Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type

We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and theory of compatible infinite dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is $(n+1)(n+2)/2$ dimensional; we describe it completely and show that it is maximal. Another has dimension $\le n+2$ and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension $n+2$ is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.


Introduction
This paper continues the Nijenhuis Geometry programme started in [1] and further developed in [2,3,20]. This programme was initially motivated by the fact that Niejnhuis operators (i.e. fields of endomorphisms L = (L i j ) with vanishing Nijenhuis torsion [27]) naturally appear in a number of different areas of geometry, algebra and mathematical physics. For this reason their normal forms, singularities and global properties deserve more systematic study than before. Although Nijenhuis operators usually serve as an auxiliary object in various mathematical constructions (two of them will be discussed in the present paper), in many cases their role is crucial and we intend to demonstrate this: the reader will notice that in many computations below we use L as a primary object and this leads to essential simplifications and new results.
We believe that the appearance of Nijenhuis operators in various, seemingly unrelated research areas, might be an evidence of a hidden relationship between them. Indeed, one may often observe similarity of ideas, techniques or clever tricks used therein. Sometimes the relationship is much deeper and is manifested in "overlapping" at the level of mathematical objects studied in these areas. In the present paper, we demonstrate such an overlap between geodesically equivalent pseudo-Riemannian metrics and compatible Poisson brackets of hydrodynamic type (see Theorem 1, 2 and 3 below). Once such a relationship is established and understood, one may try to transfer insights from one area to the other. That is what we do in Theorem 5 by using our expertise in the theory of geodesically equivalent metrics to prove a uniqueness result for a certain type of Poisson pencils of hydrodynamic type.
Also we would like to emphasise that general methods of Nijenhuis Geometry allows one to deal with singularities, i.e., those points where collisions of eigenvalues happen and L changes its algebraic type. Traditionally, such singularities are excluded and as a consequence most constructions are local and restricted to a domain where L reduces to some standard canonical form (for instance, is R-diagonalisable with simple spectrum). In many problems, however, singularities cannot be ignored. First of all, this relates to global problems, when L "lives" on a closed manifold and singularities become unavoidable. In this paper, we either impose no additional restrictions on L, or assume that L is differentially non-degenerate (this condition still allows singularities). Moreover, we give a description of all Nijenhuis operators geodesically compatible with a flat metric. Since this description is explicit, it can be used for analysis of singularities these operators may have.
Institute of Advanced Studies, Loughborough University where the first results of the paper were obtained, and also Centre International de Rencontres Mathématiques Lumini, where the paper was finished and written. We are also grateful to Jena Universität, in particular, Ostpartnerschaft programm for supporting our research on Nijenhuis Geometry for several years. The work of Alexey Bolsinov and Andrey Konyaev was supported by Russian Science Foundation (project 17-11-01303).

Basic definitions and main results
We start with introducing two classes of objects we will be dealing with: Poisson structures of hydrodynamic type and geodesically equivalent metrics.
Given a metric g of any signature on a manifold M which is always assumed to be of dimension n ≥ 2, for a function h, treated as the density of a Hamiltonian of hydrodynamic type, one can construct an operator (= (1, 1)tensor field) by the formula (we use g for index manipulations, for example Γ is j = Γ s pj g pi ).
If the metric is flat, this construction has many parallels with the construction of the Hamiltonian vector field by a function H and a Poisson bracket. Actually, a flat metric defines an infinite-dimensional Poisson bracket, the density h defines a variational functional of hydrodynamic type and the operator ∇ i ∇ j h determines the Hamiltonian flow generated by this functional. We refer to [9,10,14,24] for details. Parallels with and intuition coming from finite-dimensional Poisson brackets appeared to be very helpful in studying the following systems of partial differential equations: we view local coordinates x 1 , . . . , x n on M n as unknown functions which depend on two variables t and τ and consider the following quasilinear systems of n PDE on n unknown functions of two coordinates: Such systems are called systems of hydrodynamic type in literature.
This construction was generalised in [17,18] for metrics of constant curvature K: in this case the analog of formula (1) is and correspondingly A in (2) is given by Recall that two Poisson structures are compatible, if their sum is also a Poisson structure. In our infinite-dimensional situation, compatibility of two Poisson structures coming from constant curvature metrics g andḡ is equivalent to the following two conditions (see e.g. the survey [26]): (A) the operator L i j :=ḡ is g sj is a Nijenhuis operator, i.e., its Nijenhuis torsion vanishes, that is, for arbitrary vector fields u and v [27].
Notice that by [25,26] condition (A) is equivalent to the following property: for any α, β ∈ R such that α Id +βL is non-degenerate, the Christoffel symbols of the metricĝ = g(α Id +βL) −1 are given byΓ ij k = α Γ ij k + βΓ ij k . In the last formula, we raise the index i of the Christoffel coefficients Γ (resp. Γ,Γ) of the metric g (resp.ḡ,ĝ) by its own metric.
We will call a set of metrics Poisson-compatible, if any two metrics g and g from this set satisfy (A,B).
Similar to the finite-dimensional case, the existence of a nontrivial compatible Poisson structure provides additional tools to analyse the system (2), in particular to construct explicit solutions and to study long-time behaviour of solutions. We refer to [18,26,31] for details.
Nijenhuis operators appeared naturally in the theory of geodesically equivalent metrics. Recall that two metrics g andg on one manifold are geodesically equivalent, if they have the same geodesics considered as unparametized curves. Let us consider the (1, 1)−tensor L = L(g,g) defined by whereg ij is the (automatically, symmetric) tensor dual tog ij (in the sensẽ g isg js = δ i j ). It is known [5] that for geodesically equivalent metrics the operator L is Nijenhuis. Notice thatg can be recovered from g and L as which is equivalent to (4).
For a given g, we will call an operator L geodesically compatible with g, if (5) defines a metricg geodesically equivalent to g.
This condition is equivalent to the property that L is self-adjont with respect to g, is non-degenerate, and satisfies the system of PDEs (12) below. Sometimes speaking about geodesic compatibility of L with g one allows L to be degenerate and only requires that L is selfadjoint and satisfies (12). Actually since Id = δ i j is geodesically compatible with every g and equation (12) is linear, addition of const · Id to L makes it locally non-degenerate. In order to avoid misunderstanding, we will always either explicitly require the operator L to be non-degenerate or allow it to be degenerate.
