Invariant prolongation of the Killing tensor equation

Abstract

The Killing tensor equation is a first-order differential equation on symmetric covariant tensors that generalises to higher rank the usual Killing vector equation on Riemannian manifolds. We view this more generally as an equation on any manifold equipped with an affine connection, and in this setting derive its prolongation to a linear connection. This connection has the property that parallel sections are in 1–1 correspondence with solutions of the Killing equation. Moreover, this connection is projectively invariant and is derived entirely using the projectively invariant tractor calculus which reveals also further invariant structures linked to the prolongation.

Introduction

On a Riemannian manifold (Mg) a tangent vector field \(k\in \mathfrak {X}(M)\) is an infinitesimal automorphism (or symmetry) if the Lie derivative of the metric g in direction of k vanishes. In terms of the Levi-Civita connection \(\nabla =\nabla ^g\), this may be written as

$$\begin{aligned} \nabla _{(a}k_{b)}=0 \end{aligned}$$
(1)

where we use Penrose’s abstract index notation, \(k_a=g_{ab}k^b\), and the (ab) indicates symmetrisation over the enclosed indices. This Killing equation is generalised to higher rank \(r\ge 1\) by the Killing tensor equation

$$\begin{aligned} \nabla _{(a}k_{b \cdots c)}=0 \end{aligned}$$
(2)

where \(k_{b\cdots c}\) is a symmetric tensor, that is \(k \in \Gamma (S^r T^*M)\) and again \((ab \cdots c)\) indicates symmetrisation over the enclosed indices. Solutions of this, so-called Killing tensors, are important for the treatment of separation of variables [2, 18, 31, 36, 39], higher symmetries of the Laplacian and similar operators [1, 15, 17, 25, 34, 35], and for the theory of integrable systems and superintegrability [11, 14, 16, 32, 33]. Partly these applications arise because a solution of Eq. (2) (for any r) provides a first integral along geodesics: if \(\gamma :I\rightarrow M\) is a geodesic (where \(I\subset {\mathbb {R}}\) is an interval) and \(u:={\dot{\gamma }}\) is the velocity of this then \(\nabla _uu=0\) and therefore by dint of Eq. (2) the function \(k_{b\cdots c}u^b\cdots u^c\) is constant along \(\gamma \), cf. [40, 42, 43].

In dimensions \(n\ge 2\) (which we assume throughout) the Eq. (2) is an overdetermined finite-type linear partial differential equation. This means, in particular, that it is equivalent to a linear connection on a system that involves the Killing tensor k but also additional variables, the prolonged system [4, 41]. For example, for Eq. (1) above this prolonged system is very easily found to be

$$\begin{aligned} {\overline{\nabla }}_a\left( \begin{array}{c}k_c\\ \mu _{bc}\end{array}\right) = \left( \begin{array}{c}\nabla _a k_b- \mu _{ab}\\ \nabla _a\mu _{bc} - R^{\phantom {b}}_{bc}{}^d{}_{a}k_d ,\end{array}\right) \end{aligned}$$
(3)

where \( R^{\phantom {b}}_{bc}{}^d{}_{a}\) is the curvature of \(\nabla \) (see Sect. 4.2 below). In general such prolonged systems are not unique, but for any such connection there is a 1–1 correspondence between its parallel sections and solutions of the original equation (Eq. (2) in this case). Thus, on connected manifolds, the rank of the prolonged systems gives an upper bound on the dimension of the space of solutions and the curvature of the given connection can lead to obstructions to solving the equation, see, e.g. [5, 22, 23].

Two affine connections \(\nabla \) and \(\nabla '\) are said to be projectively equivalent if they share the same unparametrised geodesics. Connections differing only by torsion are projectively related, and we will lose no generality in our work here if we restrict to torsion-free connections, which we do henceforth. An equivalence class of \({\varvec{p}}=[\nabla ]\) of such projectively related torsion-free connections is called a projective structure, and a manifold \(M^{n \ge 2}\) equipped with such a structure is called a projective manifold. An important but not fully exploited feature of Eq. (2) is that it is projectively invariant. This will be explained fully in Sect. 2.2, but at this stage it will suffice to say the following. First when we introduced Eq. (2) above, \(\nabla \) denoted the Levi-Civita connection of a metric, but the equation makes sense and is important for any affine connection \(\nabla \), and it is in this setting that we now study it. Next the projective invariance means that Eq. (2) has a certain insensitivity and, in particular, descends to a well-defined equation on a projective manifold \((M,{\varvec{p}})\).

On a general projective manifold \((M,{\varvec{p}})\) there is no distinguished affine connection on TM. However, there is a distinguished projectively invariant connection \(\nabla ^{{\mathcal {T}}}\) on a vector bundle \({\mathcal {T}}\) that extends (a density twisting of) the tangent bundle TM:

$$\begin{aligned} 0\rightarrow {\mathcal {E}}(-1) {\mathop {\rightarrow }\limits ^{X}}{\mathcal {T}}\rightarrow TM\otimes {\mathcal {E}}(-1) \rightarrow 0 \end{aligned}$$
(4)

where \({\mathcal {E}}(-1)\) is a natural real-oriented line bundle defined in Sect. 2 below. This is the normal projective tractor connection, and it (or the equivalent Cartan connection) provides the basic tool for invariant calculus on projective manifolds. An important feature of this connection is that it is on a low-rank bundle (i.e. \(\hbox {dim}(TM)+1\)) that is simply related to the tangent bundle. The tractor calculus is recalled in Sect. 2.2.

For most applications that one can imagine it makes sense then to seek a prolongation of Eq. (2) that is itself a projectively invariant connection. For example, if this can be found, then its curvature simultaneously constrains solutions for the entire class of projectively related connections. In fact such a connection exists. Equation (2) is an example of a first BGG equation and arises as a special case of the very general theory of Hammerl et al. in [27] (see also [26]). That theory describes an algorithm for producing an invariant connection giving the prolonged system for any of the large class of BGG equations (and we refer the reader to that source for the meaning of these terms) and in this sense is very powerful. Although the algorithm of [27] produces in the end an invariant connection, it proceeds through stages that break the invariance of the given equation. For example, in treating Eq. (2) the steps of the algorithm are not projectively invariant. Moreover, beyond the case of rank 1 the explicit treatment of Eq. (2) using this algorithm seems practically intractible due to the number of steps involved. Finally, although the construction of [27] is strongly linked to the calculus of the normal tractor connection (of [3, 6, 10]), the connection finally obtained is not easily linked to the normal tractor connection. A conformally invariant prolongation of the conformal Killing form equation was developed in [24] and linked there to the normal tractor connection. However, the approach in that case is ad hoc and so does not immediately lead to a useful way to treat other equations.

The aim of this article is to produce an alternative invariant prolongation procedure that is simple, conceptual, explicit, and that reflects the invariance properties of the original equations. It is well known that for the projective BGG equations the normal tractor connection easily recovers the required prolongation in the case that the structure is projectively flat (i.e. the projective tractor/Cartan connection is flat). A motivation is to be able to produce the explicit curvature correction terms that modify the normal tractor connection to deal with general solutions on a projectively curved manifold. An explicit knowledge of these terms will enable us to deduce properties of the prolongation and so properties of solutions in general. We develop here a projectively invariant prolongation of Eq. (2) for each \(r\ge 1\). This uses at all stages the calculus of the normal projective tractor connection \(\nabla ^{{\mathcal {T}}}\) (as in [3]). The result is a connection on a certain projective tractor bundle (a tensor part of a power of the dual \({\mathcal {T}}^*\) to \({\mathcal {T}}\)) that differs from the normal tractor connection by the algebraic action of a tractor field that is projectively invariant and produced in a simple way from the curvature of the normal tractor connection and iterations of a projectively invariant operator on this. An advantage is that the construction and calculation use projectively invariant tools, and at all stages the link to the very simple normal tractor connection is manifest. As an immediate application, this approach typically simplifies the computation of integrability conditions, see Remark 18 and in particular Eq. (56).

Tensorial approaches to prolonging the Killing equation have been developed in many sources, see, e.g. [12, 28, 29, 44]. A recent approach that collects the prolongation into a connection on the prolonged system is provided by [30] (and we thank the authors of [30] for pointing out their article and several of the other sources mentioned). These do not use the projective invariance of the equations, but an important early work that does exploit this is that of Veblen and Thomas [42] (see especially Section 19 therein). Unfortunately this is difficult to use because of the way the prolongations are presented. Another projectively invariant prolongation we are aware of is the one in [13] for rank 1 Killing tensors on surfaces. This explicitly treats the holonomy obstructions and provides interesting interpretations of these. Our approach will provide a projectively invariant prolongation in any dimension and any rank, with explicit formulae in rank 2. Because the prolongation is captured by the projective tractor machinery, the results can be applied rather easily as will illustrated in subsequent works. Concerning our results for the projectively flat case in Sect. 3.1 there are necessarily some strong links to the prolongation approach of [35]. However, our route to the prolongation is very different, and it is this that is important for the development of the curved theory.

In fact there is considerable information in some of the preliminary results along the way in our treatment. For example, each Killing equation is captured in the very simple tractor equation of Proposition 6. This is part of a rather general picture, and it is clear that the theory here will generalise considerably. (In fact aspects of our treatment here were inspired by the conformally invariant prolongation of the conformal Killing equation via tractors in [21, Proposition 2.2].) This will be taken up in subsequent works. Proposition 6 also may be interpreted as showing that solutions of the Killing tensor equation on \((M,{\varvec{p}})\) correspond in a simple way to Killing tensors for the canonical affine connection on the Thomas cone over \((M,{\varvec{p}})\); the Thomas cone is discussed in, e.g. [7, 10].

Throughout we use an abstract index notation in the sense of Penrose. As mentioned above, \((ab\cdots c)\) indicates symmetrisation over the enclosed indices, while \([ab\cdots c]\) indicates skewing over the enclosed indices. Then, \({\mathcal {E}}\) is used to denote the trivial bundle, and for example, \({\mathcal {E}}_{(abc)}\) is the bundle of covariant symmetric 3-tensors \(S^3T^*M\).

