M endowed with the metric g defined by g = dr 2 + r 2 g (i.e., in the local coordinate system (r, x 1 , ..., x n ) on M , where r is the standard coordinate on R >0 , and (x 1 , ..., x n ) is a local coordinate system on M , the scalar product in g of the vectors

u = u 0 ∂ r + n i=1 u i ∂ x i and v = v 0 ∂ r + n i=1 v i ∂ x i is given by g(u, v) = u 0 v 0 + r 2 n i=1 g ij u i v i
). We will denote by D the Levi-Civita connection of g and by D the Levi-Civita connection of g. The holonomy of cones over pseudo-Riemannian is strongly related to the equation ( 1). This relation is given by the following proposition, which is almost contained in the proofs of [3, corollaire 3.3] (for an implication) and in [12, Corollary 1] (for the reciprocal). As we will use some lines from it, as those proofs have a non empty intersection, and for the convenience of the reader, we give its proof but it does not pretend to be new. Proposition 3.1. Let (M, g) be a pseudo-Riemannian manifold, let c = 1. Let ( M , g) be the cone manifold over (M, g).

Then, there exists a non-constant function α : M → R satisfying (1), if and only if there exists a non-trivial (i.e., not proportional to g) symmetric parallel (i.e., the covariant derivative vanishes) (0, 2)-tensor on ( M , g).

More precisely if α is a non-constant solution of (1) then the Hessian of the function

A : M → R defined by A(r, m) = r 2 α(m) is non-trivial and parallel (i.e., D D DA = 0). Conversely if T is a non-trivial symmetric parallel (0, 2)-tensor on M then T (∂ r , ∂ r ) does not depend on r and is a non-constant solution of (1). Moreover 2 T is the Hessian of the function A defined by A(r, m) := r 2 T (r,m) (∂ r , ∂ r ).
In the proof of proposition 3.1, we will need the following two statements; in these statements X, Y , Z will denote arbitrary vector fields on M . We will also denote by the same letters X, Y, Z the lift of these vector fields to M . Fact 3.2 (for example, [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF][START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF][START_REF] Matveev | Gallot-Tanno theorem for pseudo-Riemannian manifolds and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist[END_REF]). The Levi-Civita connection of g is given by

D X Y = D X Y -rg(X, Y )∂ r , D ∂r ∂ r = 0, D ∂r X = D X ∂ r = 1 r X. (4)
Proof. We take a point (r, m) ∈ M . Without loss of generality we can assume that

DX(m) = DY (m) = DZ(m) = 0. Using g([X, Y ], ∂ r ) = 0 and [∂ r , X] = [∂ r , Y ] = 0, we have 2 g( D X Y, ∂ r ) = -∂ r . g(X, Y ) = -2rg(X, Y ).
Similarly we have that 2 g( D X Y, Z) = r 2 g(D X Y, Z). It implies the first assertion. The two others can be shown the same way.

In the next corollary we will tautologically identify M with M 1 := {1} × M ⊂ M (the point m ∈ M will be identified with (1, m) ∈ M 1 ). By definition of the cone metric, g |M 1 = g.

Corollary 3.3 ([12]

). Let T be a symmetric parallel (0, 2)-tensor on ( M , g). Then, α := T (∂ r , ∂ r ) does not depend on r and can be considered therefore as a function on M . Moreover, for every m ∈ M and every X, Y, Z ∈ T m M we have

2 T (∂ r , X) = rDα(X) (5) 2 T (X, Y ) = r 2 (2α g(X, Y ) + DDα(X, Y )) , (6)
Moreover, α is constant if and only if T is proportional to g. Moreover, for every

(1, m) ∈ M 1 we have 2DT (X, Y, Z) = -Dα ⊗ g(Y, X, Z) -Dα ⊗ g(Z, X, Y ), ( 7
)
where T is the restriction of the tensor

T to M 1 ( tautologically ≡ M ).
Proof. Since T is parallel, we have

0 = D T (∂ r , ∂ r , ∂ r ) = ∂ r . T (∂ r , ∂ r ) -2 T ( D ∂r ∂ r , ∂ r ) (4) = ∂ r . T (∂ r , ∂ r ) = 0.
Thus T (∂ r , ∂ r ) is a function on M . The first statement of corollary 3.3 is proved.