Our first result is the following relation between geodesically equivalent and Poisson-compatible metrics: Theorem 1. Let g be a flat metric. Then the following statements hold: 1. If L is geodesically compatible with g and non-degenerate, then the metric gL −1 has constant (possibly zero) curvature and is Poissoncompatible with g.
2. If non-degenerate L 1 and L 2 are geodesically compatible with g, then the metrics gL −1 1 and gL −1 2 are Poisson-compatible.
The first statement of this theorem means that the metrics g andḡ = gL −1 define a family (pencil) of compatible Poisson brackets of hydrodynamic type, which can be used to construct integrable systems of quasilinear PDEs. One of such constructions is based on Casimir functions of the relevant Poisson structures. In the context of this paper, by a Casimir of the Poisson structure of hydrodynamic type (related to a metric g with constant curvature K, see Remark 7.1 for the definition of K) we understand a function h : M → R which, if plugged into (3), produces the zero operator: Since the property of h to be a Casimir is completely defined in terms of g, for the sake of brevity, we will refer to it as a Casimir of g. If K = 0 and x 1 , . . . , x n are flat coordinates for g, then the Casimirs are just linear combinations of the form a 0 +a 1 x 1 +· · ·+a n x n , a i ∈ R. If K = 0, the Casimirs still form a vector space of dimension n + 1 that admits a simple explicit description as soon as g is given by means of a certain canonical model. For instance, if g is the standard metric on the round sphere S n ⊂ R n+1 , then the Casimirs of g are just restrictions of linear functions from R n+1 to S n .
Let g andḡ be two Poisson-compatible metrics and h α be a Casimir of an arbitrary metric g α = (g −1 − αḡ −1 ) −1 , α ∈ R from the corresponding pencil. It is a well known fact that the Hamiltonian flows generated by h α 's w.r.t. the Poisson bracket related to g commute and this property leads to their integrability. However, if h α is a common Casimir of g α and g, then the corresponding flow vanishes and the above construction becomes trivial. For this reason it makes sense to distinguish essential Casimirs h α . The next theorem describes Casimirs and the corresponding commuting Hamiltonian flows for the pencil of Poisson-compatible metrics from Theorem 1.
Theorem 2. Let g be a flat metric and L be a non-degenerate geodesically compatible operator. Then, the following statements hold: 1. The metrics g andḡ = gL −1 have n common independent Casimirs.
3. The Hamiltonian flow generated by h α w.r.t. the Poisson bracket related to g is given (up to a factor depending on α) by the operator The flows given by A α and A β commute for all α, β ∈ R.
Remark 2.1. Instead of commuting flows given by (6) and parametrised by α ∈ R, one usually considers the coefficients of the expansion of (6) in powers of α: where σ k 's denote the coefficients of the characteristic polynomial det(Id −αL) = 1 − σ 1 α − · · · − σ n α n . The flows generated by these operators commute and hence define an integrable system of PDEs. They admit an infinite series of conservation laws given by the Casimir functions h α = det(−αL + Id) or, equivalently, by the coefficients of the α-power expansion h α = 1 + αF 1 + α 2 F 2 + . . . : Notice that the commuting flows and their conservation laws are defined in terms of a Nijenhuis operator L only, whereas the metric g in not involved in the final conclusion. In fact, this integrable system is a particular example of the so-called integrable ε-systems introduced and studied by M. Pavlov in [28] for R-diagonalisable operators L. These systems, in turn, are a particular case of a nice construction developed by P. Lorenzoni and F. Magri in [22] and based exclusively on a Nijenhuis operator L. No other ingredient is needed so that the system itself and its integrability can naturally be understood in the framework of Nijenhuis Geometry.
If we allow L to be degenerate, then the operators geodesically compatible with g form a vector space L g . Since all these operators L are Nijenhuis, we may refer to L g as a Nijenhuis pencil. According to the second statement of Theorem 1 the pencil L g , in the case of a flat metric g, automatically leads to a large family of Poisson-compatible constant curvature metrics of the form gL −1 (it is more appropriate here to consider contravariant metrics; they are given by the matrices Lg −1 = L ij ). The following theorem provides an explicit description for them. Then every operator L geodesically compatible to g with the index raised by g is given by the following formula: Here (a ij , b i , K) are constants and a ij = a ji . Conversely, every L given by (7) is geodesically compatible to g and the components of 1 2 grad g (tr L) are Moreover, near those points where L ij is non-degenerate and, therefore, defines a pseudo-Riemannian contravariant metric, the curvature of this metric is constant and equals K.
Following [26,25], we call a set of metrics compatible, if for any two of them, say g andḡ, the operator L i j :=ḡ is g sj is Nijenhuis (i.e., condition (A) from the definition of Poisson compatibility is fulfilled) and in addition for any α, β ∈ R such that the operator α Id +βL is non-degenerate, the curvature tensors of g,ḡ andĝ = g(α Id +βL) −1 satisfy the relation (In each curvature tensor we raised the index by its own metric).
Comparing this definition with the definition of Poisson-compatibility we see that Poisson compatible metrics are precisely compatible metrics of constant curvature.
Corollary 2.1. Let g = (g ij ) be a flat metric on R n and L an operator geodesically compatible with g. Then for any polynomials P and Q, the metrics gP (L) −1 and gQ(L) −1 are compatible (whenever the operators P (L) and Q(L) are invertible). In particular, if the metrics g, gL −1 , . . . , gL −k are of constant curvature (resp. flat), then {P (L)g −1 , deg P ≤ k} is a pencil of Poisson-compatible (resp. flat) contravariant metrics.
Notice that every set {g 0 , g 1 , g 2 , g 3 , . . . } of Poisson-compatible metrics (or equivalently the corresponding pencil of compatible Poisson structures of hydrodynamic type) leads to a natural Nijenhuis pencil which is the linear span of all Nijenhuis operators of the form L k = g −1 k g 0 , k = 0, 1, . . . . In a very similar way, Nijenhuis pencils appear in the area of geodesically equivalent metrics and finite-dimensional multi-Hamiltonian structures and already for this reason they deserve to be studied as a separate subject in geometry. For instance, it would be interesting to describe maximal Nijenhuis pencils, i.e., those which are not contained in any larger Nijenhuis pencil. Notice that maximality of a Nijenhuis pencil would immediately imply maximality of the corresponding Poisson pencils and families of geodesically equivalent metrics (the converse, as a rule, is not true).
