# Metrisability of three-dimensional path geometries

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## Abstract

Given a projective structure on a three-dimensional manifold, we find explicit obstructions to the local existence of a Levi-Civita connection in the projective class. These obstructions are given by projectively invariant tensors algebraically constructed from the projective Weyl curvature. We show, by examples, that their vanishing is necessary but not sufficient for local metrisability.

## Keywords

Projective differential geometry Path geometry Weyl geometry Metrisability## Mathematics Subject Classification

53A20## 1 Introduction

There are several inequivalent geometric structures that give rise to a preferred family of curves on a smooth *n*-manifold *M*. A *path geometry* on *M* is a locally defined family of unparametrised smooth curves (called *paths*), one through each point and in each direction. A path geometry is *projective* if its paths are the unparametrised geodesics of a torsion-free connection \(\nabla \) on *TM*. The corresponding projective structure \((M, [\nabla ])\) is then defined by the equivalence class of torsion-free connections sharing their unparametrised geodesics with \(\nabla \). Finally a path geometry is *metrisable* if its paths are the unparametrised geodesics of a (pseudo-)Riemannian metric *g*. In this case the path geometry is, of course, also projective since the underlying projective structure is defined by the Levi-Civita connection of *g*. The converse does not hold: a general projective structure does not contain a Levi-Civita connection of any metric.

The characterisation of metrisable projective structures is a classical problem, which goes back to the work of Roger Liouville [17]. This problem has recently been solved if \(n=2\): in this case (assuming real-analyticity for sufficiency, as one must) necessary and sufficient conditions are given by the vanishing of a set of projective differential invariants [2], the simplest of which is of differential order five in the connection coefficients of a chosen \(\nabla \in [\nabla ]\). The case \(n=2\) is special—the projective Weyl curvature vanishes on a surface. This is no longer the case if \(n=3\), where the first set of obstructions already arises at order one, and is algebraic in the projective Weyl tensor. In this paper we shall present some of these obstructions as explicit projectively invariant tensors constructed algebraically from the Weyl curvature.

As is often done in differential geometry, we shall adorn tensors with indices in order to denote the type of the tensor. Thus, we may denote a vector or vector field by \(X^a\) but \(\omega _a\) always denotes a co-vector or 1-form. This is Penrose’s *abstract index notation* [22]. The canonical pairing between vectors and co-vectors is denoted by repeating an index so that \(X^a\omega _a\) is the scalar that would often be written without indices as Open image in new window . For any tensor \(\psi _{abc}\), we shall denote its skew part by \(\psi _{[abc]}\) and its symmetric part by \(\psi _{(abc)}\).

### **Theorem 1.1**

*A*,

*B*,

*C*,

*D*,

*F*,

*J*,

*K*,

*L*be the symmetric tensors defined bywhere \(\bigodot \) denotes symmetrisation over the non-contracted indices, and letIn general, the tensor \(T^{abcdef}\) does not vanish. However, if \(\nabla \) is projectively equivalent to a Levi-Civita connection then Open image in new window .

### **Theorem 1.2**

*T*arises in a more fundamental way, already described in [2]. More specifically, one can construct from the Weyl curvature a homomorphism \(\mathrm{\Xi }:\bigodot ^2(TM)\rightarrow \bigodot ^3(TM)\) that must have a non-trivial kernel in the metrisable case and \(T^{abcdef}\) is defined to be what amounts to the determinant of this homomorphism: Open image in new window is characterised as being the determinant of the compositionfor any 1-form \(X_a\). The homomorphism \(\mathrm{\Xi }\) is constructed by forming and prolonging the metrisability equation (e.g. [7]) and is the natural first step [20] in searching for a metric in a given projective class. We should point out that these constructions are carried out having arbitrarily chosen a non-zero section \(\epsilon ^{abc}\) of the line bundle Open image in new window . However, since a different section only changes the scale of the various obstruction tensors at each point, whether they vanish or not is unaffected. In the main body of this article we shall restore precision by introducing

*projective weights*, in effect a mechanism for keeping track of the scale of \(\epsilon ^{abc}\).

