Benenti Tensors: A useful tool in Projective Differential Geometry

Abstract Two metrics are said to be projectively equivalent if they share the same geodesics (viewed as unparametrized curves). The degree of mobility of a metric g is the dimension of the space of the metrics projectively equivalent to g. For any pair of metrics (g, ḡ) on the same manifold one can construct a (1, 1)- tensor L(g, ḡ) called the Benenti tensor. In this paper we discuss some geometrical properties of Benenti tensors when (g, ḡ) are projectively equivalent, particularly in the case of degree of mobility equal to 2.


Introduction
In the present paper, the word metric is used for both Riemannian and pseudo-Riemannian metrics, unless otherwise speci ed. The Einstein summation convention will be used.
De nition 1.1. We say that two symmetric a ne connections on the same manifold M are projectively equivalent if they share the same geodesics (as unparametrized curves). The set of all connections projectively equivalent to a given connection Γ is called the projective class of such connection or the projective connection determined by Γ. Two metrics are projectively equivalent if their Levi-Civita connections are so.
We use the term projective connection mainly when we refer to a system of ordinary di erential equations (ODE) that represents a certain projective class. Below we describe how to construct this representative system.

De nition 1.2. A projective transformation is a (local) di eomorphism of M that sends geodesics into geodesics (where geodesics are to be understood as unparametrized curves). A vector eld on M is projective if its (local) ow acts by projective transformations.
Locally, two symmetric a ne connections Γ andΓ are projectively equivalent if and only if Γ a bc =Γ a bc − δ a b ϕc − δ a c ϕ b (1) where ϕ i are the components of a -form, cf. [1][2][3]. For a detailed explanation and proof of this statement see for instance [4]. From an ODE perspective, the projective class of a given symmetric a ne connection Γ can be understood as follows. Let (u , u , . . . , u N ) = (x, u , . . . , u N ) be a system of coordinates on M. Then Γ gives rise to a system of second order ordinary di erential equations obtained by eliminating the external parameter from the classical geodesic equations (see for instance [5]). We can interpret system (2) as the projective connection associated to Γ. In fact, as we said, two connections Γ andΓ are projectively equivalent if and only if they are related by (1) and connections linked by (1) give the same system (2); in other words, for any solution (u (x), . . . , u N (x)) to (2), the curve (x, u (x), . . . , u N (x)) is a geodesic of Γ up to reparametrization. From this perspective, local di eomorphisms (u , . . . , (2) ( nite point symmetries) are projective transformations of Γ as they send geodesics into geodesics. In nitesimal point symmetries of (2) are projective vector elds of Γ and generate a -parametric family of projective transformations.
Taking into account what we said so far, and the form of system (2), the following system of ODE de nes a (N-dimensional) projective connection: Note that, for N = , system (2) reduces to a single ODE, namely the classical -dimensional projective connection associated to a -dimensional metric where (x, u, ux, uxx) := (u , u , u x , u xx ). Furthermore, (3) reduces to a single ODE having on the right hand side a general polynomial of third degree in the rst derivatives, i.e., a general -dimensional projective connection, extensively studied, for instance, in [6][7][8].
For N > , the right-hand side term of (3) is not a generic third order polynomial in the rst derivatives u i x . Indeed, in any equation forming such a system, not all monomials of third order degree appear.
In [9] Benenti introduced, in the context of Riemannian manifolds, a certain conformal Killing ( , )tensor, that he called L-tensor in [10,11]. After lowering one index by using the metric, one obtains a self-adjoint ( , )-tensor, whose eigenspaces play a central role in the theory of orthogonal separation coordinates. Most importantly, there exist separable coordinate systems associated to certain subspaces of Killing tensors (so-called Killing-Stäckel spaces, see Section 3 in [12]) for which the basis can be obtained from one single L-tensor, see e.g. Formula (2.21) and Remark 2.1 in [9] or, more recently, Theorem 8.1 in [12].
A number of subsequent papers is devoted to the historical context and geometric signi cance of this class of tensors, e.g. [13][14][15], from which we take the following de nition.
De nition 1.3 ([13, 15] is a Benenti tensor for g. Benenti tensors (4) that are constructed starting from a pair of projectively equivalent metrics become particularly useful in many areas of projective di erential geometry. They have been successfully used, e.g., in the proof of the Lichnerowicz-Obata conjecture [4], the splitting-gluing construction of geodesically equivalent metrics [16], at the crossroads of geodesic equivalence and integrability [19], the solution of Lie's Second Problem [6], in the context of general relativity [4], and others.
In the present paper we show how Benenti tensors, as de ned in De nition 1.5, are helpful for studying the structure of the space of metrics projectively equivalent to a given one. In Section 2 we basically characterize such a space as a complement of an algebraic variety. Then we focus our attention to the case when the projective class of the initial metric contains, roughly speaking, also non-proportional metrics. In Section 4, under this assumption, we study the case when there exists a projective vector eld. In particular, we investigate the Lie derivative of Benenti tensors along such a vector eld. In Section 5 we use the results of Sections 2 and 4 for studying -dimensional Riemannian metrics of Levi-Civita type admitting a projective vector eld.