Background

Conventions for affine geometry

Let \((M,\nabla )\) be an affine manifold (of dimension \(n\ge 2\)), meaning that \(\nabla \) is a torsion-free affine connection. The curvature

$$\begin{aligned} R_{ab}{}^c{}_d\in \Gamma (\Lambda ^2 T^*M \otimes TM\otimes T^*M ) \end{aligned}$$

of the connection \(\nabla \) is given by

$$\begin{aligned}{}[\nabla _a,\nabla _b]v^c=R_{ab}{}^c{}_d v^d , \qquad v\in \Gamma (TM). \end{aligned}$$

The Ricci curvature is defined by \(R_{bd} =R_{cb}{}^c{}_d \).

On an affine manifold the trace-free part \(W_{ab}{}^c{}_d\) of the curvature \(R_{ab}{}^c{}_d\) is called the projective Weyl curvature and we have

$$\begin{aligned} R_{ab}{}^c{}_d= W_{ab}{}^c{}_d +2 \delta ^c_{[a}{\textsf {P}}_{b]d}+\beta _{ab}\delta ^c_d, \end{aligned}$$
(5)

where \(\beta _{ab}\) is skew and \({\textsf {P}}_{ab}\) is called the projective Schouten tensor. That \(W_{ab}{}^c{}_d\) is trace-free means exactly that \(W_{ab}{}^a{}_d=0\) and \(W_{ab}{}^d{}_d=0\). Since \(\nabla \) is torsion-free, the Bianchi symmetry \(R_{[ab}{}^c{}_{d]}=0\) holds, whence

$$\begin{aligned} \beta _{ab}=-2{\textsf {P}}_{[ab]} \qquad \text{ and } \qquad (n-1){\textsf {P}}_{ab} = R_{ab}+\beta _{ab}. \end{aligned}$$

As we shall see below, the curvature decomposition Eq. (5) is useful in projective differential geometry.

First some further notation. On a smooth n-manifold M the bundle \({\mathcal {K}}:=(\Lambda ^{n} TM)^2\) is an oriented line bundle, and thus, we can take correspondingly oriented roots of this. For projective geometry a convenient notation for these is as follows: given \(w\in {\mathbb {R}}\) we write

$$\begin{aligned} {\mathcal {E}}(w):={\mathcal {K}}^{\frac{w}{2n+2}} . \end{aligned}$$
(6)

Of course the affine connection \(\nabla \) acts on \(\Lambda ^{n} TM\) and hence on the projective density bundles\({\mathcal {E}}(w).\) As a point of notation, given a vector bundle \({\mathcal {B}}\), we often write \( {\mathcal {B}}(w) \) as a shorthand for \({\mathcal {B}}\otimes {\mathcal {E}}(w)\).

Projective geometry and tractor calculus

Two affine torsion-free connections \(\nabla ' \) and \(\nabla \) are projectively equivalent, that is they share the same unparametrised geodesics, if and only if there exists some \(\Upsilon \in \Gamma (T^*M)\) s.t.

$$\begin{aligned} \nabla '_a v^b=\nabla _a v^b +\Upsilon _a v^b+ \Upsilon _c v^c\delta _a^b \end{aligned}$$
(7)

for all \(v\in \Gamma (T^*M)\). This implies that on sections of \({\mathcal {E}}(w)\) we have

$$\begin{aligned} \nabla '_a \tau =\nabla _a\tau +w\Upsilon _a \tau , \end{aligned}$$

while on sections of \(T*M\),

$$\begin{aligned} \nabla '_a u_b=\nabla _a u_b -\Upsilon _a u_b- \Upsilon _b u_a \end{aligned}$$

It follows at once that on \(k_{a_1\cdots a_k}\in S^k T^*M(2r)\) we have

$$\begin{aligned} \nabla '_{(a_0} k_{a_1\cdots a_k)}=\nabla _{(a_0} k_{a_1\cdots a_k)}. \end{aligned}$$

Thus, for \(k\in S^k T^*M(2r)\) the Killing Eq. (2) is projectively invariant and descends to a well-defined equation on \((M,{\varvec{p}})\), where \({\varvec{p}}=[\nabla ]=[\nabla ']\), the projective equivalence class of \(\nabla \).

On a general projective n-manifold \((M,{\varvec{p}})\) there is no distinguished connection on TM. However, there is a projectively invariant connection on a related rank \((n+1)\) bundle \({\mathcal {T}}\). This is the projective tractor connection that we now describe.

Consider the first jet prolongation \(J^1{\mathcal {E}}(1)\rightarrow M\) of the density bundle \({\mathcal {E}}(1)\). (See for example, [37] for a general development of jet bundles.) There is a canonical bundle map called the jet projection map\(J^1{\mathcal {E}}(1)\rightarrow {\mathcal {E}}(1)\), which at each point is determined by the map from 1-jets of densities to simply their evaluation at that point, and this map has kernel \(T^*M (1)\). We write \({\mathcal {T}}^*\), or in index notation \({\mathcal {E}}_A\), for \(J^1{\mathcal {E}}(1)\) and \({\mathcal {T}}\) or \({\mathcal {E}}^A\) for the dual vector bundle. Then, we can view the jet projection as a canonical section \(X^A\) of the bundle \({\mathcal {E}}^A(1)\). Likewise, the inclusion of the kernel of this projection can be viewed as a canonical bundle map \({\mathcal {E}}_a(1)\rightarrow {\mathcal {E}}_A\), which we denote by \(Z_A{}^a\). Thus, the jet exact sequence (at 1-jets) is written in this notation as

$$\begin{aligned} 0\longrightarrow {\mathcal {E}}_a(1){\mathop {\longrightarrow }\limits ^{Z_A{}^a}} {\mathcal {E}}_A {\mathop {\longrightarrow }\limits ^{X^A}}{\mathcal {E}}(1)\longrightarrow 0. \end{aligned}$$
(8)

We write to summarise the composition structure in (8) and \(X^A\in \Gamma ({\mathcal {E}}^{A}(1))\), as defined in (8), is called the canonical tractor or position tractor. Note the sequence (4) is simply the dual to (8).

As mentioned above, any connection \(\nabla \in {\varvec{p}}\) determines a connection on \({\mathcal {E}}(1)\). On the other hand, by definition, a connection on \({\mathcal {E}}(1)\) is precisely a splitting of the 1-jet sequence (8). Thus, given such a choice we have the direct sum decomposition \({\mathcal {E}}_A {\mathop {=}\limits ^{\nabla }} {\mathcal {E}}(1)\oplus {\mathcal {E}}_a(1) \) and we write

$$\begin{aligned} Y_A:{\mathcal {E}}(1) \rightarrow {\mathcal {E}}_A \qquad \text{ and } \qquad W^A{}_a: {\mathcal {E}}_A\rightarrow {\mathcal {E}}_a(1), \end{aligned}$$
(9)

for the bundle maps giving this splitting of (8); so

$$\begin{aligned} X^A Y_A=1, \qquad Z_A{}^b W^A{}_a=\delta ^b_a, \qquad \text{ and } \qquad Y_A W^A{}_a=0. \end{aligned}$$

By definition X and Z are projectively invariant. The formulae for how \(Y_A\) and \(W^A_a\) transform when \(\nabla \) is replaced by \(\nabla '\), as in Eq. (7), is easily deduced and can be found in [3].

With respect to a splitting (9) we define a connection on \({\mathcal {T}}^*\) by

$$\begin{aligned} \nabla ^{{\mathcal {T}}^*}_a \left( {\begin{array}{c}\sigma \\ \mu _b\end{array}}\right) := \left( {\begin{array}{c} \nabla _a \sigma -\mu _a\\ \nabla _a \mu _b + {\textsf {P}}_{ab} \sigma \end{array}}\right) . \end{aligned}$$
(10)

Here \({\textsf {P}}_{ab}\) is the projective Schouten tensor of \(\nabla \in {\varvec{p}}\), as introduced earlier. It turns out that Eq. (10) is independent of the choice \(\nabla \in {\varvec{p}}\), and so \(\nabla ^{{\mathcal {T}}^*}\) is determined canonically by the projective structure \({\varvec{p}}\). We have followed the construction of [3, 9], but as mentioned in those sources this cotractor connection is due to T.Y. Thomas. Thus, we shall also term \({\mathcal {T}}^*={\mathcal {E}}_A\) the cotractor bundle, and we note the dual tractor bundle\({\mathcal {T}}={\mathcal {E}}^A\) has canonically the dual tractor connection: in terms of a splitting dual to that above this is given by

$$\begin{aligned} \nabla ^{\mathcal {T}}_a \left( \begin{array}{c} \nu ^b\\ \rho \end{array}\right) = \left( \begin{array}{c} \nabla _a\nu ^b + \rho \delta ^b_a\\ \nabla _a \rho - {\textsf {P}}_{ab}\nu ^b \end{array}\right) . \end{aligned}$$
(11)

Note that given a choice of \(\nabla \in {\varvec{p}}\), by coupling with the tractor connection we can differentiate tensors taking values in tractor bundles and also weighted tractors. In particular, we have

$$\begin{aligned} \nabla _aX^B=W^B{}_a, \quad \nabla _a W^B{}_b=-{\textsf {P}}_{ab} X^A , \quad \nabla _a Y_B ={\textsf {P}}_{ab}Z_B{}^b, \quad \text{ and }\quad \nabla _a Z_B{}^b = -\delta ^b_a Y_B.\nonumber \\ \end{aligned}$$
(12)

The curvature of the tractor connection is given by

$$\begin{aligned} \kappa _{ab}{}{}^C{}_{D}=W_{ab}{}^c{}_d W^C{}_c Z_{D}{}^d-C_{abd}Z_{D}{}^d X^C, \end{aligned}$$
(13)

where \(W_{ab}{}^c{}_d\) is the projective Weyl curvature, as above, and

$$\begin{aligned} C_{abc}:= \nabla _a {\textsf {P}}_{bc}-\nabla _b {\textsf {P}}_{ac} \end{aligned}$$
(14)

is called the projective Cotton tensor.