In this context, the pencil from Theorem 3 admits the following interpretation. Consider an Euclidean metric g ≃ d(x i ) 2 and try to construct a big family of Poisson-compatible metrics containing g. Let us start with the trivial family of metrics whose components are constant in coordinates x 1 , . . . , x n . The corresponding Nijenhuis pencil consists of symmetric operators with constant entries. Can this pencil be extended? And if yes, then how? The answer is that such an extension exists, is unique in the class of g-symmetric operators and coincides with the pencil from Theorem 3 (Corollary 5.1). In particular, the family of Poisson compatible metrics from Theorem 3 is maximal (Corollary 5.2). Theorem 3 combined with Theorem 1 gives us many examples of Poissoncompatible metrics and, consequently, integrable systems of hydrodynamic type. We discuss these examples in Section 6. Some of these examples has features which were not observed in integrable systems of hydrodynamic type before. In particular, we construct nontrivial examples with Jordan blocks corresponding to nonconstant eigenvalues. Corollary 2.1 suggests one more method for constructing multidimensional pencils of Poisson brackets of hydrodynamic type. Let us start with a flat metric g and an operator L geodesically compatible to it. We know that g and gL −1 are Poisson-compatible (and hence generate a pencil of dimension 2). By [29, Theorem 1] (we reprove it in Corollary 3.1) gL −1 and L are still geodesically compatible. Hence if gL −1 is flat, we can repeat this procedure and consider the metric gL −2 which will be Poisson-compatible with g and gL −1 by Corollary 2.1. If this metric is still flat, then we can make one more step and so on. If gL −k is flat but gL −k−1 has non-zero constant curvature, then we show that the process stops as the next metric gL −k−2 does not have constant curvature: Theorem 4. Let L be geodesically compatible to an Einstein metric g of constant non-zero scalar curvature and non-degenerate. If d tr(L) = 0, then the metric gL −1 is not an Einstein metric of constant scalar curvature.
In Theorem 4 we allow any dimension n ≥ 2. In dimensions 2 and 3, Einstein metrics with constant scalar curvature are just metrics of constant curvature. In dimensions ≥ 3 the scalar curvature of any Einstein metric is automatically constant.
In dimension n ≥ 3, the analog of Theorem 4 for metrics of constant curvature (which is in fact enough for our study of metrics gL −k ) is due to N. Sinjukov [29]. In dimension n ≥ 3, our proof goes along the same lines as that by Sinjukov (which was merely sketched), but essentially uses results which were not available in the time of [29]. The proof in dimension n = 2 is different, is partially based on the technology we develop in the present paper and as far as we know is new.
The example of a sequence (g, gL −1 , . . . , gL −k flat, gL −k−1 of constant non-zero curvature) with k = n was constructed in [16] and is as follows: the metric g and operator L are given in local coordinates x 1 , . . . , x n by One can slightly generalise this example to include the points at which the operator L is not diagonalisable. The generalisation is essentially due to [1]: the metric is [1, Eq. (37)] and the operator is [1, Eq. (36)]. We repeat these formulas below: These two examples are related as follows: if one writes the pair (9) in the coordinates (σ 1 , . . . , σ n ), where σ's are the coefficients of the characteristic polynomial det(t Id −L) = t n − σ 1 t n−1 − σ 2 t n−2 − · · · − σ n , one obtains (up to a sign) the pair (10). Of course, the inverse transformation, from (10) to (9), is possible only near those points where L has n different real-valued eigenvalues.
In both cases, the metrics g, gL −1 , . . . , gL −n are flat and gL −n−1 has constant non-zero curvature. All together they generate a pencil of compatible Poisson brackets of hydrodynamic type of dimension n + 2. It is not hard to see that this pencil is maximal. Our next result shows that a pencil with such properties is unique.
Theorem 5. Let L be a Nijenhuis operator which is invertible, self-adjoint with respect to a metric g and differentially non-degenerate almost everywhere. Suppose the metrics gL −k are flat for k = 0, . . . , n and the metric gL −n−1 has constant curvature K = 0. Then g and L are geodesically compatible. Moreover, in a neighborhood of every point, where L is differentially non-degenerate, the pair (g, L) is locally isomorphic (up to multiplication of g by a constant) to (10).
Remark 2.2. For dimension two, Theorem 5 follows from [18,Theorem 10]. In arbitrary dimension, this statement in a slightly different form was announced in [16], but to the best of our knowledge, the proof has never been published.
Remark 2.3. The case when also the metric gL −n−1 is flat is much easier (provided L is differentially non-degenerate and therefore is diagonalisable almost everywhere). We discuss it within the proof, in Remark 7.3.
In this case essentially only the following example and its natural modifications with complex-valued eigenvalues are possible: the metric is g = diag(g 1 (x 1 ), . . . , g n (x n )) and the operator is L = diag(ℓ 1 (x 1 ), . . . , ℓ n (x n )). In this case, generators of commuting flows coming from this multihamiltonian structure are all functions of the Nijenhuis operator L and are therefore Nijenhuis operators also. In this case, the metric g is not geodesically compatible with L. Systems of hydrodynamic type (2) such that A is a Nijunhuis operator are well-understood in the case when L is diagonalisable; they decouple in Hopf equations and can be solved (almost) explicitly. An inclusion of singular points to this situation was done in [2].
Note that if we do not assume that L is differentially non-degenerate, then there are non-trivial examples when the above explained process does not stop so that all the metrics of the form gL −k , k ∈ N are flat and we obtain an infinite-dimensional Poisson pencils of hydrodynamic type (see Example 6.1).

3
Proof of Theorem 1 Let L be an operator geodesically compatible with a metric g of any signature. We do not assume a priori that the metric is flat. We consider the function λ := 1 2 tr L, and its differential dλ whose components will be denoted by λ i = ∂λ ∂x i and we denote the components of grad g λ by λ i := g is λ s . We start with Lemma describing the relationship between Christoffel symbols of ∇ and∇, the Levi-Civita connections of the metrics g and g = gL −1 .
Lemma 3.1. Let g and L be geodesically compatible and L be non-degenerate. Then the Christoffel symbols ofḡ = gL −1 are given bȳ where Γ i jk are the Christoffel symbols of g.