Combining D T = 0 with Fact 3.2, we have:

0 = D T (X, ∂ r , ∂ r ) = Dα(X) -2 T ( D X ∂ r , ∂ r ) (4) = Dα(X) -2 r T (X, ∂ r ). ( 8
)
This shows [START_REF] Hiramatu | Riemannian manifolds admitting a projective vector field[END_REF]. Similarly, using Fact 3.2 and (5), we obtain

0 = D T (X, Y, ∂ r ) = X. T (Y, ∂ r ) -T ( D X Y, ∂ r ) -T (Y, D X ∂ r ) (4),(5) = r 2 DDα(X, Y ) + 2g(X, Y )α -1 r T (Y, X).
This shows [START_REF] Kiosak | Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two[END_REF]. If α = const, DDα = 0. Then, ( 5), [START_REF] Kiosak | Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two[END_REF], and the definition of

α implies T = α • g. If T = const • g, then T (∂ r , ∂ r ) = const • g(∂ r , ∂ r ) = const.
Similarly, using Fact 3.2 and already proved parts of corollary 3.3, we obtain (for every (1,

M ) ∈ M 1 ⊂ M ) 0 = D T (X, Y, Z) = X. T (Y, Z) -T ( D X Y, Z) -T (Y, D X Z) (4) = X.T (Y, Z) -T (D X Y, Z) -T (Y, D X Z) + g(X, Y ) T (Z, ∂ r ) + g(X, Z) T (Y, ∂ r ) (5) = DT (X, Y, Z) + 1 2 Dα ⊗ g(Y, X, Z) + Dα ⊗ g(Z, X, Y )
implying [START_REF] Kiosak | Complete Einstein metrics are geodesically rigid[END_REF].

Proof of proposition 3.1.

Equation ( 1) being tensorial we can suppose without loss of generality that DX = DY = DZ = 0. We set X := 1 r X, Y := 1 r Y , Z := 1 r Z. By Fact 3.2, we have:

D ∂r ∂ r = D ∂r X = D ∂r Y = D ∂r Z = 0, D X ∂ r = 1 r X and D X Y = -1 r g(X, Y )∂ r . (9)
Let α be a solution of (1), and A = r 2 α(m). Our first goal is to show that D D DA = 0. We have:

DA( Z) = rDα(Z) and DA(∂ r ) = 2rα. ( 10
)
Then,

D DA( Z, ∂ r ) = D DA(∂ r , Z) = ∂ r .(r Dα(Z)) -DA( D ∂r Z) = Dα(Z), (11)
and similarly

D DA(∂ r , ∂ r ) = 2α (12)
Using that, DY (m) = 0, we get Y.DA(Z) = DDA(Y, Z) and

D DA( Y , Z) = Y .(rD Z α) + 1 r g(Y, Z) DA(∂ r ) = DDα(Y, Z) + 2g(Y, Z)α. (13)
Now we can prove that D D DA = 0, we will first show that D D DA(∂ r , ., .) = D D DA(., ∂ r , .) = D D DA(., ., ∂ r ) = 0:

D D DA(∂ r , X, Y ) = ∂ r . D DA( X, Y ) (13) = ∂ r . (DDα(Y, Z) + 2g(Y, Z)) = 0, D D DA(∂ r , X, ∂ r ) = D D DA(∂ r , ∂ r , X) = ∂ r . D DA(∂ r , X) (11)
= ∂ r . (Dα(X)) = 0, D D DA(∂ r , ∂ r , ∂ r ) = ∂ r . D DA(∂ r , ∂ r ) (12) = 2∂ r .(α) = 0, D D DA( X, ∂ r , ∂ r ) = X. D DA(∂ r , ∂ r ) -2 D DA D X ∂ r , ∂ r (10),( 12
) = X. (Dα(X)) = 0. D D DA( X, ∂ r , Y ) = D D DA( X, Y , ∂ r ) (11),(9) = X. Y .α -1 r D DA( X, Y ) -1 r g(X, Y ) D DA(∂ r , ∂ r ) (12), (13)
= 0