The projective Thomas-D operator is a first-order projectively invariant differential operator, or more accurately family of such operators. Given any tractor bundle \({\mathcal {V}}\) (including the trivial bundle \({\mathcal {E}}\)) and any \(w\in {\mathbb {R}}\) it provides an operator on the weighted tractor bundle \({\mathcal {V}}(w)\)

$$\begin{aligned} {\mathbb {D}}: {\mathcal {V}}(w) \rightarrow {\mathcal {T}}^*\otimes {\mathcal {V}}(w-1) \end{aligned}$$

given by

$$\begin{aligned} {\mathbb {D}}_A V= w Y_A V + Z_A{}^a\nabla _a V , \end{aligned}$$
(15)

where \(\nabla _a\) is the connection induced on the weighted bundle \({\mathcal {V}}\) from the tractor connection \(\nabla _a^{{\mathcal {T}}^*}\) and the connection on \({\mathcal {E}}(1)\) coming from a representative in \({\varvec{p}}\). Note that from this definition and Eq. (12) follows

$$\begin{aligned} {\mathbb {D}}_A X^B= \delta _{A}{}^B, \qquad \text{ and } \qquad X^A {\mathbb {D}}_A V =w V, \end{aligned}$$
(16)

for \(V\in \Gamma ({\mathcal {V}}(w) )\). Also from the definition it follows that \({\mathbb {D}}\) satisfies a Leibniz rule, in that if \({\mathcal {U}}(w)\) and \({\mathcal {V}}(w')\) are tractor (or density) bundles of weights w and \(w'\), respectively, then for sections \(U\in \Gamma ({\mathcal {U}}(w))\) and \(V\in {\mathcal {V}}(w')\) we have

$$\begin{aligned} {\mathbb {D}}(U\otimes V)= ({\mathbb {D}}U) \otimes V+ U\otimes {\mathbb {D}}V. \end{aligned}$$

Thus, from Eq. (16), when commuting \({\mathbb {D}}_A\) with the tensor product with \(X^B\), we get the commutator identity

$$\begin{aligned}{}[{\mathbb {D}}_A, X^B]=\delta _A{}^B . \end{aligned}$$
(17)

In view of the last property, as an operator on weighted tractor fields, the commutator \([{\mathbb {D}}_A,{\mathbb {D}}_B]\) is a “curvature” in that it acts algebraically. We will treat it this way by writing,

$$\begin{aligned}{}[{\mathbb {D}}_A,{\mathbb {D}}_B] V^C=W_{AB}{}^C{}_D V^D \end{aligned}$$
(18)

for its action on \(V\in \gamma ({\mathcal {T}}(w))\). For this reason and for convenience we will refer to \(W_{AB}{}^C{}_D\) as the W-curvature. Investigating this, consider \({\mathbb {D}}\) on projective densities \(\tau \in \Gamma ({\mathcal {E}}(w))\) to form \({\mathbb {D}}_B\tau \). Using Eq. (12) we have

$$\begin{aligned} {\mathbb {D}}_A{\mathbb {D}}_B\tau&= (w-1)Y_A {\mathbb {D}}_B \tau + Z_A{}^a\nabla _a {\mathbb {D}}_B\tau \\&= w(w-1)Y_AY_B\tau + 2(w-1)Y_{(A} Z_{B)}{}^b\nabla _b\tau + Z_A{}^a Z_B{}^b \nabla _a\nabla _b\tau , \end{aligned}$$

which we note is symmetric. Phrased alternatively, we have on sections of density bundles

$$\begin{aligned}{}[{\mathbb {D}}_A,{\mathbb {D}}_B]\tau =0 . \end{aligned}$$
(19)

So \({\mathbb {D}}\) is “torsion free” in this sense, and from the Jacobi identity we have at once the Bianchi identities

$$\begin{aligned} W_{[AB}{}^C{}_{D]}=0 \qquad \text{ and }\qquad {\mathbb {D}}_{[A}W_{BC]}{}^E{}_F=0 . \end{aligned}$$
(20)

To compute \(W_{AB}{}^C{}_D\) it suffices to act on a section \(V\in \Gamma ({\mathcal {T}})\). Note from Eq. (12)

$$\begin{aligned} {\mathbb {D}}_A{\mathbb {D}}_B V^C = -Y_{A}{\mathbb {D}}_B V^C -Y_{B}{\mathbb {D}}_A V^C + Z_{A}{}^{a}Z_{B}{}^{b} \nabla _a\nabla _b V^C . \end{aligned}$$

Thus

$$\begin{aligned} W_{AB}{}^C{}_D= Z_{A}{}^{a}Z_{B}{}^{b}\kappa _{ab}{}^C{}_D , \end{aligned}$$
(21)

where \(\kappa \) is the tractor curvature given above, and in particular

$$\begin{aligned} X^AW_{AB}{}^C{}_D=X^BW_{AB}{}^C{}_D=X^DW_{AB}{}^C{}_D=0, \end{aligned}$$
(22)

as well as

$$\begin{aligned} Z_C{}^c W_{AB}{}^C{}_D=Z_A{}^aZ_B{}^bZ_D{}^d W_{ab}{}^c{}_d, \quad Y_C W_{AB}{}^C{}_D=-Z_A{}^aZ_B{}^bZ_D{}^d C_{abd} \end{aligned}$$
(23)

The action of the W-tractor, as on the right-hand side of Eq. (18), extends to tensor products of \({\mathcal {T}}\) and \({\mathcal {T}}^*\) by the Leibniz rule, and we use the shorthand \(W_{AB}\sharp \) for this. For example, for any (possibly weighted) 2-cotractor field \(T_{CD}\) we have

$$\begin{aligned} W_{AB}\sharp T_{CD}= -W_{AB}{}^E{}_C T_{ED} -W_{AB}{}^E{}_D T_{CE}. \end{aligned}$$

Remark 1

The W-curvature \(W_{AB}{}^C{}_D\) satisfies, of course, stronger properties if the projective structure includes the Levi-Civita connection of a metric. An interesting case is when, in particular, the metric is Einstein but not scalar flat, as in this case there is a parallel (non-degenerate) metric on the projective tractor bundle. This can be used to raise and lower tractor indices [9], and it follows easily that the W-curvature \(W_{AB}{}^C{}_D\) has the same algebraic symmetries as a conformal Weyl tensor. This is potentially important for applications, but we will not exploit these observations in the current work.

Young diagrams and some algebra

For a real vector space \({\mathbb {V}}\) of dimension N we consider irreducible representations of \(SL({\mathbb {V}})\cong SL(N,{\mathbb {R}})\) within \(\otimes ^{m}{\mathbb {V}}^*\) for \(m\in {\mathbb {Z}}_{\ge 0}\). Up to isomorphism, these are classified by Young diagrams [19, 20] and we assume an elementary familiarity with this notation. Each diagram is (equivalent to) a weight \((a_1,a_2,\cdots ,a_{N})\) where \(m \ge a_1\ge \ldots \ge a_{N} \ge 0\) with \(\sum _{i=1}^{k}a_i=m\). We usually omit terminal strings of 0, strictly after \(a_1\), that is for \(s\ge 2\) we usually omit \(a_s\) from the list if \(a_s=0\). In particular, the trivial representation of \(SL({\mathbb {V}})\) on \({\mathbb {R}}\) (so \(m=0\)) will be denoted (0) rather than \((0,\cdots ,0)\) and the dual of the defining (or fundamental) representation of \(SL({\mathbb {V}})\) on \({\mathbb {V}}^*\) (so \(m=1\)) will be denoted (1) rather than \((1,0,\cdots ,0)\). Given this notation for weights the representation space for the representation \((a_1,\cdots ,a_h)\) will usually be denoted \({\mathbb {V}}_{(a_1,\cdots ,a_h)}\), or by the weight \((a_1,\cdots ,a_h)\), simply, if \({\mathbb {V}}\) is understood. We will term h the height of the diagram.

In fact for our current purposes we shall only need the Young diagrams of height at most 2, and \({\mathbb {V}}\) will be \({\mathbb {R}}^{n+1}\) with its standard representation of \(SL(n+1,{\mathbb {R}})\). The symmetric representations \(S^m{\mathbb {V}}^*\) have the diagram (m), while \((k,\ell )\) with \(k +\ell =m\ge 1\), \(k\ge \ell \ge 1\), can be realised by tensors \(T_{B_1\ldots B_{k} C_1 \ldots C_\ell }\) on \({\mathbb {V}}\) which are symmetric in the \(B_i\)’s, also symmetric in the \(C_i\)’s, and such that symmetrisation over the first (equivalently any) \(k+1\) indices vanishes:

$$\begin{aligned} T_{B_1\ldots B_{k} C_1 \ldots C_\ell }= T_{(B_1\ldots B_{k}) (C_1 \ldots C_\ell )} \quad \text{ and } \quad T_{(B_1\ldots B_{k} C_1)C_2 \ldots C_\ell }=0 . \end{aligned}$$
(24)

In this article we will call these particular realisations Young symmetries and \({\mathbb {V}}_{(k,\ell )}\) will mean the \(SL({\mathbb {V}})\)-submodule of \(\otimes ^m {\mathbb {V}}\) consisting of tensors on \({\mathbb {V}}\) with these Young symmetries.

The key algebraic fact we need is then the following.

Proposition 2

The map of \(SL({\mathbb {V}})\) representations

$$\begin{aligned} {\mathbb {V}}_{(r+1)}\otimes {\mathbb {V}}_{(r)} \rightarrow {\mathbb {V}}_{(r)} \otimes {\mathbb {V}}_{(r+1)} \end{aligned}$$
(25)

given by

$$\begin{aligned} T_{B_1\ldots B_rB_{r+1} C_1 \ldots C_r}\mapsto T_{B_1\ldots B_r(B_{r+1} C_1 \ldots C_r)} \end{aligned}$$

is an isomorphism.