Proof. We use the equation of geodesic compatibility (see e.g. [5, Theorem 2]): which we rewrite raising two indexes by g as Notice that L ij =ḡ ij (whereḡ isḡ sj = δ i j as usual). Hence (13) gives This means thatḡ is parallel w.r.t. the symmetric connection whose Christoffel symbolsΓ i jk are defined by (11). Thus, this is the Levi-Civita connection forḡ, as stated.
Proof. Using the identity ∇g = 0 and Lemma 3.1 we get It remains to notice that∇ k (L s iḡ sj ) = λ iḡjk + λ jḡik is exactly the condition (12) forḡ and L. Proof. For a metric g, we consider the following system of linear partial differential equations on the function h: This equation naturally appeared in different parts of differential geometry and is well understood; let us recall the following known property of this equation: If the space of solutions of (14) is at least n + 1-dimensional, then the metric has constant curvature (for n ≥ 3 , see [8, end of page 343]).
We did not find the case n = 2 in the literature so let us give a proof. Consider the canonical complex structure J = J i j (which is the g-volume form with one index raised) and observe that (14) implies that the "skew gradient" so that the Killing equation is fulfilled. The existence of three linearly independent solutions of (14) implies the existence of two linearly independent Killing vector fields so the curvature is constant, as claimed.
For each function h in view of (11), its secondḡ-covariant derivative reads: Since g is flat, locally it has n + 1 linearly independent functions h such that In flat coordinates y 1 , . . . , y n for g (i.e., such that Γ i jk = 0), these functions take the form h = α 0 + α 1 y 1 + · · · + α n y n . In view of (15) they also satisfȳ ∇ j∇i h = λ s (∇ s h)ḡ ji which immediately implies (14) with respect to the metricḡ. Thus, the dimension of the space of solutions of (14) forḡ is at least n + 1 and henceḡ has constant curvature. Lemma 3.2 proves the first part of the first statement of Theorem 1. By construction, the operator L is Nijenhuis (as an operator geodesically compatible with g), so the property (A) from the definition of Poisson-comptible metrics is automatically fulfilled. It remains to check property (B), we will do it at the end of this section. Now let us discuss the second statement of Theorem 1. Since L 1 and L 2 are geodesically compatible with g, the metrics g 1 = gL −1 1 and g 2 = gL −1 2 have constant curvature by Lemma 3.2. Recall that operators compatible with a fixed metric g are all Nijenhuis and form a vector space (Nijenhuis pencil). In particular, αL 1 + βL 2 is a Nijenhuis operator for all α, β ∈ R. We, however, need to check this property for the operator (g 1 ) −1 g 2 = L 1 L −1 2 . To that end, we prove the following general fact from Nijenhuis geometry which is important on its own. Lemma 3.3. Consider a pair of non-degenerate Nijenhuis operators L 1 and L 2 . Then αL 1 + βL 2 is Nijenhuis operator for arbitrary constants α, β ∈ R if and only if L 1 L −1 2 is Nijenhuis operator.
Proof. It is well known that if L 1 and L 2 are both Nijenhuis, then αL 1 +βL 2 is Nijenhuis if and only if the Frolicher-Nijenhuis bracket of L 1 and L 2 vanishes (see [12] and e.g. [1]): Here v, w are arbitrary vector fields and [ , ] stands for standard Lie bracket of vector fields. Let us prove the following identity, which immediately implies the statement of the lemma: We have as stated.
To complete the proof of Theorem 1 (both items 1 and 2), it remains to show that the curvature of the metric gL −1 linearly depends on L, where L is now understood as an element for the vector space (Nijenhuis pencil) of all operators geodesically compatible with g. This will be implied by the following Lemma: Lemma 3.4. Let g be a flat metric, L be a geodesically compatible nondegenerate operator and K ∈ R be the curvature of gL −1 . Then, Proof. This identity will be derived from the following algebraic relation between the curvature tensor R of a metric g and an operator L geodesically compatible to it, see e.g. [4,Eq. (13) and Theorem 7]: Here W.l.o.g. we will assume that L is not proportional to the identity (recall that if L = f (x) Id, then f (x) = const [32] and the statement becomes trivial). Since in our case g is flat, then R(v, u) = 0 implying M = ρ Id (as M commutes with any matrix of the form u ⊗ g(v) − v ⊗ g(u)). To show that ρ = −K, we apply (18) once again forḡ = gL −1 and L (as they are still geodesically compatible by Corollary 3.1): Sinceḡ is of constant curvature K, thenR(v, u) = K(v ⊗ḡ(u) − u ⊗ḡ(v)).
On the other hand, formula (15) gives: or, in our case,M = ρL + λ s λ s Id .
Hence, (19) can be rewritten in the form Now consider two operators L 1 and L 2 compatible with g and apply Lemma 3.4 for the linear combination L = αL 1 + βL 2 . We denote the curvatures of L, L 1 and L 2 by K, K 1 , K 2 , and similarly λ = 1 2 tr L, λ 1 = 1 2 tr L 1 , λ 2 = 1 2 tr L 2 . Then Lemma 3.4 gives Hence, K = αK 1 + βK 2 , as required. This completes the proof of the second statement of Theorem 1. For the first statement, the end of proof is exactly the same. One just need to replace L 1 and L 2 with Id and L. Theorem 1 is proved.

Proof of Theorem 2
We consider a flat metric g and a non-degenerate operator L geodesically compatible to it. From Theorem 1 we already know thatḡ = gL −1 is a metric of constant curvature K.
Recall that Casimir functions h for g are those satisfying the equation whereas the equation for Casimirs ofḡ = gL −1 is In view of (15), the latter equation can also be rewritten in the form We first show that the metrics g andḡ have "many" common Casimirs. The next Lemma is equivalent to the first statement of Theorem 2. Proof. Let h be a Casimir ofḡ = gL −1 . Taking into account thatḡ si = L si , we can rewrite (22) in the form with ρ = −λ s ∇ s h − Kh. Thinking of ρ as an unknown function, we will derive some additional conditions for it.