Theorem 1.2 is established by a different route, which seemingly creates a proliferation of projectively invariant obstructions to metrisability. For example, if one considers tensors such as \(T^{abcdef}\), taking values in \(\bigodot ^6(TM)\) and of degree 6 in the projective Weyl tensor, equivalently in the tensor Open image in new window , then one finds an 8-dimensional space of obstructions in the 11-dimensional space of projective invariants of this type. Indeed, there is already a quadratic obstruction as follows.

### **Theorem 1.3**

And if one prefers a scalar obstruction, then there is one of degree 3 as follows.

### **Theorem 1.4**

Besides proving these theorems, we shall provide a systematic way of creating many more invariants. Nevertheless, we shall show by examples, in Sects. 2.3 and 2.4, that this proliferation of invariants is insufficient to characterise the metrisable projective structures. In Sect. 3 we shall reformulate the problem for path geometries in terms of systems of two second order ODE for the unparametrised paths.

We would like to thank Katharina Neusser for pointing out the projectively invariant pairing (12) as a useful device in understanding the metrisability equation and also for helpful discussions concerning the form of the tensor Open image in new window for the Egorov and Newtonian structures in Sects. 2.3 and 2.4.

Finally, as detailed in Sect. 2.3, we would like to thank Vladimir Matveev for drawing our attention to an alternative proof [15] of the non-metrisability of the Egorov structure and also for his pertinent comments concerning the possible values of the degree of mobility of Riemannian and Lorentzian metrics in three dimensions.

## 2 Projective structures and metrisability

*M*be a smooth manifold. Let us consider an equivalence class \([\nabla ]\) of torsion-free connections on

*TM*, where to say that \(\nabla \) and \(\widehat{\nabla }\) belong to \([\nabla ]\) is to say that there is a 1-form \(\mathrm{\Upsilon }_a\) such thatThis is the condition for the geodesics sprays of \(\nabla \) and \(\widehat{\nabla }\) on

*TM*to have the same projection to \(\mathbb {P}(TM)\). Therefore, it is exactly the condition that all connections in \([\nabla ]\) share the same unparametrised geodesics on

*M*. In other words, the equivalence class \([\nabla ]\) operationally defines what is a projective structure.

*projective Schouten tensor*and Open image in new window is the

*projective Weyl tensor*. If we change connection in the projective class using (5) thenwhilst Open image in new window remains unchanged.

*M*is 3-dimensional. Computing the effect of a change of connection (5) on a 3-form \(\eta _{abc}\), we find thatand so if

*M*is oriented (as we shall suppose henceforth) and \(\eta _{abc}\) is chosen to be everywhere non-vanishing (we say that \(\eta _{abc}\) is a

*choice of scale*), then we may specify \(\mathrm{\Upsilon }_a\) by requiring that Open image in new window , thus obtaining a unique connection \(\widehat{\nabla }\) in the projective class such that Open image in new window . We shall refer to the connections obtained in this way as

*special*. From (6), we find thatand conclude that \(\beta _{ab}=0\) for special connections and hence that the Schouten tensor \(\mathrm {P}_{ab}\) is symmetric. (If

*M*is not oriented, then we define a choice of scale to be a nowhere-vanishing section of Open image in new window instead, where Open image in new window is the bundle of 3-forms on

*M*.) For any \(w\in {\mathbb {R}}\), it is convenient to denote by \({\mathcal {E}}(w)\) the line bundle Open image in new window , invariantly defined as the bundle whose fibre at \(p\in M\) is the 1-dimensional vector spacewhere Open image in new window denotes the 3-forms at

*p*positive with respect to the orientation, and we shall refer to a section \(\rho \) of \({\mathcal {E}}(w)\) as a