Proof

This is a straightforward consequence of the well-known Littlewood–Richardson rules for decomposing the tensor product \(U_{C_1\cdots C_r}\otimes V_{B_1\cdots B_{r+1}}\in {\mathbb {V}}_{(r)}\otimes {\mathbb {V}}_{(r+1)}\) into its direct sum of irreducible parts, and then the properties of these irreducibles in terms of Young symmetries as explained in [19, 20, 38]. Each of the summands is a representation equivalent to either \({\mathbb {V}}_{(2k+1)}\) or \({\mathbb {V}}_{(k,\ell )}\), with \(\ell \ge 1\), \(k+\ell =2r+1\), and each projection to such a component may be factored through the map (25). \(\square \)

This yields the following consequence.

Corollary 3

For \(r\in {\mathbb {Z}}_{\ge 1}\) and \(k\ge \ell \ge 1\) with \(k+\ell =r+1\),

$$\begin{aligned} ({\mathbb {V}}_{(r+1)}\otimes {\mathbb {V}}_{(r)} )\cap ({\mathbb {V}}_{(r)}\otimes {\mathbb {V}}_{(k,\ell )})=\{0\}. \end{aligned}$$

Proof

The irreducible components of \(\otimes ^{r+1} {\mathbb {V}}^*\) isomorphic to \({\mathbb {V}}_{(k,\ell )}\), with \(k\ge \ell \ge 1\) and \(k+\ell =r+1\), all lie in the kernel of the map

$$\begin{aligned} \otimes ^{r+1} {\mathbb {V}}^* \rightarrow {\mathbb {V}}_{(r+1)} \end{aligned}$$
(26)

However, from Proposition 2 the kernel of the map (25) is trivial. \(\square \)

In fact the kernel of the map (26) is spanned by the irreducible components of \(\otimes ^{r+1} {\mathbb {V}}^*\) isomorphic to \({\mathbb {V}}_{(k,\ell )}\), with \(k\ge \ell \ge 1\) and \(k+\ell =r+1\). Thus, it is clear that in fact Corollary 3 is equivalent to Proposition 2.

Another fact that will be useful is the following.

Lemma 4

Suppose that \(T_{B_1\cdots B_rC_1\cdots C_r}=T_{(B_1\cdots B_r)(C_1\cdots C_r)}\in {\mathbb {V}}_{(r,r)}\). Then

$$\begin{aligned} T_{B_1\cdots B_rC_1\cdots C_r}=(-1)^r T_{C_1\cdots C_rB_1\cdots B_r} . \end{aligned}$$
(27)

Proof

The projector \(P_{(r,r)}:\otimes ^{2r}{\mathbb {V}}^*\rightarrow {\mathbb {V}}_{(r,r)}\) is given by

$$\begin{aligned} P_{(r,r)} T= S_{(1,\ldots , r)}\circ S_{(r+1,\ldots , 2r)}\circ S_{[1,r+1]}\circ \cdots \circ S_{[r,2r]}(T), \end{aligned}$$
(28)

where \(S_{(1\ldots r)}\) denotes symmetrisation over the first r indices, \(S_{(r+1,\ldots , 2r)}\) denotes symmetrisation over the last r indices, \(S_{[i,j]}\) denotes anti-symmetrisation over the two indices in, respectively, the ith and jth positions.

The claim in the Lemma is an immediate consequence. \(\square \)

In the following we extend these conventions, notations, and definitions to vector bundles (with fibre \({\mathbb {V}}\)) in the obvious way.

Killing equations: prolongation via the tractor connection

Here we treat the Killing-type equations

$$\begin{aligned} \nabla _{(a_0}k_{a_1\cdots a_r)}=0 , \end{aligned}$$
(29)

on an affine manifold with an affine connection \(\nabla \). For simplicity we assume \(\nabla \) is torsion free, but this plays almost no role. There is such an equation for each \(r\in {\mathbb {Z}}_{>0}\), and as discussed above the equations are each projectively invariant if we take the symmetric rank r tensor to have projective weight 2r, i.e. \(k_{b\cdots c}\in \Gamma ({\mathcal {E}}_{(b\cdots c)}(2r))\). In the following, we denote by \({\mathcal {T}}_{(k,\ell )}\) the tractor bundle with fibre \({\mathbb {V}}_{(k,\ell )}\) where \({\mathbb {V}}={\mathbb {R}}^{n+1}={\mathcal {T}}|_p\). Moreover, we include the weight w in the notation as \({\mathcal {T}}_{(k,\ell )}(w)\).

Via the cotractor filtration sequence (8) we evidently have the following.

Lemma 5

There is a projectively invariant bundle inclusion

$$\begin{aligned} S^r T^*M (2r)\rightarrow S^r{\mathcal {T}}^* (r)={\mathcal {T}}_{(r)}(r) \end{aligned}$$

given by

$$\begin{aligned} S^r T^*M (2r) \ni k_{b\cdots c}\mapsto K_{B\cdots C}:=Z_{B}{}^{b}\cdots Z_{C}{}^{c} k_{b\cdots c}\in {\mathcal {T}}_{(r)}(r). \end{aligned}$$
(30)

Note that for K as here we have

$$\begin{aligned} X^BK_{B \cdots C}=0. \end{aligned}$$
(31)

Moreover, if \(K\in {\mathcal {T}}_{(r)}(r)\) satisfies Eq. (31), then it is in the image of the map (30).

This enables a tractor interpretation of the Killing-type equations, as follows.

Proposition 6

For each rank r Eq. (29) is equivalent to the tractor equation

$$\begin{aligned} {\mathbb {D}}_{(A}K_{B\cdots C)}=0, \end{aligned}$$
(32)

where \(K_{B\cdots C}\) is given by Eq. (30).

Proof

From the tractor formulae Eq. (12) and Eq. (15) we have

$$\begin{aligned} {\mathbb {D}}_{A_0}K_{A_1\cdots A_r} =&\ rY_{A_0}K_{A_1A_2\cdots A_r} - Y_{A_1}K_{A_0A_2\cdots A_r}- \cdots - Y_{A_r}K_{A_1A_2\cdots A_{r-1}A_0}\\&+ Z_{A_0}{}^{a_0}Z_{A_1}{}^{a_1}\cdots Z_{A_r}{}^{a_r}\nabla _{a_0}k_{a_1\cdots a_r} , \end{aligned}$$

which implies

$$\begin{aligned} {\mathbb {D}}_{(A_0}K_{A_1\cdots A_r)} = Z_{(A_0}{}^{a_0}Z_{A_1}{}^{a_1}\cdots Z_{A_r)}{}^{a_r}\nabla _{a_0}k_{a_1\cdots a_r} , \end{aligned}$$

from which the result follows immediately. \(\square \)

In the following \(K_{A_1\cdots A_r}\) will always refer to a weight r symmetric tractor as given by Eq. (30). We now define a projectively invariant operator

$$\begin{aligned} {\mathcal {L}}: S^r T^*M (2r)\rightarrow P_{(r,r)}( \otimes ^{2r}{\mathcal {T}}^*) = {\mathcal {T}}_{(r,r)}, \end{aligned}$$
(33)

where \(P_{(r,r)}\) is the (rr) Young symmetry as described in expression (28), by applying the Young projection \(P_{(r,r)}\) to \({\mathbb {D}}^{r}K\), as follows

$$\begin{aligned} k_{c_1\cdots c_r}\mapsto P_{(r,r)}({\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r}), \end{aligned}$$

with \(K_{C_1\cdots C_r}=Z_{C_1}{}^{c_1}\cdots Z_{C_r}{}^{c_r} k_{c_1\cdots c_r}\).

Proposition 7

The operator \({\mathcal {L}}: S^r T^*M (2r)\rightarrow {\mathcal {T}}_{(r,r)}\) of (33) is a differential splitting operator.

Proof

We claim that

$$\begin{aligned} X^{B_1}\cdots X^{B_r} W^{C_1}{}_{c_1}\cdots W^{C_1}{}_{c_1}P_{(r,r)}({\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r}) = c k_{c_1\cdots c_r} , \end{aligned}$$
(34)

where c is a nonzero constant. It clearly suffices to show that

$$\begin{aligned} X^{B_1}\cdots X^{B_r} P_{(r,r)}({\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r}) = c K_{C_1\cdots C_r} . \end{aligned}$$
(35)

Contract \(X^{B_1}\cdots X^{B_r} \) into the explicit expansion of \(P_{(r,r)}({\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r})\). Use (i) \([{\mathbb {D}}_A,X^B]=\delta ^B_A\), (ii) \(X^A {\mathbb {D}}_A f=w f\), for any tractor field V of weight w [see Eq. (16)], and that (iii) \(X^AK_{A\cdots C}=0\), to eliminate all occurrences of \(X^A\). It follows easily that the result is \(c K_{C_1\cdots C_r}\) for some constant c, since there is no way to include a term involving \({\mathbb {D}}\)s that has the correct valence (i.e. the tractor rank r). That \(c\ne 0\) is found by explicit computation or more simply the fact that it is not zero in the case that the affine connection \(\nabla \) is projectively flat, as we shall see below. \(\square \)

The above definition is motivated by the projectively flat case where the situation is particularly elegant. (It is easily verified that the operator \({\mathcal {L}}\) above is a so-called first BGG splitting operator, as discussed in, e.g. [8], and see references therein. We will not use this fact however.)