Taking ∇ k derivative of the first formula in (23) and applying (13) we obtain: ( Now subtract the same relation with indices j and k interchanged, use the fact that ∇ k ∇ s ∇ j h = ∇ j ∇ s ∇ k h (since g is flat) and substitute ∇ i ∇ j h from the second formula in (23): This implies ∇ k ρ = −ρ λ β (L −1 ) β k or, equivalently, in more invariant way: If ρ ≡ 0, then (23) becomes ∇ i ∇ j h = 0, i.e. h itself is a Casimir of g. Otherwise, we use the following general property of Nijenhuis operators [1, which gives dρ det L and, finally, where c h is a constant function depending only on the choice of the Casimir function h ofḡ. In other words, we obtain a natural map h → c h from the vector space of the Casimirs ofḡ to R. This map is obviously linear so that the common Casimirs h ofḡ and g are defined by one single linear relation c h = 0, which is equialent to the statement of the Lemma.
Our next goal is to verify the second statement of Theorem 2.
Lemma 4.2. Assume that (g, L) is a geodesically compatible pair, the metric g is flat and det L > 0. Then √ det L is a Casimir of the Poisson structure corresponding to gL −1 .
Proof. We first rewrite relation (22) for Casimirs ofḡ = gL −1 in a slightly different way. Taking into account thatḡ si = L si and using (13), we obtain the following expression for the l.h.s. of (22): Once again we use the general property (24) of Nijenhuis operators that gives as required (at the very last step we used Lemma 3.4).
The third statement of Theorem 2 now immediately follows from Lemmas 4.1 and 4.2. Indeed for any Casimir ofḡ, and in particular for h = √ det L, formulas (23) and (25) give: It remains to replace L with the linear combination −αL + Id.

Proof of Theorem 3 and Corollary 2.1 and more properties of the related pencils
Proof of Theorem 3. Let x 1 , . . . , x n be flat coordinates for the metric g. In these coordinates, the Christoffel symbols vanish so that the covariant deriva-tive coincides with the usual one. Then formula (17) reads with a constant K. Hence, for some constants b 1 , . . . , b n .
Next, using the fact that ∇ k = ∂ ∂x k , we obtain from (13): This system of equations implies L ij = a ij +b i x j +b j x i −Kx i x j (with constants a ij that are necessarily symmetric with respect to i and j) as we claimed. Clearly every such L ij satisfies (30) so it is geodesically compatible with g.
To finish the proof, recall that near the points where L ij is non-degenerate, the metric gL −1 has constant curvature K by Lemma 3.4, and that L ij = g is L j s is the contravariant inverse ofḡ = gL −1 . Theorem 3 is proved.
Proof of Corollary 2.1. As above, g denotes a flat metric and L is an operator geodesically compatible with g. Let P and Q be polynomials such that P (L) and Q(L) are non-degenerate. The operator connecting the metrics gP (L) −1 and gQ(L) −1 is a rational function of the Nijenhuis operator L and, therefore, is Nijenhuis also by [1, Proposition 3.1], so the first condition from the definition of compatibility is fulfilled. We need to prove the "curvatureadditivity" condition (8).
In order to do it, we observe that in local coordinates from Theorem 3, the curvature tensors of gP (L) −1 and gQ(L) −1 are given by rational functions in N := n + (n + 1)n 2 + (n + 1)(n + 2) 2 (31) variables: the first n variables are x 1 , . . . , x n , the next (n+1)n 2 are g ij and the last (n+1)(n+2) 2 are the data (a ij , b i , K). The "curvature-additivity" condition (8) is then equivalent to a system of algebraic relations on these N variables.
If it is fulfilled in an non-empty open subset of R N , it is fulfilled everywhere. Let us explain why such a subset exists.
Recall that by [25,26] if two metrics g andḡ are related by a Nijenhuis operator L = L i j =ḡ is g sj with n = dim M distinct real eigenvalues, then the curvature tensors of g andḡ satisfy (8) automatically 1 . This statement also follows from our proof of Theorem 5, see Remark 7.2 below. Next, notice that (9) gives us an example of L having n different real eigenvalues. More precisely, letǧ ij ,ǎ ij ,b i andǨ be the data from Theorem 3 corresponding to the example (9) and at some pointx ∈ R n the corresponding L has n different real eigenvalues. At every point of a small neighborhood of (x i ,ǧ ij ,ǎ ij ,b i ,Ǩ) in R N , the corresponding operator L still satisfies the property that its eigenvalues are real and different. Using the above mentioned result from [25,26], we conclude that in this neighbourhood, (8) is fulfilled which implies that it is fulfilled identically for all L and at all points whenever it makes sense.
We conclude this section with discussing some more properties of the pencil L g of Nijenhuis operators L i j geodesically equivalent to g(equivalently, pencil L of Poisson compatible contravariant metrics L ij ) from Theorem 3. Notice that the relation between these two pencils can be written as L g = Lg −1 . Also notice that L does not depend on the choice of g whereas L g does.
First observe that dim L g equals n(n+1) 2 + n + 1 = (n+2)(n+1) 2 , which is the dimension of the space of symmetric (n + 1) × (n + 1)-matrices. As the next theorem shows, this is not a coincidence: to each L ij of form (7) one can uniquely assign such a matrix by a natural geometric procedure.
Proposition 5.1. Consider R >0 × R n (the coordinate on R >0 will be denoted by x 0 and those on R n will be x 1 , . . . , x n ) with the symmetric affine connection ∇ = Γ α βγ (with α, β, γ ∈ {0, . . . , n}) such that the only non-zero Christoffel symbols are as follows: Then∇ is flat. Moreover, for any constants (a ij , b i , K) with i, j = 1, . . . , n and a ij = a ji , the symmetric contravariant (2, 0) tensor A αβ on R >0 × R n given by is parallel with respect to∇. Furthermore, in the "new" coordinates the Christoffel symbols of this connection vanish. In these coordinates, the matrix of A becomes constant with components Proof. One verifies Proposition 5.1 by direct calculations. First we substitute the Christoffel symbols (32) into the formula (45) for the curvature tensor and see that it vanishes. Next we substitute (33) in the relation∇A = 0 and see that it is equivalent to (30). Again by direct calculations we observe that ∇-Hessians of the functions y α vanish and, therefore, in the y α -coordinates the components of A are constants. Finally, calculating the Jacobi matrix ∂y ∂x and using it in the transformation rule for (2,0)-tensors finishes the proof.