*projective density of weight w*. There are canonical isomorphisms \({\mathcal {E}}(k)={\mathcal {E}}(1)^{\otimes k}\) for \(k\in {\mathbb {Z}}\). Also, by construction, there is an identification Open image in new window , which we shall write as Open image in new window for \(\rho \) of projective weight Open image in new window . Equivalently, we have a canonical volume form \(\epsilon _{abc}\) of weight 4, that is to say a canonical section of Open image in new window , the tensor product Open image in new window . Having done this, a scale may be alternatively specified as a nowhere-vanishing density \(\sigma \) of projective weight 1, so that \(\eta _{abc}=\sigma ^{-4}\epsilon _{abc}\) is the corresponding volume form. This is the viewpoint we shall adopt henceforth. In summary, we are working with special connections specified by a choice of projective density \(\sigma \) of weight 1. Choosing a different scale, say \(\widehat{\sigma }=\mathrm{\Omega }^{-1}\sigma \) for some nowhere-vanishing function \(\mathrm{\Omega }\), induces a projective change of connection (5) where \(\mathrm{\Upsilon }=\mathrm{\Omega }^{-1}d\mathrm{\Omega }\). In the presence of a scale \(\sigma \in \mathrm{\Gamma }(M,{\mathcal {E}}(1))\), we may view a projective density \(\rho \in \mathrm{\Gamma }(M,{\mathcal {E}}(w))\) as a smooth function, specifically \(f=\rho /\sigma ^w\), but if we change scale \(\sigma \mapsto \widehat{\sigma }=\mathrm{\Omega }^{-1}\sigma \), then this function changes according to Open image in new window . Finally, for any scale \(\sigma \in \mathrm{\Gamma }(M,{\mathcal {E}}(1))\), the line bundles \({\mathcal {E}}(w)\) inherit connections characterised by \(\nabla \sigma ^w=0\) and then, if \(\rho \) is a projective density of weight

*w*, we see thatand note that this is consistent with (7) and the identification Open image in new window . Otherwise said, the tautological 3-form \(\epsilon _{abc}\) of weight 4 is covariant constant ( Open image in new window ) for any special connection on Open image in new window . We shall denote by \(\epsilon ^{abc}\) the induced canonical section of the dual bundle Open image in new window normalised so that

*S*from Theorem 1.4 is a projective density of weight Open image in new window .

*degree*of such a bundle as(It is the action of the

*grading element*normalised as in [3].) The point about the degree is that it simply adds under tensor product, for example,and changes sign when taking duals, for example,Also notice that our previous discussion concerning projective weights and the tautologically defined tensors \(\epsilon _{abc}\) and \(\epsilon ^{abc}\) is implicitly incorporated into this notation. For example, the identification Open image in new window in (9) is given by \(\omega _{ab}\mapsto \epsilon ^{abc}\omega _{bc}\). Fundamental for this article is (1), giving Open image in new window of projective weight Open image in new window . This irreducible tensor is every bit as good as the unweighted projectively invariant Weyl tensor Open image in new window , the inverse to (1) being given by Open image in new window .

### 2.1 Metrisability

*g*on

*M*gives rise to a projective structure \([\nabla ]\), namely the one that contains the Levi-Civita connection of

*g*. Hence we obtain a first order non-linear operatorwhich carries a metric to its associated projective structure, where Open image in new window is a Zariski-open subbundle of the rank 24 first jet bundle (e.g. [24]), and the affine bundle Open image in new window of projective structures on

*M*is modelled on Open image in new window , which has rank 15.

### **Lemma 2.1**

*M*be an

*n*-dimensional manifold and let \(p\in M\) be an arbitrarily chosen point. Then there is a torsion-free connection on

*TM*whose projective Weyl curvature at

*p*is Open image in new window .

### *Proof*

where the right hand side of this equation is an instruction written in LiE [16].

Henceforth, we shall use the terminology *metric* to mean (pseudo-)Riemannian metric. The signature plays no essential rôle in our considerations and can be discussed separately.

One can attack the metrisability problem directly, asking for a metric \(g_{ab}\) such that its Levi-Civita connection be projectively equivalent to a given connection. Although the resulting partial differential equations on \(g_{ab}\) are projectively invariant by construction, they are also non-linear. A surprising observation, essentially due to Liouville [17], is that there is a non-linear change of variables that turns this system into a linear one. For the convenience of the reader, we summarise the conclusions in three dimensions here and refer to [7, 19] for detail.

### **Theorem 2.2**

*metrisability operator*and a projectively invariant differential pairinggiven by

*metrisability equation*then the pairing with its determinant vanishes:Furthermore, wherever \(\det \sigma \) is non-zero, the weight zero tensor Open image in new window defines a metric whose Levi-Civita connection lies in the given projective class. Finally, up to sign, all metrics in a given projective class arise in this manner.