We conclude this section with an observation. It shows, in particular, that sections of \({\mathcal {T}}_{(r,r)}\) that are parallel for the usual tractor connection determine solutions of Eq. (29). These are the so-called normal solutions (see, e.g. [8]):

Proposition 8

Let \((M,{\varvec{p}})\) be a projective manifold (not necessarily flat) and let \(L\in \Gamma ({\mathcal {T}}_{(r,r)})\) such that

$$\begin{aligned} 0= X^{B_1}\cdots X^{B_r}{\mathbb {D}}_A L_{B_1\cdots B_rC_1\cdots C_r}. \end{aligned}$$
(36)

Then, \(K_{C_1\cdots C_r}\in \Gamma ({\mathcal {T}}_{(r)})\) defined by \(K_{C_1\cdots C_r}=X^{B_1}\cdots X^{B_r}L_{B_1\cdots B_r C_1\cdots C_r}\) satisfies Eq. (32). If we assume in addition that

$$\begin{aligned} 0= {\mathbb {D}}_A L_{B_1B_2\cdots B_{r}C_1\cdots C_r}, \end{aligned}$$
(37)

then L defines a rank r Killing tensor via Eq. (34) such that L is a constant multiple of \({{\mathcal {L}}}(k)\).

Proof

The proof is a direct rewriting of Eq. (36),

$$\begin{aligned} 0= & {} X^{B_1}\cdots X^{B_r}{\mathbb {D}}_{A_1} L_{B_1\cdots B_rC_1\cdots C_r} \nonumber \\= & {} X^{B_2}\cdots X^{B_r}\left( {\mathbb {D}}_{A_1} (X^{B_1} L_{B_1,\cdots B_rC_1\cdots C_r})- L_{{A_1} B_2\cdots B_rC_1\cdots C_r} \right) \nonumber \\= & {} -\,r X^{B_2}\cdots X^{B_r} L_{{A_1} B_2\cdots B_rC_1\cdots C_r} + {\mathbb {D}}_{A_1} K_{C_1\cdots C_r}, \end{aligned}$$
(38)

where we successively apply \([{\mathbb {D}}_A,X^B]=\delta _A{}^B\) to commute and eliminate X’s and \({\mathbb {D}}\)’s and use the symmetries of L. Note that this computation does not require any mutual commutations of \({\mathbb {D}}_A\)’s. Now since \(L_{B_2\cdots B_r(AC_1\cdots C_r)}=0\) this equation implies Eq. (32). Moreover, because of the symmetries of L, we also have that

$$\begin{aligned} X^{C_i}K_{C_1\cdots C_r}=X^{C_i}X^{B_1}\cdots X^{B_r}L_{B_1\cdots B_rC_1\cdots C_r}=0, \end{aligned}$$

for each \(i=1,\ldots , r\). This implies that K is given by a k as in Eq. (30).

Applying \({\mathbb {D}}_{A_r}, \ldots , {\mathbb {D}}_{A_2} \) successively to Eq. (38), commuting with the X’s successively by \([{\mathbb {D}}_A,X^B]=\delta ^B_A\) and finally using the additional hypothesis Eq. (37), shows that \({\mathbb {D}}_{A_r}\cdots {\mathbb {D}}_{A_1}K_{C_1\cdots C_r}\) is a nonzero constant multiple of \(L_{{A_r}\cdots {A_1}C_1\cdots C_r}\). Hence, L is a constant multiple of \(\mathcal {{\mathcal {L}}}(k)\). \(\square \)

Projectively flat structures

In this subsection we restrict to affine (or projective) manifolds that are projectively flat, i.e. where the projective tractor curvature vanishes. According to Eq.  (21) this also means that the Thomas-\({\mathbb {D}}\) operators mutually commute when acting on weighted tractor sections.

In the projectively flat setting we obtain a nice characterisation of Killing tensors.

Proposition 9

Let \((M,{\varvec{p}})\) be a projectively flat manifold. Let \(k_{c_1\cdots c_r}\in \Gamma (S^r T^*M(2r))\) and define \(K_{C_1\cdots C_r}:= Z_{C_1}{}^{c_1} \cdots Z_{C_r}{}^{c_r}k_{c_1\cdots c_r}\), as in Eq. (5). Then, k satisfies the Killing equation Eq. (29) if and only if

$$\begin{aligned} {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r}\in \Gamma ({\mathcal {T}}_{(r,r)} ). \end{aligned}$$
(39)

In particular, on a projectively flat manifold there is a nonzero constant c so that

$$\begin{aligned} {{\mathcal {L}}}(k)= c\ {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r}, \end{aligned}$$

if and only if k solves Eq. (29).

Proof

(\(\Rightarrow \)) Since we work in the projectively flat setting the Thomas-\({\mathbb {D}}\) operators commute. So

$$\begin{aligned} {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r}\in \Gamma ({\mathcal {T}}_{(r)}\otimes {\mathcal {T}}_{(r)}) \end{aligned}$$

Suppose that Eq. (29) holds. Then, Eq. (32) holds, so symmetrising the left-hand side of the display over any \(r+1\) indices that include \(C_1\cdots C_r\) results in annihilation and so we conclude (39) from the definition of \({\mathbb {V}}_{(r,r)}\) and hence of \({\mathcal {T}}_{(r,r)} \) in Eq. (24).

(\(\Leftarrow \)) If Eq. (39) holds then

$$\begin{aligned} {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_{r-1}}{\mathbb {D}}_{(B_r} K_{C_1\cdots C_r)}=0 \end{aligned}$$

so

$$\begin{aligned} X^{B_1}\cdots X^{B_{r-1}}{\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_{r-1}}{\mathbb {D}}_{(B_r} K_{C_1\cdots C_r)} = (r-1)!\ {\mathbb {D}}_{(B_r} K_{C_1\cdots C_r)} =0, \end{aligned}$$

from Eq. (16), thus we obtain the result from Proposition 6. \(\square \)

Here and throughout, as above, \(K\in \Gamma ({\mathcal {T}}_{(r)}(r))\) is the image of some \(k\in \Gamma (S^r T^*M(2r)) \) as in formula (30).

Proposition 10

The constant c in Eq. (34) is not 0.

Proof

In the case that the structure is projectively flat this is immediate from Proposition  9, since \(X^{B_1}\cdots X^{B_r}\) contracted into \({\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r}\) gives \(r!\ K_{C_1\cdots C_r} \). But it is clear from the argument in the proof of Proposition 7 that c does not depend on curvature, as no commutation of \({\mathbb {D}}\)s is involved. \(\square \)

Theorem 11

Let \((M,{\varvec{p}})\) be projectively flat manifold. Then, the splitting operator \({\mathcal {L}}\) gives an isomorphism between Killing tensors of rank r and sections of \({\mathcal {T}}_{(r,r)}\) that are parallel for the projective tractor connection.

Proof

Since \({\mathcal {L}}\) is a splitting operator, it does not have a kernel. Moreover, using that \(\nabla _a L=0\) is equivalent to \({\mathbb {D}}_A L=0\), Proposition 8 shows that every parallel section of \({\mathcal {T}}_{(r,r)}\) arises as \({\mathcal {L}}(k)\) for a Killing tensor k. So it remains to show that \({\mathcal {L}}(k)\) is a parallel section of the projective tractor connection whenever k is a Killing tensor: suppose that Eq. (29) holds. Then, by Proposition 9,

$$\begin{aligned} {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r} ={{\mathcal {L}}}(k), \end{aligned}$$

and \({{\mathcal {L}}}(k)\) has weight 0 so

$$\begin{aligned} {\mathbb {D}}_{A}{{\mathcal {L}}}(k)= Z_A{}^a\nabla _a {{\mathcal {L}}}(k). \end{aligned}$$

Thus, it suffices to show that \({\mathbb {D}}_{A}{{\mathcal {L}}}(k)=0\). But

$$\begin{aligned} {\mathbb {D}}_{A}{{\mathcal {L}}}(k)= {\mathbb {D}}_{A}{\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r} =0, \end{aligned}$$

because of the identity \([{\mathbb {V}}_{(r+1)}\otimes {\mathbb {V}}_{(r)}]\cap [{\mathbb {V}}_{(r)}\otimes {\mathbb {V}}_{(r,1)}]=\{0\} \) from Corollary 3 (where we have used Eq. (32) which implies that \({\mathbb {D}}K\) is a section of \({\mathcal {T}}_{(r,1)}(2r-1)\)). \(\square \)

As a final note in this section, we observe that it is easy to “discover” the projectively invariant Killing equation using the tractor machinery, as follows. Consider a symmetric rank r covariant tensor field \(k_{c_1\cdots c_r}\) of projective weight 2r. Form

$$\begin{aligned} K_{C_1\cdots C_r}\in S^r{\mathcal {T}}^*(r) \end{aligned}$$

by Lemma 5. We wish to prolong this to a parallel tractor. This requires a tractor field of weight 0. Thus, we apply the r-fold composition of \({\mathbb {D}}\). Altogether we have the projectively invariant operator

$$\begin{aligned} k\mapsto {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} Z_{C_1}{}^{c_1}\cdots Z_{C_r}{}^{c_r} k_{c_1\cdots c_r}= {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r}, \end{aligned}$$

and the image has weight zero. Thus, we can form

$$\begin{aligned} \nabla _a {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} Z_{C_1}{}^{c_1}\cdots Z_{C_r}{}^{c_r} k_{c_1\cdots c_r}, \end{aligned}$$

by construction it is projectively invariant and we can ask what it means for this to be zero. Equivalently, we seek the condition on k determined by

$$\begin{aligned} {\mathbb {D}}_A {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r}=0 . \end{aligned}$$

But this implies \(X^{C_1}\cdots X^{C_r} {\mathbb {D}}_A {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r}=0\) and from Eq. (36) in the proof of Theorem 11 it follows that

$$\begin{aligned} {\mathbb {D}}_{(A} K_{B_1\cdots B_r)}= 0, \qquad \text {implies} \qquad \nabla _{(a}k_{b_1\cdots b_r)}=0 \end{aligned}$$

where we again used Proposition 6.