The last thing to check is

D 3 A(X, Y , Z) = X.( D DA( Y , Z)) -D D( D X Y , Z) -D D( Y , D X Z) = X.(DDα(Y, Z) + 2g(Y, Z)α) + D DA 1 r g(X, Y )∂ r , Z + D DA 1 r g(X, Z)∂ r , Y = DDDα(X, Y, Z) + 2g(Y, Z)Dα(X) + g(X, Y )Dα(Z) + g(X, Z)Dα(Y ) = 0.
Thus, D D DA = 0. The proposition is proved in the "=⇒" direction.

Let us now prove the proposition in the "⇐=" direction. We take a point (1, m) ∈ M 1 . Covariantly differentiating [START_REF] Kiosak | Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two[END_REF] with the help of D and substituting [START_REF] Kiosak | Complete Einstein metrics are geodesically rigid[END_REF], we obtain 0 = -DT (X, Y, Z) + 2g(Y, Z) X.α + DDDα(X, Y, Z)

(7) = 2(Dα ⊗ g)(X, Y, Z) + (Dα ⊗ g)(Y, X, Z) + (Dα ⊗ g)(Z, X, Y ) + DDDα(X, Y, Z).
Comparing corollary 3.3 and ( 11), [START_REF] Matveev | Gallot-Tanno theorem for pseudo-Riemannian manifolds and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist[END_REF], [START_REF] Matveev | Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics[END_REF] we see that 2 T is the Hessian of A := r 2 α. Let us recall that a pseudo-Riemannian manifold is said to be decomposable if it possess a nontrivial parallel non-degenerate (i.e., the restriction of the metric to it is nondegenerate) distribution.

By [START_REF] Wu | On the de Rham decomposition theorem[END_REF], if a pseudo-Riemannian manifold is decomposable, then the manifold can be locally written as the product (M 1 , g 1 ) × (M 2 , g 2 ) of two pseudo-Riemannian manifolds; the tangent space of M 1 naturally embedded in the tangent space of the product is precisely the parallel distribution. The tangent space of M 2 is the orthogonal complement to the parallel distribution, which is itself also a nondegenerate parallel distribution.

Contrarily to the Riemannian case, the existence of a parallel symmetric (0, 2)-tensor on a pseudo-Riemannian manifold does not imply that the manifold is decomposable. It is a consequence of the fact that the self-adjoint endomorphism associated to such a tensor and the metric can not always be simultaneously diagonalized. However, the situation is more simple for cones over closed manifolds as shows the following Proposition 3.4. Let (M, g) be a closed pseudo-Riemannian manifold and c = 1. If the equation (1) has a non-constant solution then ( M , g) is decomposable.

Proof. Let α be a non-constant solution of (1) on M . As M is closed there exists two critical points m -and m + of α associated to distinct critical values (i.e., Dα(m ± ) = 0 and α(m -) = α(m + )). As Dα(m ± ) = 0, it follows from [START_REF] Matveev | Proof of the Lichnerowicz Conjecture[END_REF] and ( 12) that, for any r > 0, D DA (r,m ± ) (∂ r , .) vanishes on T M and takes the value 2α(m ± ) on ∂ r . It means that

D DA (r,m ± ) (∂ r , .) = 2α(m ± )g (r,m ± ) (∂ r , .). ( 14
)
Since the eigenvalues of the parallel tensor are constants, 2α(m -) = 2α(m + ) are two different eigenvalues of the field of self-adjoint endomorphisms associated to D DA at every points. Since the field of self-adjoint endomorphisms associated to D DA is also parallel, its characteristic spaces (=generalized eigenspaces) provide a parallel orthogonal decomposition of T M . Indeed, they are clearly nondegenerate; as there are at least two distinct eigenvalues each characteristic space is non trivial. Then, ( M , g) is decomposable.