### *Proof*

As set forth in the statement of this theorem, these claims are straightforwardly verified from the definitions, the only further observation required being that (14) can be rewritten on Open image in new window as \(\widehat{\nabla }(g^{ab})=0\) where Open image in new window is projectively equivalent to Open image in new window according to (5) if we take Open image in new window , where \(g_{ab}\) is the inverse to \(g^{ab}\). (We have taken the opportunity here, following a suggestion of Katharina Neusser, to streamline the exposition in [7] by highlighting the rôle of (14).) \(\square \)

*metrisability equation*(13), there being a 2–1 correspondence between non-degenerate solutions of this equation and

*positive*metrics in the projective class (note that \(\sigma ^{ab}\) and Open image in new window give rise to the same metric, that these metrics have positive determinant (we call them

*positive*), and that conversely if \(g^{ab}\) is such a metric, then

### **Theorem 2.3**

### *Proof*

### **Theorem 2.4**

### *Proof of Theorem 1.1*

*A priori*this might always vanish but a suitable Weyl tensor Open image in new window is exhibited in [2, Sect. 8] with non-zero determinant (and this is realised by a projective structure in accordance with Lemma 2.1). It remains only to check that (2) gives a formula for

*T*but this is easily accomplished with the aid of computer algebra. \(\square \)

### 2.2 An elementary construction of obstructions

### **Lemma 2.5**

For a metric connection Open image in new window .

### *Proof*

*ab*as may be verified by computing

### *Remark 2.6*

At first glance, it may seem that Lemma 2.5 cannot be useful in restricting the possible Weyl curvature of a metrisable projective structure because the metric is already involved in the formula for Open image in new window especially via the tensor Open image in new window . It turns out, however, that there are non-trivial projective covariants that necessarily vanish for Open image in new window of this special form no matter what metric and no matter what symmetric form \(R^{da}\) are chosen. The simplest example is Open image in new window from Theorem 1.3.

### *Proof of Theorem 1.3*

*wiring diagrams*as in [22]:Now, we must show that \(Q_{ab}{}^{c}\) vanishes in the metrisable case. Well, if Open image in new window has the form given in Lemma 2.5, then we computewhich is evidently skew in

*ab*, as required. \(\square \)

Alternatively, we may compute in a preferred basis, the projective invariance ensuring that it does not matter what basis is chosen. In the Riemannian setting, for example, we may choose an orthonormal basis so that \(g_{ab}\) is represented by the identity matrix and in addition choose \(\epsilon _{abc}\) to be the associated volume form. We may also diagonalise \(R^{ab}\) and, optionally, remove its trace since \(g^{ab}\) does not contribute to Open image in new window . We leave the resulting verification to the reader. It is also straightforward to instruct a computer to work with these normalisations and this is our preferred method for analysing more complicated projective invariants. Finally, it is sufficient to work in the Riemannian setting:

### **Proposition 2.7**

Working at a point (so that the following statement is purely algebraic) suppose a projective covariant constructed from Open image in new window vanishes for all tensors of the form Open image in new window constructed from a fixed positive definite symmetric form \(g^{ab}\), an associated volume form \(\epsilon _{abc}\), and an arbitrary trace-free symmetric form \(R^{ab}\). Then the same covariant also vanishes for any non-degenerate \(g^{ab}\).

The statement is clear by complexification.

In particular, when instructing a computer, it is sufficient to assume that \(g^{ab}\) and \(R^{ab}\) are simultaneously diagonalised: even though this might not be possible to arrange in the Lorentzian setting for example, as a statement of pure algebra it is densely true and this is good enough. In any case, we shall henceforth suppose that all metrics we encounter are positive definite.

### **Proposition 2.8**

Up to scale, there are exactly five distinct quadratic covariants that may be constructed from Open image in new window , only one of which vanishes in the metrisable case.

### *Proof*

Unfortunately, we only know how to prove Proposition 2.8 by direct calculation. Whilst we have no theoretical justification for why we might expect obstructions created in this way, the method of proof given above allows a systematic though computationally intensive method of finding many more as the following proofs show.