Note that the construction of the connection∇ on R >0 × R n does not involve g or L, so it is the same for all L and all g from Theorem 3. In fact it is motivated by different 'conification' procedures in projective (see e.g. [11,13,15]) and other Cartan parabolic geometries. Proposition 5.1 reduces the classification of geodesically compatible pairs (g, L) with flat g to the classification of pairs of symmetric (n + 1) × (n + 1)matrices which is, of course, well known (see e.g. [21]). Note that the metric g itself is included in the family (7) from Theorem 3 with L = Id, i.e., a ij = g ij , b i = 0 and K = 0. We see that the first column and first row of the corresponding matrix A g vanish, so that in terms of [21] we are only interested in those pairs A g , A ∈ Sym((n + 1) × (n + 1)) for which A g has rank n.
Our next observation is maximality of the pencils L g and L and their relations with other (simpler) pencils. Let x 1 , . . . , x n be canonical coordinates for the Euclidean metric g ≃ d(x i ) 2 . Consider the family S of operators A given in these coordinates by symmetric matrices with constant entries. Obviously every A ∈ S is a Nijenhuis operator and therefore S is a Nijenhuis pencil.
Can one extend S to get a larger Nijenhuis pencil? To answer this question, we need to describe Nijenhuis operators L such that for each A ∈ S the sum L + A is still Nijenhuis. Analytically, this condition means that L commutes with all A ∈ S in sense of Frolicher-Nijenhuis bracket [12], that is, for any vector fields ξ and η. If we temporarily ignore the fact that L itself is a Nijenhuis operator, then the problem becomes linear and can be solved by straightforward computation that we omit.

Proposition 5.2. Let [[L,
A]] = 0 for all A ∈ S. Then for n ≥ 3, in the coordinates x 1 , . . . , x n , the matrix of L takes the following form: A is an arbitrary constant matrix and K ∈ R (cf. formula (7) from Theorem 3).
In dimension 2, L is the sum of an opertor (35) and arbitrary skew- Notice that in order for (35) to be a Nijenhuis operator, the parameters A, b, c and K must satisfy additional (non-linear) relations. However, the following fact is straightforward.
Corollary 5.1. There is a unique Nijenhuis pencil L g satisfying the following conditions: (iii) all operators from L g are g-symmetric.
The matrices of operators L ∈ L g in coordinates x 1 , . . . , x n are given by (35) with b = c and symmetric A. In other words, L g is the Nijenhuis pencil from Theorem 3 with g ij = δ ij .
If we omit condition (iii), then there exist other maximal Nijenhuis extensions of S. For instance, we may take all operators L of the form L(x) = A + xb ⊤ with arbitrary constant A (not necessarily symmetric!).
For a flat metric g of arbitrary signature, Corollary 5.1 can be generalised as follows. Let S be a Nijenhuis tensor that consists of covariantly constant g-symmetric operators A, then there exists a unique Nijenhuis pencil L g satisfying the above conditions (i), (ii) and (iii). This pencil coincides with that from Theorem 3. The maximality of L g immediately implies This pencil admits the following alternative description. Let g be a flat metric and ∇ be its Levi-Civita connection. Consider all the metricsĝ covariantly constant w.r.t. ∇. Obviously, they are flat and pairwise Poissoncompatible.
Corollary 5.3. Assume thatḡ is almost compatible with every covariantly constantĝ in the sense of [26,25], i.e., L =ḡ −1ĝ is Nijenhuis. Thenḡ has constant curvature. Moreover, any two metricsḡ 1 andḡ 2 satisfying this condition are Poisson compatible. All together they form the pencil L from Theorem 3.
Throughout this section, g denotes a flat metric and L is an operator geodesically compatible with g.
From Lemma 3.4 (and Theorem 3) we see that the metric gL −1 is flat if and only if K = 0. The next two statements answer the natural question on necessary and sufficient conditions for the metric gL −2 , and more generally, all the metrics gL −k with k ≤ k 0 to be flat. Lemma 6.1. Suppose g is flat and L is geodesically compatible with g and non-degenerate. Let gL −1 be flat, then λ j is a parallel vector field, i.e., ∇ i λ j = 0, implying that g ij λ i λ j is a constant. Moreover, the metric gL −2 is flat if and only if λ j is null, that is, g ij λ i λ j = g ij λ i λ j = 0.
Proof. Since gL −1 is flat, K in Lemma 3.4 is zero. Hence, (17) implies that λ i is ∇-parallel. Of course, the same is seen from Theorem 3, since in this case λ i = a i in flat coordinates and is clearly parallel.
In order to prove the second claim, we use (17) for the metricḡ = gL −1 which is still compatible with L:∇ i λ j +Kδ i j = 0. In view of (15), we havē ∇ i λ j = L i s ∇ s λ j − λ s λ s δ i j . Since ∇ s λ j = ∇ j λ s = 0, we see thatK = 0 if and only if λ s λ s = g is λ i λ s = 0. Corollary 6.1. Suppose g is flat, L is geodesically compatible with g and non-degenerate.
The necessary and sufficient conditions for flatness of each of the metrics g, gL −1 , . . . , gL −k (k ≥ 2) are as follows: the 1-form d 1 2 tr L = λ i is parallel with respect to ∇ = ∇ g and Moreover, if each of the metrics g, gL −1 , . . . , gL −k (k ≥ 2) is flat, then λ i is parallel with respect each Levi-Civita connection related to g, gL −1 , . . . , gL −k+1 .
Here by (gL −m ) ij we denote the dual tensor to (gL −m ) ij . In other words, in matrix notation, (gL −m ) ij are the components of the inverse matrix (gL −m ) −1 = L m g −1 .
Proof. Corollary 6.1 follows from iterative application of Lemma 6.1.
This corollary allows us to construct examples of pairs (g, L) such that gL −k is flat for all k > 0 and d tr(L) = 0. The simplest example of this kind is as follows. Let b = b i = 0 be a constant vector which is null with respect to a non-degenerate symmetric matrix g ij . We view g as a flat metric and take a ij = (g −1 ) and K = 0. The (2, 0)-tensor L ij constructed by (7) for these a ij , b i , K evidently has the property that λ i is parallel with respect to ∇ g and that (gL −k ) ij λ i λ j = 0 for every k.
Let us give a more interesting example based on the same linear-algebraic idea.