Proposition 3.4 does not say that a cone over a closed manifold with interesting holonomy is automatically decomposable. For example, the cone may admit anti-symmetric parallel (0, 2)-tensors. The reader can consult Alekseevsky et al [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF] for a more systematic study of the holonomy of cones.

4 Decomposable cones over closed manifolds and the proof of Theorem 1.

The goal of this section is to prove Proposition 4.1. Let (M, g) be a closed connected pseudo-Riemannian manifold such that the cone ( M , g) is decomposable. Then, g is the Riemannian flat metric, and g is the Riemannian metric of constant curvature 1.

Proof. Let V 1 and V 2 := V ⊥ 1 be the complementary nondegenerate parallel distribution on M . Let T 1 and T 2 be the symmetric (0, 2)-tensors on M defined for i ∈ {1, 2} by

T i (v, u) = g(v i , u),
where the v i 's are the factors of the decomposition of v according to the splitting

T M = V 1 ⊕ V 2 .
Clearly, T 1 + T 2 = g. Since the distributions V i are parallel, then the tensors T i are also parallel.

We set

α i := T i (∂ r , ∂ r ). Since g(∂ r , ∂ r ) = 1, we have α 1 + α 2 = T 1 (∂ r , ∂ r ) + T 2 (∂ r , ∂ r ) = g(∂ r , ∂ r ) = 1.
As in section 3, we define the functions A 1 and A 2 on M by A i (r, m) = r 2 α i (m).

Applying proposition 3.1 to the tensors T i we obtain that the following statements hold for every i = 1, 2:

( * ) α i is a non constant function on M and it is a solution of the equation [START_REF] Alekseevsky | Cones over pseudo-Riemannian manifolds and their holonomy[END_REF].

( * * ) The Hessian of A i is 2 T i .
Let us prove that the only possible critical values of α i are 0 and 1. Moreover, for any m ∈ M , we have 0 ≤ α i (m) ≤ 1, the extremal values being reached.

Since α 1 + α 2 = 1, it is sufficient to prove this statement for α 1 . Let m ∈ M be a critical point of α 1 . As we already saw, at [START_REF] Obata | Riemannian manifolds admitting a solution of a certain system of differential equations[END_REF], it implies that at the point (r, m)

2α 1 (m)g(∂ r , .) = D DA 1 (∂ r , .) ( * * ) = 2 T 1 (∂ r , .).
Then, ∂ r (r, m) is an eigenvector of the self-adjoint endomorphism associated to T 1 , and α 1 (m) is the eigenvalue of this endomorphism. Since the only eigenspaces of T 1 are V 1 (with eigenvalue 1) and V 2 (with eigenvalue 0), then α 1 (m) = 0 or α 1 (m) = 1. Thus, the only critical values of α 1 are 0 and 1.

Since M is closed, there exists m 1 , m 0 ∈ M 2 such that α 1 (m 1 ) = max m∈M α 1 (m) and α 1 (m 0 ) = min m∈M α 1 (m). Then, dα 1 (m 1 ) = dα 1 (m 0 ) = 0, implying α 1 (m 1 ) = 1 and α 1 (m 0 ) = 0.

Let us prove that the tensor T 1 is nonnegatively definite. We take the point (r, m 0 ) ∈ M . This point is a minimum of the function A 1 . Indeed, A 1 (r, m) = r 2 α 1 ≥ 0, and

A 1 (r, m 0 ) = r 2 α 1 (m 0 ) = 0. Since (r, m 0 ) is a minimum, at this point D DA 1 is nonnegatively defined. Since 2 T 1 ( * * ) = D DA 1 ,
T 1 is nonnegatively defined at the point (r, m 0 ). Since T 1 is parallel, it is also nonnegatively defined at every point of M .