### *Proof of Theorem 1.4*

*S*, however, is to consider the decompositionimmediately obtained from LiE, write each of them as a linear combination of contractions, and then test each of these potential obstructions by substituting Open image in new window . This quickly leads to

*S*as stated in Theorem 1.4. Finally, the veracity of our claimed formula for

*S*may be instantly tested (with a computer) by simply calculating the result in our preferred variables (21), obtaining Open image in new window , and observing that it is non-zero. \(\square \)

### *Proof of Theorem 1.2*

We have just shown that \(C^{abc}-2B^{abc}=0\) in the metric case, either by direct computation or as a consequence of the vanishing of the quadratic covariant \(Q_{ab}{}^c\) from Theorem 1.3. Generally, it is non-zero. The remaining claims in Theorem 1.2 may be straightforwardly checked by direct computation (with a computer). \(\square \)

It is unclear whether all obstructions in Theorem 1.2 may be written in terms of \(Q_{ab}{}^{c}\). More generally, it is straightforward to generate many more invariant obstructions all of which may yet arise from the basic obstruction \(Q_{ab}{}^c\). We leave this question for a future investigation and content ourselves with the following complete determination of the sextic obstructions taking values in Open image in new window (as does \(T^{abcddef}\) from Theorem 1.1).

### **Theorem 2.9**

There is an 11-dimensional space of covariants of Open image in new window of degree 6 taking values in Open image in new window , an 8-dimensional subspace of which vanishes in the metrisable case.

### *Proof*

*wiring diagrams*as was done in the \(19^{\mathrm {th}}\) century [6] (see also [22]). Thus, the covariants \(A^{ab}\!,B^{abc}\), and \(C^{abc}\) from Theorem 1.1 are written asRecall from (23) that Open image in new window occurs with multiplicity 2 in Open image in new window that \(B^{abc}\) and \(C^{abc}\) span the covariants of this type. At quartic level,

*A*,

*B*,

*C*,

*D*,

*E*,

*F*,

*J*,

*K*,

*L*. Therefore, amongst the 11-dimensional space of covariants of degree 6, the subspace comprising those that vanish in the metrisable case is 8-dimensional and is spanned byfor example. \(\square \)

### 2.3 Egorov’s projective structure

A *projective symmetry* is a vector field whose local flow maps unparametrised geodesics to unparametrised geodesics. Such symmetries form a Lie algebra \({\mathfrak {g}}\) under Lie bracket and, for connected 3-dimensional projective structures, we have \(\dim {\mathfrak {g}}\leqslant 15\) with equality if and only if Open image in new window in which case the structure is projectively flat, equivalently Open image in new window .

*A*,

*B*,

*C*. Since all solutions are degenerate, we have shown:

### **Proposition 2.10**

The Egorov projective structure is not metrisable.

### *Another proof*

As soon as the dimension of the solution space to (13) reaches 3, the projective structure cannot be metrisable unless it is projectively flat (it is easy to check that (26) solves (13) and that the projective Weyl curvature is non-vanishing without knowing that (26) is the general solution). In general, the *degree of mobility* of a metric is the dimension of the solution space of (13) for the associated projective structure and in three dimensions it can only be 1, 2, or 10 (as shown in [14] in the Riemannian case and [12] in the Lorentzian case (see [13, 23] in the Riemannian setting and [9] in the Lorentzian setting for a detailed analysis concerning possible values of the degree of mobility in all dimensions (in [9], a detailed analysis is conducted under the assumption that there are at least two metrics in the projective class whose corresponding Levi-Civita connections are different but if this is not the case, then this is a sufficient imposition on the projective Weyl curvature that it must vanish))). Alternatively, Kruglikov and Matveev [15] consider the dimension of the space of local projective symmetries to conclude that the Egorov structure is not metrisable. Specifically, in three dimensions they show that the dimension of this space is bounded by 5 if there is a Riemannian metric inducing the projective structure and 6 if there is a Lorentzian metric inducing the projective structure (whereas, as noted above, the local projective symmetries are 8-dimensional for the Egorov structure). \(\square \)

### *Yet another proof*

### 2.4 Newtonian projective structures

*Newtonian*projective structures because they are created as limits of metrisable structures as follows. In local coördinates \((x^1\!,x^2\!,x^3)\), consider the metricwhere \(\epsilon \not =0\) is constant and \(f(x^1\!,x^2)\) is an arbitrary smooth function. The corresponding projective structures are metrisable by definition but if we let \(\epsilon \rightarrow 0\), then these metric connections have a perfectly good limit, namely(whose geodesic equations are Newton’s equations for a particle in the \((x^1\!,x^2)\)-plane moving under the influence of the potential \(f(x^1\!,x^2)\) with \(x^3=\) ‘time’) whereas

### **Proposition 2.11**

Unless projectively flat, the Newtonian projective structures (28) are not metrisable.