Restoring curvature

We return now to the general curved case and seek the generalisations of the results in the previous subsection. First we observe the following first generalisation of Proposition 9:

Proposition 12

Let \(k\in \Gamma (S^rT^*M(2r))\) on a general affine manifold \((M,\nabla )\) (or projective manifold \((M,{\varvec{p}})\)) and \(K= K(k)\in \Gamma ({\mathcal {T}}_{(r)} (r)) \), as in Eq. (30). Then, k is a Killing tensor, i.e. a solution of Eq. (29), if and only if we have

$$\begin{aligned} {{\mathcal {L}}}(k)_{B_1\cdots B_rC_1\cdots C_r}={\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r} + {\mathbf{Kurv}}(K)_{B_1\cdots B_rC_1\cdots C_r}, \end{aligned}$$
(40)

where \({\mathbf{Kurv}}\) is a specific projectively invariant linear differential operator on \(\Gamma ( {\mathcal {T}}_{(r)} (r))\), of order at most \((r-2)\), constructed with the W-curvature and the Thomas-\({\mathbb {D}}\) operators and such that the W-curvature and its \({\mathbb {D}}\)-derivatives appear in the coefficients of every term.

Proof

(\(\Rightarrow \)) Suppose that k solves Eq. (29). We have

$$\begin{aligned} {{\mathcal {L}}}(k)= P_{(r,r)} ({\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r} ). \end{aligned}$$

We expand out this expression on the right-hand side using the definition of the operator \(P_{(r,r)}\) in Eq. (28). We would like to show that the resulting terms can be combined and rearranged to yield Eq. (40). We have the identity Eq. (32) available. In the projectively flat case we also have the identity \([{\mathbb {D}}_A,{\mathbb {D}}_B]=0\) as an operator on (weighted) tractors. In the flat case the two identities are enough to conclude Eq. (40) (with \( {\mathbf{Kurv}}(K)=0\)), according to the proof of Proposition 9. In the curved case we perform the same formal computation but keep track of the curvature, i.e. replace each \([{\mathbb {D}}_A,{\mathbb {D}}_B]\) with \(W_{AB}\sharp \) (instead of 0). The order statement follows by construction (or elementary weight arguments), so this proves the result in this direction and generates a specific formula for \({\mathbf{Kurv}}(K)\).

(\(\Leftarrow \)) Now we suppose that \(k \in \Gamma (S^rT^*M(2r))\) is any section such that

$$\begin{aligned} {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r} + {\mathbf{Kurv}}(K)_{B_1\cdots B_rC_1\cdots C_r} \end{aligned}$$

is a section of \({\mathcal {T}}_{(r,r)}(r)\). Then, in particular

$$\begin{aligned} {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_{r-1}}{\mathbb {D}}_{(B_r} K_{C_1\cdots C_r)} + {\mathbf{Kurv}}(K)_{B_1\cdots B_{r-1}(B_rC_1\cdots C_r)} =0, \end{aligned}$$

according to Eq. (24). As in the proof of Proposition 9, we contract now with \(X^{B_1}\cdots X^{B_{r-1}}\). This contraction annihilates the second term in the display as follows. Each of the \(X^{B_i}\)’s is contracted into either a \({\mathbb {D}}_{B_i}\), into K, or into the curvature W. Thus, every \(X^{B_i}\) can be eliminated using the identities (16), that \(X^BK_{B\cdots C}=0\), and that similarly \(X^B\) contracted into any of the lower indices of the curvature W is zero. But, by the construction of the operator \({\mathbf{Kurv}}\), in any term there are at most \((r-2)\)\({\mathbb {D}}\) operators (either applied to the curvature or directly to the argument) and so the identities (16) remove only \((r-2)\) of the \((r-1)\)X’s. This means that in every term produced we have a contraction of the form \(X^BK_{B\cdots C}=0\), so that term vanishes, or X into W so also that term vanishes. Thus, we are left with

$$\begin{aligned} 0=X^{B_1}\cdots X^{B_{r-1}}{\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_{r-1}}{\mathbb {D}}_{(B_r} K_{C_1\cdots C_r)} =(r-1)! \ {\mathbb {D}}_{(B_r} K_{C_1\cdots C_r)}, \end{aligned}$$

as in the proof of Proposition 9. \(\square \)

Proposition 13

Let \(k\in \Gamma (S^rT^*M(2r))\) on a general affine manifold \((M,\nabla )\) (or projective manifold \((M,{\varvec{p}})\)) and \(K= K(k)\in \Gamma ({\mathcal {T}}_{(r)} (r)) \), as in Eq. (30). Then, k is a solution of Eq. (29) if and only if we have

$$\begin{aligned} {\mathbb {D}}{{\mathcal {L}}}(k)= {\mathbf{Curv}}(K), \end{aligned}$$
(41)

where \({\mathbf{Curv}}\) is a projectively invariant linear differential operator, of order at most \((r-1)\), on \(\Gamma ({\mathcal {T}}_{(r,r)}(r))\) given by a specific formula constructed with the W-curvature, and the Thomas-\({\mathbb {D}}\) operator such that the W-curvature and its derivatives appear in the coefficients of every term. Moreover, if \({\mathcal {L}}(k)\) satisfies equation Eq. (41), then

$$\begin{aligned} X^{B_1}\cdots X^{B_r}{\mathbb {D}}_A{\mathcal {L}}(k)_{{B_1}\cdots _{B_r} C_1\cdots C_r}=0. \end{aligned}$$
(42)

Proof

(\(\Rightarrow \)) Suppose that k solves Eq. (29). We apply \({\mathbb {D}}_A\) to both sides of Eq. (40). This yields

$$\begin{aligned} {\mathbb {D}}_A{{\mathcal {L}}}(k)= {\mathbb {D}}_A {\mathbb {D}}_{B_1}\cdots {\mathbb {D}}_{B_r} K_{C_1\cdots C_r} + {\mathbb {D}}_A \mathbf{Kurv}(K)_{B_1\cdots B_rC_1\cdots C_r} . \end{aligned}$$
(43)

In the case when \(\nabla \) is projectively flat the first term on the right can be shown to be zero by a formal calculation using just the identities \([{\mathbb {D}}_A,{\mathbb {D}}_B]=0\) and \({\mathbb {D}}_{(A_0}K_{A_1\cdots A_r)}=0\). This follows from the proof of Theorem 11. Performing the same formal calculation, but now instead replacing the commutator of \({\mathbb {D}}\)’s with \([{\mathbb {D}}_A,{\mathbb {D}}_B]=W_{AB}\sharp \) and combining the result with the second term on the right-hand side yields the result: \({\mathbb {D}}{{\mathcal {L}}}(k)\) is equal to a specific formula for a linear differential operator \(\mathbf{Curv}\) on K that is constructed polynomially, and with usual tensor operations, involving just the W-curvature, and the Thomas-\({\mathbb {D}}\) operator. Thus, by construction it is projectively invariant, and also by construction (or weight arguments) the order claim follows.

(\(\Leftarrow \)) We suppose now that Eq. (41) holds with \(k\in \Gamma (S^rT^*M(2r))\), K as in Eq. (30) and with the operator \(\mathbf{Curv}\) given by the formula found in the first part of the proof. So we have

$$\begin{aligned} {\mathbb {D}}_A{{\mathcal {L}}}(k)_{B_1\cdots B_rC_1\cdots C_r}= \mathbf{Curv}(K)_{AB_1\cdots B_rC_1\cdots C_r} . \end{aligned}$$

Note that contraction of \(X^{C_1}\cdots X^{C_r}\) annihilates the right-hand side by an easy analogue of the argument used in the second part of the proof of Proposition 12 above: in this case there are at most \((r-1)\) many \({\mathbb {D}}\) operators in any term but we are contracting in \(\otimes ^r X\), so in each term an X is contracted directly into and undifferentiated K or W. The result now follows by the argument used in second part of the proof of Theorem 11 for the projectively flat case. Thus, we have just shown that we have Eq. (42). Then, the result follows from the first part of Proposition 8. \(\square \)

For the proof of the main theorem we recall the following fact, which follows from the theory of overdetermined systems of PDE.

Lemma 14

For every \(T\in {\mathcal {T}}_{(r,r)}|_x\), where \(x\in M\), there is a local section \(k\in \Gamma ( {\mathcal {S}}^r T^*M|_U)\), such that \(T={{\mathcal {L}}}(k)|_x\).

Proof

In the case of (projectively) flat \((M,{\varvec{p}})\) this follows at once from the fact that in the flat case for \(L\in \Gamma (T_{(r,r)})\) we have shown that \(\nabla L=0\) implies \(L={{\mathcal {L}}}(k)\).

For the general case the result then follows as the formula for the operator \({{\mathcal {L}}}(k)\) generalises that from the flat case by the simply the addition (at each order) of lower-order curvature terms.\(\square \)

Now we state and prove the main results of the paper.

Theorem 15

Let \((M,{\varvec{p}})\) be a projective manifold. Then, there is a specific section \({\mathcal {R}}_A\sharp \in {\mathcal {T}}^*M\otimes \hbox {End}({\mathcal {T}}_{(r,r)})\) (where we suppress the endomorphism indices) such that \(X^A{\mathcal {R}}_A\sharp =0\) and such that the differential splitting operator \({\mathcal {L}}:\Gamma (S^rT^*M(2r)) \rightarrow \Gamma ( {\mathcal {T}}_{(r,r)}) \) gives an isomorphism between Killing tensors of rank r and sections L of the bundle \({\mathcal {T}}_{(r,r)}\) that satisfy the equation

$$\begin{aligned} {\mathbb {D}}_A L = {\mathcal {R}}_A \sharp L. \end{aligned}$$
(44)

Proof

Again, the splitting operator \({\mathcal {L}}\) is injective. Hence, we have to show the following:

  1. (A)

    For every Killing tensor k the image \({\mathcal {L}}(k)\) satisfies Eq. (44) with a specific \({\mathcal {R}}_A\sharp \in {\mathcal {T}}^*M\otimes \hbox {End}({\mathcal {T}}_{(r,r)})\) that will be determined;

  2. (B)

    \({\mathcal {L}}\) restricted to Killing tensors [i.e. the solutions of Eq. (29)] is surjective onto the sections L that satisfy Eq. (44), where the right-hand side is as determined in (A).