Similarly, one can prove that T 2 is nonnegatively defined: instead of the point (r, m 0 ) one should take the point (r, m 1 ) where the function A 2 accepts its minimum.

Since g = T 1 + T 2 , it is also nonnegatively defined. Since it is nondegenerate, it is positively defined, i.e., is a Riemannian metric. As we recalled in the introduction, the Riemannian version of theorem 1 was proved by Gallot [START_REF] Gallot | Équations différentielles caractéristiques de la sphère[END_REF] and Tanno [START_REF] Tanno | Some differential equations on Riemannian manifolds[END_REF]. Thus, by Gallot-Tanno Theorem, g has constant curvature equal to 1, and g is the Riemannian flat metric.

Remark 4.2. The hypothesis of compactness in Proposition 4.1 is only use to obtain that the function α i defined during the proof has a minimum and a maximum. Hence, we could replace the hypothesis of compactness by this weaker one.

Proof of theorem 1. The case c = 0 was done in section 2. By remark 2.1, we can assume c = 1. By proposition 3.4, the existence of a nonconstant solution of (1) implies that the cone ( M , g) is decomposable. By proposition 4.1, g is a Riemannian metric of constant curvature 1.

5 Application I: geodesic rigidity of Einstein manifolds and projective Obata conjecture.

The set of metrics geodesically equivalent (the definition is in §1.2) to a metric g is in one-to-one correspondence with the set of nondegenerate symmetric (0, 2)-tensors T such that for any vector fields

X, Y, Z on M DT (X, Y, Z) = 1 2 (Dtrace(T ) ⊗ g(Y, X, Z) + Dtrace(T ) ⊗ g(Z, X, Y )) , (15)
where the trace and the covariant derivative are taken according to g, see for example [7, §2.2] for details (in the "tensor" notations, the equation ( 15) reads T ij,k = 1 2 T p p,i g jk + T p p,j g ik .) Since this equation is linear, the space of its solutions is a linear vector space. Its dimension is called the degree of mobility of g.

Locally, the degree of mobility of g coincides with the dimension of the set (equipped with natural topology) of metrics geodesically equivalent to g.

It is easy to see that if α is a solution of the equation ( 1) then the tensor defined by ( 13) is a solution of [START_REF] Obata | Certain conditions for a Riemannian manifold to be isometric with a sphere[END_REF]. Indeed, one can check it directly, or one can use that for the covariantly-constant by proposition 3.1 tensor T = D DA the tensor ( 13) is precisely the tensor T from corollary 3.3. Then, it satisfies the equation [START_REF] Kiosak | Complete Einstein metrics are geodesically rigid[END_REF], which is equivalent to [START_REF] Obata | Certain conditions for a Riemannian manifold to be isometric with a sphere[END_REF].

In some cases the reciprocal is true, hence theorem 1 has the following corollaries.

Corollary 5.1. Let g be an Einstein (i.e., the Ricci tensor is proportional to g) pseudo-Riemannian metric on an (n > 2)-dimensional closed connected manifold. Assume that ḡ is geodesically equivalent to g, but is not affinely equivalent to g. Then for a certain constant c = 0 the metric c • g is the Riemannian metric of constant curvature 1.

Proof. By [7, Corollary 3], if the metric g is Einstein and if there exists a geodesically equivalent, but not affine equivalent metric ḡ, then the equation ( 1) admits a non-constant solution. The corollary therefore follows from theorem 1.

Corollary 5.2. Let g be a pseudo-Riemannian metric on an (n > 1)-dimensional closed connected manifold. Then, if the metric ḡ on M is geodesically equivalent to g, but not affinely equivalent to g, then the degree of mobility of g is precisely 2 or for certain constants c = 0 = c the metrics c • g and c • ḡ are Riemannian metrics of constant curvature 1.