Degree of mobility, metrizable projective connections and Benenti tensors
A classical question is to see if a projective connection is metrizable, i.e., if there exists a Levi-Civita connection whose corresponding projective connection is the given one.
In local coordinates, the projective connection (3) is metrizable if there exists an N-dimensional metric g such that (2), where Γ = Γ i jk is the Levi-Civita connection of g, is equal to (3). This is equivalent to the existence of a solution to the following system of N(N − )(N + ) PDEs where the Christo el symbols are given by System (5) is highly non-linear in the unknown functions g ij , but, it turns out that if we perform the substitution we obtain a linear system in the unknown variables σ ij . Of course, (6) does not make sense if g is negativede nite. Therefore, without further mentioning, in (6) and in the remainder of the paper, the fractional exponent implies that we use the absolute value of the base expression unless stated otherwise. We have the following theorem.

Theorem 2.1 ([17]). A metric g on an N-dimensional manifold lies in the projective class of a given connection Γ if and only if σ ij de ned by (6) is a solution of
where ∇ is the covariant derivative of Γ.
Note that Theorem 2.1 implicitly contains the assumption det(σ) ̸ = , as σ ij otherwise does not correspond to a metric. Furthermore, note that, since σ is a weighted tensor eld, where ", a" stand for the derivative w.r.t. the a-coordinate. Of course, the set of solutions to the linear system of PDEs (7) form a linear space. Note that in terms of solutions of system (7), taking into account (6), Formula (4) takes the compact form L(σ,σ) :=σσ − . (7) is denoted by A(g) (or A in short when there is no risk of confusion) and its dimension is called the degree of mobility of g.

De nition 2.2. Let g be a metric and Γ its Levi-Civita connection. The (linear) space of solutions to system
Since the degree of mobility is the same for any choice of metric g within a given projective class, it is also reasonable to call the degree of mobility of g the degree of mobility of its projective class (or of its projective connection). Let us recall that, in general, a solution σ to (7) can be such that det(σ) = , implying that there is no metric corresponding to σ via (6). In the current section, we would like to understand the structure of the solution space of (7). We begin by showing that the 'desired' solutions lie dense among all solutions of the system (7). From now on we assume a metrizable projective connection.

Proposition 2.3. Assume a metrizable projective connection and let A denote the (linear) space of solutions to (7). Then there is a basis of A made up of full-rank solutions and the set
Proof. By the hypothesis, there exists a metric g such that its Levi-Civita connection lies in the initial projective class. This means that, taking into account formula (6), c · σ is a full-rank solution to system (7) for each c ∈ R \ { }. This (punctured) line lies in an open subset of A as the condition to be of maximal rank is an open one. This implies that we can choose k linearly independent solutions to system (7), where k is the degree of mobility of the metric g. Denote those solutions by (σ , . . . , σ k ) and consider the equation The above equation de nes an algebraic variety in A which is a zero-measure set in A ∼ R k , so that its complement is dense in A.
Note that, in view of Proposition 2.3, the degree of mobility of a metric g is a "measure of the size" of the projective class of g.
Let us now consider the case when the projective class has degree of mobility 2, i.e. dim(A) = .
The assumption of degree of mobility 2 is justi ed for the following reasons: If the degree of mobility is less than 2, the situation is somehow trivial since all projectively equivalent metrics are proportional. Secondly, for compact connected manifolds of dimension N ≥ , in [18] it is proved, under mild assumptions, that either the degree of mobility of the projective class is at most 2 or the projective class is made of a nely equivalent metrics. Lastly, for -dimensional metrics, there always exists a (at most) 2-dimensional subspace of A, invariant under the action of an essential (i.e., non-homothetic) projective vector eld, and for -dimensional metrics of non-constant curvature, the degree of mobility is at most 2 (see again [18]).
We shall see that the properties of Benenti tensors are closely linked to lower-rank solutions of (7). (g , g ) be a pair of non-proportional, projectively equivalent metrics and L their Benenti tensor. Assume that L admits the constant eigenvalue t. Then any linear combination (7), and vice versa.

Proposition 2.4. Consider a metrizable projective connection with degree of mobility 2. Let
Proof. Let σ i be obtained from g i as in (6). Eigenvalues of L are roots of the characteristic polynomial det(L − t Id). Multiplying by det(σ ) ̸ = , we get The backwards direction of the proof follows by the same argument.
We Proof. The rst part is easily con rmed by checking the properties in De nition 1.3. For the second part, let g be a metric projectively equivalent to g. Since the degree of mobility is 2, we have that any Benenti tensor of a pair with g as its rst entry is given by an element of the family where we obtain, respectively, σ,σ via (6). Therefore L αβ = α Id +βL with L being the Benenti tensor eld of (g,ḡ), which proves the assertion.