### *Proof*

*c*. Already, all solutions to (13) are degenerate so the structure is not metrisable. In fact, one can go on to check that

*A*,

*B*,

*C*. As for the Egorov example, this 3-dimensional space of solutions also precludes metrisability. \(\square \)

### *Yet another proof*

As for the Egorov example, the form of the projective Weyl curvature (29) conflicts with Lemma 2.5 to provide yet another proof. This time Open image in new window has the form \(X^{ab}Y_c\) for non-zero \(X^{ab}\) and \(Y_a\) (with \(X^{ab}\) symmetric but never simple). Lowering the indices with the purported metric gives \(V_{(abc)}=X_{(ab}Y_{c)}\not =0\), contrary to (27). \(\square \)

### 2.5 A Weyl metrisable but not metrisable projective structure

*M*consists of a conformal structure [

*g*] together with a torsion-free connection

*D*that is compatible with the conformal structure in a sense thatfor \(g\in [g]\) and some 1-form \(\omega \), this compatibility condition being invariant under the transformationwhere \(\Theta \) is a non-zero function on

*M*. For any metric

*g*in the conformal class, the 1-form \(\omega \) determines the connection

*D*.

*D*is proportional to

*g*. Let [

*D*] be the projective structure defined by

*D*.

*T*from Theorem 1.1 also does not vanish. A convenient way to present

*T*is to regard it as a ternary sextic. Setting \(X_{a}=(X, Y, Z)\) we findup to a non-zero multiplicative constant.

From either of these obstructions, we therefore conclude that the projective structure [*D*] is not metrisable. It is nevertheless Weyl metrisable by construction. In dimension two all projective structures are locally Weyl metrisable [18]. We expect this not to be the case in dimension three, where, up to diffeomorphism, a general real-analytic projective structure depends on twelve arbitrary functions of three variables, but a Weyl structure only depends on five such functions. Characterising projective connections that are Weyl metrisable is an interesting open problem, which we do not pursue here. In general, we also do not know which Weyl metrisable structures are genuinely metrisable. If the Einstein–Weyl equations hold, however, then we have a satisfactory answer as follows.

### 2.6 Einstein–Weyl projective structures

*D*, [

*g*]) as outlined at the beginning of the previous section. In general, the Ricci tensor of

*D*contains both symmetric and skew parts, the latter being proportional to Open image in new window . The 2-form Open image in new window is an invariant of the Weyl structure, often called the

*Faraday form*. The Weyl structure is called

*Einstein–Weyl*if

*D*(noting that removing the trace of a symmetric tensor depends only on the conformal class [

*g*]).

### **Theorem 2.12**

Let (*D*, [*g*]) be an Einstein–Weyl structure in dimension 3, and let [*D*] be the projective structure defined by *D*. Then [*D*] is metrisable if and only if its Faraday form \(F_{ab}\) vanishes.

### *Proof*

In fact, if the Faraday form vanishes, and locally we choose a metric connection in the projective class according to (30), then the Einstein–Weyl equations (31) revert to the Einstein equations. Since we are in three dimensions, the Einstein equations imply that the metric is constant curvature. Therefore, the only metrisable Einstein–Weyl structures in three dimensions are projectively flat.

## 3 Path geometries and systems of ODEs

*F*,

*G*, specified uniquely by a point and a direction in

*U*.

*TU*are [10]To establish this result it is enough to consider the geodesic equations for a given \(\nabla \), and eliminate the affine parameter

*s*between the three equationswhere \(x^a=(x, y, z)\). This yields (32), withwhereNote that the expressions for

*A*,

*B*,

*C*,

*D*are invariant under (5). Conversely, imposing (33) on system (32) yields (34) as in [4]. For example the Egorov projective structure (24) corresponds to a system

*F*,

*G*, and their derivatives gives point invariants of system (32).

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