Proof. Assume first that n = dim(M ) ≥ 3. Under this assumption, by [START_REF] Kiosak | Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two[END_REF] if the degree of mobility of g is ≥ 3, then for every solution T of ( 15), the function f := trace(T ) is a solution of (1). More precisely, [6, Lemma 3 and Corollary 4] implies that in a neighborhood of almost every point there exists a constant c such that f is a solution of (1). Now, by [6, Lemma 7 in §2.3.4] the constant c is actually universal (implying that the equation ( 1) is fulfilled on the whole manifold). In this case the result follows therefore from Theorem 1. Now, the case n = dim(M ) = 2 follows from [8, Theorem 5.1] for the signatures (+,+) and (-,-), and from [13, Corollary 1] for the signature (+, -).

Corollary 5.2 is related to the following classical conjecture:

Projective Obata conjecture. Let G be a connected Lie group acting on a closed connected pseudo-Riemannian or Riemannian manifold (M, g) of dimension n > 1 by projective transformations. Then it acts by affine transformations or there exists a constant c = 0 such that (M, c • g) is the quotient of a Riemannian round sphere.

By corollary 5.2, we have:

Corollary 5.3. If (M n , g
) is a counter-example to the projective Obata conjecture, then n := dim(M n ) ≥ 3 and the degree of mobility of g is precisely 2.

Proof. The existence of a projective nonaffine transformation for g implies the existence of a metric that is geodesically equivalent to g, but is not affine equivalent to g. By corollary 5.2, if n = dim(M n ) ≥ 3, the degree of mobility of g is 2.

Now, by [START_REF] Matveev | Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics[END_REF]Theorem 6], the projective Obata conjecture is true in dimension two.

In the Riemannian case, projective Obata conjecture was proved in [10, Theorem 1] for dimension 2 and in [11, Corollary 1] for dimensions ≥ 3. The natural idea to prove the conjecture in the pseudo-Riemannian case is to mimic the Riemannian proof for pseudo-Riemannian metrics. The (Riemannian) proof contains two parts: (i) proof for the metrics with the degree of mobility 2 ([11, Theorem 15]), (ii) proof for the metrics with the degree of mobility ≥ 3 [START_REF] Matveev | Proof of the Lichnerowicz Conjecture[END_REF]Theorem 16]). We expect that it is possible, though nontrivial, to generalize (i) for the pseudo-Riemannian case. On the other side, one can not expect to generalize (ii) for pseudo-Riemannian metrics, because (ii) is based on Riemannian results that are no more true in a pseudo-Riemannian setting. Hence corollary 5.3 proves the part that was, a priori, the most difficult part of the projective Obata conjecture for pseudo-Riemannian metrics.

Moreover, the next corollary shows that the group of projective transformations of a closed manifold coincides with the group of affine transformations, or the group of isometries has codimension one in the group of projective transformations.

Corollary 5.4. Let (M, g) be a closed connected (n > 1)-dimensional pseudo-Riemannian manifold. Assume that for no constant c ∈ R \ {0} the metric c • g is the Riemannian metric of constant curvature 1. Then, every projective vector field is an affine vector field, or certain nontrivial linear combination of every two projective vector fields is a Killing vector field.

Proof. Indeed, it is well known (see, for example [START_REF] Matveev | Proof of the Lichnerowicz Conjecture[END_REF], or more classical sources acknowledged therein) that a vector field X is projective if the tensor

T := L X g -1 n+1 trace(L X g) • g (16)
is a solution of [START_REF] Obata | Certain conditions for a Riemannian manifold to be isometric with a sphere[END_REF], where L X is the Lie derivative with respect to X. Moreover, the projective vector field is affine, if and only if the trace of T is constant. Suppose the degree of mobility of g is not 2. Then, corollary 5.2 implies that all projective vector fields are actually affine, which is one of the possibilities in corollary 5.4. Now, suppose the degree of mobility of g is precisely 2. Let X and Y be projective vector fields. We consider the solutions T = L X g -1 n+1 trace(L X g) g and T ′ = L Y g -1 n+1 trace(L Y g) • g of ( 15). Since the degree of mobility is 2, T , T ′ , and g are linearly dependent, i.e., for certain constants k, k ′ , l we have k T + k ′ T ′ = l g. Since the mapping