Example 6.1. Take (Here e m denotes the m-th basis vector so that all components of b vanish except for the m-th one which is equal to 1). Now if m ≥ n/2 + 1, then the necessary and sufficient flatness conditions for g, gL −1 , . . . , gL −k from Corollary 6.1 are fulfilled for any k. This implies, in particular, that for any real analytic function f of one variable, the metric g(f (L)) −1 is also flat (at those points where f (L) is non-degenerate) so that we obtain an infinite-dimensional pencil of contravariant flat metrics of the form f (L)g −1 .
This example may possibly be of interest in the study of infinite-dimensional systems of hydrodinamic type for the following reason. In this theory one customary assumes that the operators defining these systems are diagonalisable and this special case is well studied. The case of other Segre characteristics is generally considered to be much harder and, in fact, there are only very few such examples in the literature. In the above example, the corresponding operator L i j = L is g sj has nontrivial Jordan blocks. For instance, in the 4-dimensional case with m = 3 and g and L given by: the operator L i j has two Jordan 2 × 2 blocks with nonconstant eigenvalues. After a suitable coordinate transformation, g and L become This phenomenon survives for all dimensions. The appearing Jordan blocks are 2 × 2, the number of such blocks depends on m and in the case of even n and m = n/2 + 1 equals n/2. The corresponding eigenvalues are not constant.
As far as we know, such examples (with many Jordan blocks with nonconstant eigenvalues) were not known before and may open a door to other Segre characteristics in the theory of integrable systems of hydrodynamic type. Now, if in Example 6.1 we set m < n/2 + 1, then it is a simple exercise in Linear Algebra to check that for a certain k 0 , the metric gL −k 0 −1 has constant non-zero curvature, whereas g, gL −1 , . . . , gL −k 0 are still flat. The maximal value of such k 0 is n. Let us give the corresponding example.
Example 6.2. On R n with coordinates x 1 , . . . , x n , consider the metric g and the tensor L ij given by (7) with the following data A, b and K: Then the metrics gL −1 , . . . , gL −n are flat and gL −n−1 has constant non-zero curvature.
In this example L i j has n different nonconstant eigenvalues at a generic point. From Theorem 5 it follows that this pair (g, L) is locally isomorphic (modulo multiplication of L by a constant) to that from (10) (or (9), at almost every point).
Proof of Theorem 4. We assume that g is Einstein with a constant non-zero scalar curvature, L is geodesically compatible to g, det L > 0 and d tr(L) = 0. Byg we denote the metric (det L) −1 gL −1 that is geodesically equivalent to g.
Starting from this point, our proof depends on dimension. We first consider the case n > 2. In this case the first two terms on the right hand side of (40) are proportional tog. Ifḡ is Einstein, the left hand side is also proportional tog, so∇ i∇j ψ is proportional tog. But this leads to a contradiction with (38) and (39). Indeed, then R ij is proportional tog ij . Hence, g ij is proportional tog ij , so L i j is proportional to δ i j . By the Weyl Theorem [32], this implies that d tr L = 0 which is forbidden by our assumptions.
Let us now consider the remaining dimension n = 2. From Corollaries 1 and 2 of [19] we see that if a metric g is Einstein of constant scalar curvature (n − 1)K and L is compatible with it, then in addition to (12) the following equation holds for a certain constant C: Note that though in [19] one generally assumes that n = dimM ≥ 3, formula (41) holds in dimension 2 as well; in fact the statement ∇ i λ j = µδ i j − KL i j (which is Corollary 1 in [19]) follows from and is equivalent to (37), and the proof of the relation ∇ i µ = −2Kλ i , which is Corollary 2 in [19], is straightforward and works also in dimension 2. Note also that (41) for metrics of constant curvature follows directly from [11,Theorem 5.1] or from [30,Theorem 1].
Next, in view of (3.1) we havē Now we use the fact that in dimension 2, we have (L 2 ) i j −tr L·L i j +det L·δ i j = 0 for any operator L (Cayley-Hamilton theorem). This allows us to replace (L 2 ) i j in the above formula by tr L · L i j − det L · δ i j and we obtain We see that the left hand side of this formula is as in the formula (41) written for the metricḡ; this implies that (−2Ktr L + C) = const which is forbidden by the assumptions K = 0 and d tr L = 0. The obtained contradiction proves Theorem 4.
Combining Theorem 4 with Lemma 6.1 we immediately see that for positive definite flat metrics the number k 0 such that g, gL −1 , . . . , gL −k 0 are flat is at most two and the number k 0 such that g, gL −1 , . . . , gL −k 0 are of constant curvature is at most three. Then there is the following alternative: either gL −1 has constant nonzero curvature and gL −2 is not of constant curvature, or gL −1 is flat, gL −2 has constant non-zero curvature and gL −3 is not of constant curvature. In particular, gL −2 cannot be flat.
Proof. If λ i is parallel, then in the positively definite case it is not null (recall that the condition d tr L = 0 implies that λ i = 0). Applying Lemma 6.1 we obtain the required claim.

Proof of Theorem 5
Proof. Suppose L is differentially-non-degenerate, then at almost every point it has n different eigenvalues. We will work in a small neighbourhood of such a point and first consider the case when the eigenvalues are real. In this case, there exists a coordinate system x 1 , . . . , x n such that L is given by Let the metrics gL −k with k ∈ {0, . . . , n + 1} satisfy the assumptions of Theorem 5. Since the (0,2)-tensor gL −1 is symmetric, g is diagonal: where g i = g i (x 1 , . . . , x n ) are some functions. Let us show that in this coordinate system the metric is given (up to a constant) by (9), which implies that L is geodesically compatible to g.
We need the following technical Lemma: Lemma 7.1. Let g be diagonal as in (43) and with the function h i depending on the coordinate x i only.
Then the Christoffel symbols Γ i jk of the metric g are as follows: • Γ k ij = 0 for pairwise different i, j and k, ∂x j for arbitrary k, j, ∂x k for arbitrary k = j.
Consequently, the Christoffel symbolsΓ i jk of the metricḡ = gL −1 are as follows: •Γ k ij = 0 for pairwise different i, j and k, •Γ k kj = Γ k kj for arbitrary k = j, •Γ k jj = h k h j Γ k jj for arbitrary k = j, (each h k is a function of one variable, so h ′ k in the latter formula and below, in e.g. (44), is just the usual derivative, h ′ k = ∂h k ∂x k ).