We prove (A): assume that k solves Eq. (29). Then, we have Eq. (41),

$$\begin{aligned} {\mathbb {D}}{{\mathcal {L}}}(k)= {\mathbf{Curv}}(K). \end{aligned}$$

from Proposition 13. The operator \({\mathbf{Curv}}\) is given by a formula polynomial in the W-curvature, its \({\mathbb {D}}\) derivatives, and the Thomas-\({\mathbb {D}}\) operators up to order \((r-1)\). Now observe that each term of the form \({\mathbb {D}}_{B_1} \cdots {\mathbb {D}}_{B_s}K_{C_1\cdots C_r}\), for \(0\le s <r\) can be replaced using Eq. (40) from Proposition 12,

$$\begin{aligned} \begin{aligned} {\mathbb {D}}_{B_1} \cdots {\mathbb {D}}_{B_s}K_{C_1\cdots C_r} = c X^{B_{s+1}} \cdots X^{B_r} {\mathcal {L}}(k)_{B_1\cdots B_r C_1\cdots C_r} + {\mathbf {Curv}}^{(s)}(K), \end{aligned} \end{aligned}$$

where \( {\mathbf{Curv}}^{(s)}\) is a differential operator given by a formula polynomial in the W-curvature, its \({\mathbb {D}}\) derivatives, and the Thomas-\({\mathbb {D}}\) operators up to order \((s-2)\). In this way we can successively eliminate all applications of \({\mathbb {D}}\) to K by terms algebraic in \({\mathcal {L}}(k)\) arriving at an equation of the form

$$\begin{aligned} {\mathbb {D}}_A {{\mathcal {L}}}(k)={{\mathcal {R}}}_A\sharp {\mathcal L}(k),\quad \text {with}\quad {\mathcal {R}}_A\in \Gamma ({\mathcal {T}}^* \otimes \hbox {End}(\otimes ^{2r}{\mathcal {T}}^*)), \end{aligned}$$
(45)

given by a polynomial in the W-curvature and its \({\mathbb {D}}\)-derivatives. Now we have to verify:

  1. (i)

    that \({{\mathcal {R}}}_A\sharp \) is indeed a section of \({\mathcal {T}}^* \otimes \hbox {End}({\mathcal {T}}_{(r,r)})\), and

  2. (ii)

    that for every \(L\in {\mathcal {T}}_{(r,r)}\), the contraction of \({{\mathcal {R}}}_A\sharp L\) with \(X^A\) is equal to zero.

In order to verify (i) and (ii) we have to make a key observation: although we phrased the discussion above in a naive way that supposes there is a solution to Eq. (29), in fact to derive Eq. (45), we do not actually require that there exist solutions, even locally, to the Eq. (29). Equation (45) simply expresses relations on the jets, of a section \(k\in \Gamma (S^rT^*M(2r))\) that are formally determined by a finite jet prolongation of the Killing Eq. (29). It is clear that we can derive Eq. (45) at any point \(x\in M\) by working with just the \(r+1\)-jet, \(j^{r+1}_xk\), of k at x. Following the argument as above, but working formally with such jets and assuming Eq. (29) holds to order r at x, we come to

$$\begin{aligned} {\mathbb {D}}_A L|_x= {\mathcal {R}}_A \sharp {{\mathcal {L}}}(k)|_x \end{aligned}$$
(46)

where all curvatures and their derivatives are evaluated at x. From the results in the projectively flat case we know that this is exactly the point where the prolongation of the finite-type PDE Eq. (29) has closed: the prolongation up to order r may be viewed as simply the introduction of new variables labelling the part of the jet that is not constrained by the equation, and these are exactly parametrised by the elements in the fibre \({\mathcal {T}}_{(r,r)}|_x\). At the next order the derivative of these variables is expressed algebraically in terms of the variables from \({\mathcal {T}}_{(r,r)}|_x\). That is (a key part of) the content of Eq. (46). Viewing this as a computation in slots (via a choice of \(\nabla \in {\varvec{p}})\) the computation is the same in the curved case as in the projectively flat case except that additional curvature terms may enter when derivatives are commuted. It follows that \({{\mathcal {L}}}(k)|_x\) may be an arbitrary element L of \({\mathcal {T}}_{(r,r)}|_x\). Using this, and since contraction with \(X^A\) annihilates the left-hand side of Eq. (46) it follows that it annihilates the right-hand side for any \(L\in T_{(r,r)}|_x\). Similarly since the left-hand side of Eq. (46) is an element of \(({\mathcal {T}}^*\otimes T_{(r,r)})|_x\) so is the right-hand side, for arbitrary \(L={{\mathcal {L}}}(k)|_x\) and thus (ii) also follows.

Now we prove (B): suppose that \(L\in \Gamma ({\mathcal {T}}_{(r,r)})\) satisfies Eq. (44) for the specific \({\mathcal {R}}_A\in \Gamma ( {\mathcal {T}}^*\otimes {\mathcal {T}}_{(r,r)}) \) obtained from the argument above. We now claim that

$$\begin{aligned} X^{B_1}\cdots X^{B_r} ( {\mathcal {R}}_C \sharp L)_{B_1 \cdots B_r C_1\cdots C_r}=0. \end{aligned}$$
(47)

Indeed, in the case that \(L={\mathcal {L}}(k)\) for a tensor k that solves Eq. (29), we know from Proposition 13 that \(X^{C_1}\cdots X^{C_r}\) annihilates the right-hand side of Eq. (44) for \({\mathcal {L}}(k)\), because then it is simply a rewriting of the right-hand side of Eq. (41). However, as mentioned above, at a point \(x\in M\) and for k satisfying Eq. (29) to order r at x, any element of \({\mathcal {T}}_{(r,r)}|_x\) can arise as \({{\mathcal {L}}}(k)|_x\) because this is the full prolonged system for the overdetermined PDE Eq. (29). Thus, it follows that \(X^{C_1}\cdots X^{C_r}\) must annihilate the right-hand side of Eq. (44) for L even if L is not \(\mathcal L(k)\) for a \(k\in \Gamma (S^rT^*M(2r))\) satisfying Eq. (29).

Having established Eq.  (47), we can apply the first part of Proposition 8 to ensure that L determines a Killing tensor k. Then, we have that \(L={\mathcal {L}}(k)\) unless the map

$$\begin{aligned} L_{B_1\cdots B_rC_1\cdots C_r}\mapsto K_{B_1\cdots B_r}=X^{C_1}\cdots X^{C_r}L_{B_1\cdots B_rC_1\cdots C_r}\in \mathcal T_{(r)}(r). \end{aligned}$$

has a kernel. To exclude this possibility, assume there is a section L of \({\mathcal {T}}_{(r,r)}\) that satisfies Eq. (44) and such that

$$\begin{aligned} X^{C_1}\cdots X^{C_r}L_{B_1\cdots B_rC_1\cdots C_r}=0. \end{aligned}$$
(48)

The following lemma shows that this implies the vanishing of L.

Lemma 16

Let \(L_{B_1\cdots B_rC_1\cdots C_r}\) be a section of \({\mathcal {T}}_{(r,r)}\) that satisfies Eq. (44) for the specific \({\mathcal {R}}_A\sharp \in \Gamma ({\mathcal {T}}^*\otimes {\mathcal {T}}_{(r,r)})\). Then, we have the following implication: if

$$\begin{aligned} X^{B_1}\cdots X^{B_k}L_{B_1\cdots B_k\cdots B_rC_1\cdots C_r}=0\quad \text { for a } k\in \{1,\ldots , r\}, \end{aligned}$$
(49)

then

$$\begin{aligned} X^{B_1}\cdots X^{B_{k-1}}L_{B_1\cdots B_{k-1} \cdots B_rC_1\cdots C_r}=0, \end{aligned}$$

and hence \(L_{B_1 \cdots B_rC_1\cdots C_r}=0\).

Proof

Assume that Eq. (49) holds. Applying \({\mathbb {D}}_A\), the Leibniz rule for \({\mathbb {D}}_A\) gives

$$\begin{aligned} 0=c \ X^{B_1}\cdots X^{B_{k-1}}L_{B_1\cdots B_{k-1}A B_{k+1}\cdots B_rC_1\cdots C_r}+ X^{B_1}\cdots X^{B_k}{\mathbb {D}}_A L_{B_1\cdots B_rC_1\cdots C_r}, \end{aligned}$$
(50)

with a nonzero constant c. Hence, we have to show that Eq. (49) implies

$$\begin{aligned} X^{B_1}\cdots X^{B_k}{\mathbb {D}}_A L_{B_1\cdots B_k\cdots B_rC_1\cdots C_r}=0, \end{aligned}$$
(51)

by using Eq. (44) and the specific form of \({\mathcal {R}}_A\sharp \). The proofs of the previous propositions and of (A) provide us with the following information about \({\mathcal {R}}_A\sharp \): in Proposition 13 we have seen that the expression \({\mathbf{Curv}}(K)\) was of order at most \((r-1)\) in \({\mathbb {D}}\) and is a linear combination of terms of the form \({\mathcal {A}}^{(s-1)}\otimes {\mathbb {D}}^{r-s} K\) for \(1\le s\le r\) and where \({\mathcal {A}}^{(s-1)}\) is a tractor of valence s containing at most \(s-1\) applications of \({\mathbb {D}}\) to the tractor curvature W. Then, in (A) of the present proof we have expressed the terms \({\mathbb {D}}^{r-s} K\) by an s-fold contraction of \({\mathcal {L}}(k)\) with X. Hence \(\mathcal R_A\sharp L\) is a linear combination of terms of the form

$$\begin{aligned} {\mathcal {A}}^{(s-1)}\otimes {\mathcal {B}}^{(s)}, \end{aligned}$$
(52)

where \({\mathcal {B}}^{(s)}\) is of the form \(X^{E_1}\cdots X^{E_s}L_{E_1\cdots E_s E_{s+1}\cdots E_rC_1\ \cdots C_r}\). Because of Eq. (49), the only terms that are nonzero in \({\mathcal {R}}_A\sharp L\) are those of the form (52) with \(s<k\). Hence the terms \(A^{(s-1)}\) contain at most \((k-2)\)\({\mathbb {D}}\)-derivatives of the tractor curvature. Now since \(X^AW_{AB}=0\) and therefore \(X^A {\mathbb {D}}_{B}W_{AC}= - W_{BC}\), each of the \({\mathcal {A}}^{(s-1)}\) is annihilated by s contractions with X. Hence the only terms of the form (52) that are nonzero when contracted with k many X’s must have at least \((k+1-s)\) contractions with X at \(B^{(s)}\), which already is obtained by s contractions with X. Hence the only terms \({\mathcal {B}}^{(s)} \) that may remain nonzero when contracted with \((k+1-s)\) many X’s are of the form

$$\begin{aligned} X^{B_1}\cdots X^{B_s}X^{C_1}\cdots X^{C_{k+1-s}} L_{B_1\cdots B_s \cdots B_rC_1 \cdots C_{k+1-s}\cdots C_r}. \end{aligned}$$
(53)