X → L X g -1 n+1 trace(L X g) g
is linear, we have

L k X+k ′ Y g -1 n+1 trace(L k X+k ′ Y g) g = l g implying L k X+k ′ Y g = (n + 1)l g,
i.e., k X + k ′ Y is a homothety vector field (if l = 0) or a Killing vector field (if l = 0). Since M is closed, it admits no homotheties implying k X + k ′ Y is a Killing vector field.

6 Application II: Holonomy groups of closed constant curvature manifolds.

Let X be a manifold and G be a Lie group acting analytically on X. A (G, X)-structure on a manifold M is given by an atlas (U i , ϕ i ) such that each ϕ i takes values in X and each transition function ϕ i • ϕ -1 j is the restriction of the action of an element of G on X. If M has a (G, X)-structure then there exists (see for example [19, pp. 140,141] for details) a local diffeomorphism δ : M → X, where M and X are the universal covers of M and X respectively, and a morphism ρ : π 1 (M ) → G (where π 1 (M ) denotes the fundamental group of M , which acts as the group of deck transformations of the covering M → M , and G denotes the covering of G that acts on X) such that, for any γ ∈ π 1 (M ) and any m ∈ M , we have δ(γ. m) = ρ(γ).δ( m). The map δ is called the developing map and the morphism ρ is called the holonomy morphism. The image of π 1 (M ) with respect to ρ is called the holonomy group of the (G, X)-manifold M , it contains a lot of informations about the geometry of M .

We denote by R p+1,q the space R p+q+1 equipped with the standard pseudo-Euclidean metric of signature (p + 1, q), and consider the pseudo-sphere S p,q = {x ∈ R p+1,q | x, x = 1} and O(p + 1, q) ⋉ R p+q+1 the isometry group of R p+1,q . We recall that S p,q is simply connected if and only if p = 1.

It is well known that every pseudo-Riemannian manifold of signature (p, q) and constant curvature equal to 1 is a manifold with a (O(p+1, q), S p,q )-structure and that the flat pseudo-Riemannian manifolds of signature (p + 1, q) are the manifolds having a (O(p + 1, q) ⋉ R p+q+1 , R p+1,q )-structure.

The holonomy groups of those (G, X)-manifolds are not the usual pseudo-Riemannian holonomy groups (even if they are closely related). For now on we will only consider holonomy groups of (G, X)-structures. Proposition 4.1 implies Corollary 6.1. If q = 0, the action of the holonomy group of a closed manifold endowed with a (O(p + 1, q), S p,q )-structure (i.e. endowed with a pseudo-Riemannian metric with constant curvature equal to 1) on R p+1,q does not preserve any non-degenerate splitting. Corollary 6.1 was known under the additional assumption that the manifold is complete, see [START_REF] Zeghib | On closed anti de Sitter spacetimes[END_REF]Fact 2.3]. Since by [START_REF] Klingler | Complétude des variétés lorentziennes à courbure constante[END_REF] any constant curvature Lorentz manifold is complete, corollary 6.1 was also known for closed manifolds of Lorentz signature. If q = 0, the sphere itself is a counterexample.

Proof of corollary 6.1. In order to simplify the notation, we will suppose p = 1, i.e., that S p,q is simply connected. Anyway, if p = 1, M is Lorentzian, therefore complete by [START_REF] Klingler | Complétude des variétés lorentziennes à courbure constante[END_REF] and the corollary follows from [START_REF] Zeghib | On closed anti de Sitter spacetimes[END_REF]. Moreover, it is easy to adapt what follows to the case p = 1.