We leave the proof as a simple exercise for the reader. One needs to substitute the components of g in the formula Γ i jk = 1 2 g si ∂g sk ∂x j + ∂g sj ∂x k − ∂g jk ∂x s for the Christoffel symbols of g and then look how multiplications of g i with 1/h i affects the formula.
Lemma 7.2. The components of the curvature tensors R i jkℓ andR i jkℓ of the connections Γ andΓ from Lemma 7.1 are as follows: (no summation over repeating indices) • R s ijk = 0 for arbitrary pairwise different i, j, k and s, • R i ijk = 0 for arbitrary i, j, k, •R s ijk = 0 for arbitrary pairwise different i, j, k and s, •R i ijk = 0 for arbitrary i, j, k, •R j ijk = R j ijk for arbitrary j = i and i = k. • For arbitrary i = j, We again leave the proof of the Lemma as an exercise (which was done many times before us, see e.g. [25, §4]) for the reader. One needs to substitute Γ i jk given by Lemma 7.1 into the standard formula for the curvature and then carefully implement the changes in the resulting formula induced by replacing Γ byΓ via formulas in Lemma 7.1.
Remark 7.1. In our conventions used throughout the paper, the metric has constant curvature K, if (after raising the second index by g) it is given by: Note also that because of algebraic symmetries of curvature tensors, the components listed in Lemma 7.2 are essentially all components.
By our assumptions, g is flat which implies that the componentsR k iji and R j ijk are zero. Thus, the components ofR which are "interesting for us" (in the sense that only they have chance to be non-zero) are those considered in (44). It is convenient to raise the index i byḡ (sinceḡ is diagonal, this operation is just the multiplication by h i g i ). We obtain: Remark 7.2. To be used in the proof of Corollary 2.1, let us observe that the equation above is linear in h i . This was expected in view of results of [25,26], and in fact the calculations above are similar to those in the corresponding places in [25]. The same is true if part of eigenvalues is complex-conjugated, see the discussion at the end of the proof.
Next, let us return to our L given by (42) and employ the condition that gL −k are flat for k = 0, . . . , n and gL −n−1 has constant curvature K. Because the equations (46) are linear in h, and since for a polynomial P the matrix P (L) is diagonal diag(P (x 1 ), . . . , P (x n )), we have that for a polynomial P (such that P (L) is non-degenerate) the "interesting components" of the curvature tensor of the metric gP (L) −1 are given by (46) with all h i replaced by P (x i ).
By our assumptions the metric gP (L) −1 is flat for P (t) = t k with k ≤ n and has constant curvature K for P (t) = t n+1 . Using the linearity we can combine these conditions as follows: for any polynomial P = a n+1 t n+1 + · · ·+ a 0 (of degree ≤ n + 1) we have Let us view (47) as an equation on the functions Γ i jk , the following trick allows us to find those Γ i jk from this equation which are not automatically zero by Lemma 7.1.
At the point p = (x 1 , . . . ,x n ) as the polynomial P we take For both polynomials we have P (x i ) = 0 for each i; for the polynomial P 0 we have and for the polynomial P 1 we have Substituting P 0 into (47) and evaluating the result at p = (x 1 , . . . ,x n ) we see that most terms vanish and obtain the equation Similarly, doing the same with P 1 , we obtain The equations (48, 49) are two linear equations on two unknowns Γ j ij and Γ j ii . Solving them at the point p = (x 1 , . . . ,x n ) and taking into account that this point is arbitrary, we obtain: Plugging them in the formulas from Lemma 7.1 we obtain (at every point (x 1 , . . . , x n )): (Notice that swapping i and j in (51) we obtain the formula equivalent to (52) which was of course expected).
Remark 7.3. From (51) we see that for K = 0 each g i depends on the variable x i only, as we claimed in Remark 2.3. In this case, the pair (g, L) is not geodesically compatible.
In what follows we assume that the dimension n ≥ 3. Recall that for dimension two Theorem 5 was proved in [18,Theorem 10] and in [25, Theorem 6.2]. See also Remark 7.4 below. Take k ∈ {i, j} and consider the polynomial P = s =k (t −x s ) of degree n − 1. For this polynomial, formula (47) evaluated at the point p = (x 1 , . . . ,x n ) takes the form: For our P , we have Substituting these and (50) into (53), we obtain: In this equation, the factor g i K cancels out and we obtain the following explicit formula for g k K: Kg k (x 1 , . . . ,x n ) = − 1 Since this formula is fulfilled at every p = (x 1 , . . . ,x n ), it remains true if we replace allx i by x i . We see that the metric is (up to a constant factor) as in example in Section 2 so g and L are geodesically compatible as we claimed.
Let us now comment on the case when not all eigenvalues of L are real. We may again assume that all eigenvalues have algebraic multiplicity 1. Let k of them are real and the remaining n − k are partitioned into complex conjugate pairs. Since L is g-symmetric, in these coordinates we also have g = diag(g 1 , ..., g k , g k+1 ,ḡ k+1 , ..., g n−k 2 ,ḡ n−k 2 ), where g i is a functions of (x 1 , ..., x k , z 1 , ...,z n−k 2 ) andḡ i is its complex conjugate. Since g is a usual real-valued metric, g 1 , ..., g k are real.
Let us denote the formal coordinates z 1 , ...,z n−k 2 by x k+1 , ..., x n . Then, the Christoffel symbols and the curvature tensor are given by the same formulas as in the "real" case, so the analogs of Lemmas 7.1, 7.2 survive and we come to the system (47). Solving the system in exactly same way as in the real case (where we used only differentiation of polynomials and algebraic manipulations, i.e., those operations that are perfectly defined over complex coordinates too) we conclude that the metric is given, up to a constant factor, by (9). Next, observe that by [6] the metric and operator of the form (9) are geodesically compatible even if x k+1 = z 1 , ..., x n =z n−k 2 , i.e., some pairs of coordinates are complex conjugate as shown. Thus, complex roots do not affect the construction and final conclusion. Now assume that L is given by (10). Under the assumptions of Theorem 5, L is geodesically compatible to g: indeed, above we have proved that it is geodesically compatible at almost every point which implies that it is geodesically compatible at every point. Since the metric g is flat, the pair (g, L) is real-analytic in some coordinates (this is known and will follow from a description of geodesically compatible (g, L) with g flat). As we explained in Section 2, the form (9) in the coordinates σ 1 , ..., σ n is (10). Theorem 5 is proved.