Comparison with Benenti's terminology
Let us brie y comment on De nitions 1.3 and 1.5. De nition 1.3 is deeply rooted in Riemannian geometry as, in fact, the construction of (orthogonal) separable coordinates is strongly related to the existence of real eigenvalues of the ( , )-tensor L. In the present paper, however, pseudo-Riemannian metrics are usual rather than exceptional. De nition 1.5 takes account of this setting, and corresponds to the de nition of a J-tensor in [10] by Benenti. The L-tensor of De nition 1.3 is a special J-tensor, compare also Section 11 of [10].

. Tensors introduced by Benenti
In addition to J and L, there are other objects in the literature that are closely related to the objects studied here. We devote this subsection to a comparison of these objects with the ones used in the present paper. To keep the exposition clear and brief, without going to far astray from the actual purpose of this paper, we shall focus on reference [10] and compare the objects studied there with those introduced in the previous sections. Basically, [10] discusses four di erent kinds of special ( , )-tensors, called A, B, J and L (the ( , )-tensors obtained by index-shifting are denoted by boldface letters A, B, J, L in the reference). To facilitate an easy comparison between conventions, we use the notation of [10] within this subsection, but not outside it. As already brie y touched upon, the Benenti tensors we introduced in De nitions 1.5 and 1.3 correspond, respectively, to J-tensors and L-tensors of [10].
As is remarked in item (i) on page 39 of [10], J-tensors form a linear space, while A and B do not. The reason is that J-tensors are very similar to the (weighted) tensors σ that appear after linearization of the metrizability problem, see (7) and also [6,7,17]. In this sense, the objects σ are more "natural" in this context.
Below we write a more speci c correspondence between the objects we introduced, especially σ and (4), and the conformal Killing tensors introduced in [10]. To begin with, tensors J ij of [10] satisfy similar equations as σ ij (see Formula (6)), cf. Equation 8(c) and Remark 3.6 of [10] and Equation (2.3) of [17], and in addition Equation (7). In [10], as we said above, the ( , )-tensor J is introduced by lowering one index of J ij : J i k = g kj J ji . Now, letḡ be a projectively equivalent metric to g. In Sections 4 and 5 of [10], in this case, it is introduced the tensor B = µḡ, where µ = det(J). Also, J = B − . This allows us to compute the tensor J in terms of g andḡ. More precisely so that which, substituted into (8), gives (4).
In [10], another class of tensors is considered, de ned by A = adj(J) := det(J)J − . This A-tensor links J-tensors to integrals of motion, by grace of the following formula (see, e.g., [19]): The function (9) is an integral of motion for the Hamiltonian H = g(ξ , ξ ), where ξ i = g ij p j , with p j denoting momenta, p ∈ T * M.

. Pencils of Benenti tensors
Propositions 2.4 and 2.5 use a certain freedom in the de nition of the Benenti tensor. If we x a point σ ∈ A then any non-proportional σ ∈ A de nes a (non-trivial) Benenti tensor L = L(σ , σ ) by (4). Moreover, we can rescale σ and σ by non-zero constants. This de nes a pencil which gives rise to a family of integrals of motion, This formula works, too, if −t ∈ R coincides with an eigenvalue of L, since the (classical) adjoint of a matrix is well-de ned even for matrices with vanishing determinant. However, often certain values must be excluded to ensure that L s,t is non-degenerate (i.e., to make sure it is linked to a metric), see also item (ii) after Remark 3.2 in [10]. Note that, following De nition 1.5, a Benenti tensor is non-degenerate. Nontheless, the pencil (10) yields a degenerate tensor if −t ∈ R is an eigenvalue of L. A more thorough discussion of the pencils (10) can be found, for instance, in [10,11] (see also references therein). We mention that there is a related notion of L-pencils, which is discussed in detail in [11].
Finally, let us remark that the freedom in (10) is not the only freedom that we allow. Indeed, we do not need to x σ ∈ A (up to scaling), and according to De nition 1.5, an arbitrary change of the bases is possible (as long as we do not use degenerate solutions). Most importantly we are thus, in contrast to (10), free to swap the basis entries. This latter freedom is made use of in many of our computations.