Let (M, g) be a closed pseudo-Riemannian manifold with constant curvature equal to 1. We denote by M the universal cover of M and by M the cone over its universal cover. It is well-known (and follows from Fact 3.2) that the curvature of the cone metric g = dr 2 + r 2 g is given by

R(X, Y )Z = R(X, Y )Z -g(Y, Z)X + g(X, Z)Y,
where R and R are the curvatures of g and of g. It implies that ( M , g) is flat. We identify S p,q , the cone over the pseudosphere S p,q , with {x ∈ R p+1,q | x, x > 0}. Let δ : M → S p,q be a developing map of the induced (O(p + 1, q), S p,q )-structure on M . The map δ : M → S p,q defined by δ(r, m) := (r, δ(m)) is a developing map of the (O(p + 1, q) ⋉ R p+q+1 , R p+1,q )structure of the flat manifold M . The holonomy morphisms associated to δ and δ are clearly the same. We denote them by ρ.

Let T 0 be a symmetric parallel (0, 2)-tensor on R p+1,q (i.e. a symmetric bilinear form) invariant with respect to the holonomy group ρ(π 1 ( M )). Let T = δ * T 0 be the pull back of T 0 by δ. Let γ be an element of π 1 ( M ) seen as the group of deck transformations of the universal covering. We have γ * T = γ * ( δ * T 0 ) = δ * (ρ(γ) * T 0 ), but as we supposed that ρ(γ) * T 0 = T 0 it implies that T is invariant by the action of π 1 (M ). It means that T is the pull-back of a parallel tensor T on M . By propositions 3.1, 3.4 and 4.1, the tensor T is proportional to the metric g. Thus T 0 also is proportional to the metric of R p+1,q . It means that the holonomy group of a closed (O(p + 1, q), S p,q )-manifold does not preserve any symmetric bilinear form on R p+1,q which is not proportional to the metric. In particular it does not preserve any non-degenerate splitting of R p+1,q .

Note that a stronger version of corollary 6.1 exists for flat pseudo-Riemannian manifolds. More precisely, by Goldman et al [START_REF] Goldman | The radiance obstruction and parallel forms on affine manifolds[END_REF] the holonomy of a closed affine manifold admitting a parallel volume form (for example pseudo-Riemannian and flat) does not preserve any non trivial subspace.

As the following example shows, corollary 6.1 is no more true for degenerate splittings. We identify R 2,2 with M (2, R) the space of order 2 square matrices endowed with the determinant (seen as a quadratic form). The pseudo-sphere is then identified with SL(2, R). For any v ∈ R 2 \ {0}, we define the set V v by V v = {M ∈ M (2, R) ; M.v = 0}. They are 2-dimensional totally degenerate subspaces. If v and w are not colinear, we have R 2,2 = V v ⊕ V w . Furthermore, SL(2, R) clearly acts isometrically by left multiplication on M (2, R). This action preserves V v and V w . Now, let Γ be a cocompact lattice in P SL(2, R). The manifold P SL(2, R)/Γ is a closed 3 dimensional anti de Sitter manifold whose holonomy lies in SL(2, R) and therefore preserves certain totally degenerate splittings of R 2,2 .

However, up to the authors knowledge, these examples (and some of their deformations cf. [START_REF] Salein | Variétés anti-de Sitter de dimension 3 possédant un champ de Killing non trivial[END_REF]) are the only known examples of closed pseudo-Riemannian manifolds of constant curvature whose holonomy preserves a non trivial degenerate subspace. Moreover, the main proposition of [START_REF] Zeghib | On closed anti de Sitter spacetimes[END_REF] is actually that the holonomy group of a closed anti de Sitter manifold of dimension greater than 3 is irreducible.

Acknowledgement: V.M. thanks Deutsche Forschungsgemeinschaft (Priority Program 1154 Global Differential Geometry and research training group 1523 -Quantum and Gravitational Fields) and FSU Jena for partial financial support, and Vicente Cortes and Dmitri Alekseevsky for useful discussions.

* partially supported by DFG (SPP 1154 and GK 1523)