Derivative of Benenti tensors along projective vector elds
Recall that we work in degree of mobility 2. Let (g,ḡ) be a pair of non-proportional, projectively equivalent metrics, with corresponding solutions σ,σ in A. Let L = L(g,ḡ) be their Benenti tensor. In addition to this pair we are now going to assume the existence of a projective vector eld w. In general, if we have pairs (g , g ) and (g , g ) with all g i , i ∈ { , . . . , }, in the same projective class, the respective Benenti tensors can be very di erent. Let (σ , σ ) correspond to (g , g ) via (6), then the Benenti tensor with respect to another pair (α σ + β σ , α σ + β σ ), α β − α β ̸ = and det(α i σ + β i σ ) ̸ = , with α i and β i ∈ R, for i = , , is So the Benenti tensor (4) heavily depends on the choice of the pair (g,ḡ). In the current section, we pose the question whether there is a 'best' choice of the pair (g,ḡ) within their projective class. We consider this question under the assumption of additional structure on the projective connection. Particularly, we assume the existence of a projective vector eld. For the Lie derivative of a Benenti tensor w.r.t. a projective vector eld, the following is well-known: Lemma 4.2 (e.g. [16]). Consider a metrizable projective connection with degree of mobility 2 and Lie derivative Lw. Let w be a non-zero projective vector eld. Moreover, let L be a Benenti tensor obtained from a pair of metrics (g , g ). Then LwL is a quadratic polynomial in L.
Proof. We start with the invariance of the Equations (7) under Lw (see Remark 4.1), which implies for some A, B, C, D ∈ R where σ i correspond to g i via (6). The Benenti tensor corresponding, via (6), to the pair (σ , σ ) is L = σ σ − . Using Lwσ − = −σ − (Lwσ) σ − , we thus have There is an obvious generalisation of Lemma 4.2 for cases with degree of mobility k ≥ , yielding k(k− ) multivariate quadratic polynomials in k(k− ) Benenti tensors. Can we modify the basis (σ , σ ) in such a way that the polynomial on the right-hand side of LwL = aL + bL + c becomes particularly simple?  Proof. Let σ ij = det(g) N+ g ij as in Formula (6). Then Hence we can nd a basis (σ , σ ) of A such that with A, C, D ∈ R. This proves the claim in view of Formula (11). Proof. In view of Lemma 4.2, LwL = aL + bL + c Id for some a, b, c ∈ R . Since a projective vector eld is de ned up to multiplication by a non-zero constant, and in view of Proposition 2.5, let us consider the transformations w → qw , where q, r ∈ R \ { } and s ∈ R is di erent from any eigenvalue of L. For the new L and w we obtain Let us rst assume a ̸ = . In this case we can choose q = ar, such that we obtain Id .
• If b ̸ = , let us further choose r = − b as (this choice narrows down our remaining freedom by s ̸ = ). We get so we arrive to normal form 3.
• If b = , we choose s = . If also c = , we have LwL = L . If c ̸ = , let r = c a , such that we obtain again an instance of normal form 3, LwL = L ± Id . Now let us assume a = .
In this case we set s = , which gives Id .
• (with any choice of r ̸ = ) to obtain LwL = L . For the second part of the claim, we can (also) proceed as follows: Lw is, possibly after a change of basis, represented by one of the matrices, with λ, µ ∈ R, for which by Formula (11) the corresponding polynomials are, respectively, LwL = (µ − λ) L , LwL = −L , LwL = µ (L + Id) .
The normal forms are obtained after a rescaling of w.
Corollary 4.6. Consider a metrizable projective connection with degree of mobility 2. If the projective algebra is at least 2-dimensional, there is a projective vector eld w such that Lw has a full-rank eigenvector and the normal form LwL = L + Id cannot be realized.
Proof. Let (σ , σ ) be a basis of A such that det(σ i ) ̸ = . Since the projective algebra is at least 2-dimensional, we nd two non-proportional projective vector elds u, v that satisfy equations We can assume a ̸ = and b ̸ = , otherwise w would be either u or v and σ the full-rank eigenvector we are looking for. Then, de ne w = b u − a v. We have that Observe that this equation does, by Theorem 4.5, not allow for the normal form LwL = L + Id.

Example: Levi-Civita metrics in 3D
We illustrate the discussion of the previous sections by the example of Levi-Civita metrics, particularly for a 3dimensional manifold. We can restrict to degree of mobility 2 by Corollary 3 of [18], as for degree of mobility 1 all metrics are proportional and the projective vector eld is homothetic. Metrics of the type discussed in this section have rst been constructed by Levi-Civita [20], see [21] for an English translation. Reference [22] discusses these metrics from the viewpoint of projective di erential geometry. In [4] a description is given how to obtain Levi-Civita metrics via the method of splitting-gluing [16]. Locally, a Levi-Civita metric can be written in the form where ε i ∈ {± } and where X i (x i ) are functions of one variable for each i. Particularly, in dimension 3, one obtains, for < X < X < X , the Riemannian metric A metric projectively equivalent, but non-proportional to g can be written, maybe after a change of coordinates,ḡ = n i= ε i j ̸ = i (X i − X j ) X i k X k dx i