Now an induction over s shows that these terms are actually zero. In fact, for \(s=1\) this follows from the assumption Eq. (49). If \(s>1\) we use that \(L\in {\mathcal {T}}_{(r,r)}\) to get

$$\begin{aligned}&X^{B_1}\cdots X^{B_s}X^{C_1}\cdots X^{C_{k+1-s}} L_{B_1\cdots B_rC_1 \cdots C_r} \\&\quad =-\sum _{i=1}^r X^{B_1}\cdots X^{B_s}X^{C_1}\cdots X^{C_{k+1-s}} L_{B_1\cdots B_{s-1}C_iB_{s+1} \cdots B_rC_1 \cdots C_{i-1}B_s C_{i-1}\cdots C_r}\\&\quad = -(k+1-s)\ X^{B_1}\cdots X^{B_s}X^{C_1}\cdots X^{C_{k+1-s}} L_{B_1\cdots B_rC_1 \cdots C_r} \end{aligned}$$

by the induction hypothesis. This shows that the terms in (53) are indeed zero and finishes the proof of the lemma. \(\square \)

This shows that every \(L\in \Gamma ({\mathcal {T}}_{(r,r)})\) that satisfies Eq. (44) is the image of a Killing tensor under the splitting operator \({\mathcal {L}}\). This finishes the proof of (B) and hence of the theorem. \(\square \)

Rewriting the result of this theorem in terms of the tractor connection gives:

Corollary 17

Let \((M,{\varvec{p}})\) be a projective manifold. Then, there is a projectively invariant section \({\mathcal {Q}}_a\sharp \in \Gamma ( T^*M\otimes \hbox {End}({\mathcal {T}}_{(r,r)})\) such that the splitting operator \({\mathcal {L}}\) gives an isomorphism between weighted Killing tensors of rank r and sections \(L\in \Gamma ( {\mathcal {T}}_{(r,r)})\) that satisfy the equation

$$\begin{aligned} \nabla ^{\mathcal {T}}_a L= {\mathcal {Q}}_a\sharp L, \end{aligned}$$
(54)

or equivalently, sections L that are parallel for connection

$$\begin{aligned} \nabla ^{\mathcal {T}}_a- {\mathcal {Q}}_a\sharp . \end{aligned}$$
(55)

Proof

This follows by contracting Eq. (44) with \(W^{A}{}_{a}\) yielding Eq. (54) with some \({\mathcal {Q}}_a\sharp \in \Gamma ( T^*M\otimes \hbox {End}({\mathcal {T}}_{(r,r)})\). Moreover, since \(X^A{\mathcal {R}}_A\sharp =0\), the resulting \({\mathcal {Q}}_a\) is projectively invariant. \(\square \)

Remark 18

As a final remark, we note that there is a considerable gain in understanding the prolongation of Eq. (29) in the form Eq. (54) [or equivalently Eq. (55)], rather than simply as some (possible invariant) connection \({\tilde{\nabla }}\) on \({\mathcal {T}}_{(r,r)}\) without the structure (55) (or some equivalent) made explicit. An obvious example of such a gain is for the explicit computation of integrability conditions. Given such a connection the standard way to compute integrability conditions is via the curvature of \(\tilde{\nabla }\), since this must annihilate any section of \({\mathcal {T}}_{(r,r)}\) that corresponds to a solution of Eq. (29). However, because the bundle \({\mathcal {T}}_{(r,r)}\) has very high rank (e.g. for \(r=2\) it has rank \(n^2(n^2-1)/12\)) and the prolongation connection is necessarily very complicated, computing such curvature is typically out of reach without the development of specialised software. However, given Eq. (54) we obtain integrability conditions immediately from the curvature \(\kappa \) [see Eq. (13)] of the normal tractor connection: differentiating Eq. (54) with the latter and skewing in the obvious way we obtain

$$\begin{aligned} 2\nabla ^{{\mathcal {T}}}_{[b}\nabla ^{\mathcal {T}}_{a]} L= \kappa _{ba}\sharp L= \nabla _{[b} ({\mathcal {Q}}_{a]}\sharp L). \end{aligned}$$
(56)

Then, using similar ideas to the treatments above, we can expand the (far) right-hand side by replacing any instance of \(\nabla ^{\mathcal {T}}_b L\) with \(Q_{b}\sharp L\) and thus, by subtracting \(\kappa _{ba}\sharp L\), obtain at once a projectively invariant 2-form with values in \({\text {End}}(T_{(r,r)})\), that must annihilate any L(k) for k solving Eq. (29). Thus, the existence of solutions to Eq. (29) constrains the rank of this natural projective invariant constructed from the tractor curvature and its derivatives. From there one can compute invariants that must vanish following standard ideas, as in, e.g. [23, Section 3] (applied there to a different problem).

Explicit results for low rank

The curved rank \(r=1\) case

The rank one case is well known, and here we compare it to our approach. We construct the connection corresponding to the equation

$$\begin{aligned} \nabla _{(a}k_{b)}=0 \qquad \nabla \in {\varvec{p}}\end{aligned}$$
(57)

on \(k_b\in \Gamma (T^*M(2))\) on a projective manifold \((M,{\varvec{p}})\). Following Lemma 5 we form \(K_C=Z_{C}{}^{c} k_c\in {\mathcal {T}}^*(1)\), where \(k_c\) is a solution of Eq. (29), and then according to the definition (33), set

$$\begin{aligned} {{\mathcal {L}}}(k)_{BC}: ={\mathbb {D}}_{[B}K_{C]}. \end{aligned}$$

Consider the case that k is a solution of Eq. (57). Then, from Proposition 6,

$$\begin{aligned} {\mathbb {D}}_BK_C\in \Gamma (\Lambda ^2{\mathcal {T}}^*) , \end{aligned}$$

and because the W-tractor satisfies the algebraic Bianchi identity \(W^{\phantom {A}}_{AB}{}^{E}{}_C+W^{\phantom {A}}_{BC}{}^{E}{}_A+W^{\phantom {A}}_{CA}{}^{E}{}_B=0\) we have \({\mathbb {D}}_{[A}{\mathbb {D}}_{B}K_{C]}=0\), that is

$$\begin{aligned} {\mathbb {D}}_A{\mathbb {D}}_BK_C=[{\mathbb {D}}_C,{\mathbb {D}}_B]K_A = -W^{\phantom {A}}_{CB}{}^{E}{}_AK_E . \end{aligned}$$

So for solutions k we have

$$\begin{aligned} {\mathbb {D}}_{A}{\mathbb {D}}_{[B}K_{C]} - W_{BC}{}^{E}{}_A X^F{\mathbb {D}}_{[F}K_{E]}=0. \end{aligned}$$

So \(\nabla _a {{\mathcal {L}}}(k)_{BC}+ W_{BC}{}^{E}{}_A W^A_aX^F {{\mathcal {L}}}(k)_{EF}=0\). But for any \(k\in \Gamma (T^*M(2))\)

$$\begin{aligned} X^F{\mathbb {D}}_{[F}K_{E]} =X^F {{\mathcal {L}}}(k)_{FE}= K_E . \end{aligned}$$

Thus, the projectively invariant connection on \(\Lambda ^2{\mathcal {T}}^*\) is given by

$$\begin{aligned} \nabla _a V_{BC}+ W^{\phantom {A}}_{BC}{}^{E}{}_A W^{A}{}_{a}X^F V_{EF} . \end{aligned}$$

It is easily checked that this agrees with the formula Eq. (3) from the introduction (and so that connection \({\overline{\nabla }}\) is projectively invariant).

The curved rank \(r=2\) case

Here we consider the case \(r=2\). We will make the computations in Sect. 3.2 explicit and in particular provide explicit formulae for the curvature tractor fields \({\mathcal {R}}_A\sharp \) and \({\mathcal {Q}}_a\sharp \).

The first observation was established as part of a more involved argument in the second part of the proof of Proposition 13:

Lemma 19

If \(K_{DE}\in \Gamma ({\mathcal {T}}_{(2)} (2)) \), then

$$\begin{aligned} X^DX^E{\mathbb {D}}_A{\mathbb {D}}_B{\mathbb {D}}_CK_{DE}=6{\mathbb {D}}_{(A}K_{BC)}. \end{aligned}$$

In particular, \( X^EX^D{\mathbb {D}}_A{\mathbb {D}}_B{\mathbb {D}}_CK_{DE}\) is totally symmetric.

Proof

A direct computation using Eq. (17) implies

$$\begin{aligned} X^C{\mathbb {D}}_A V_{CB\cdots } =\left[ X^C,{\mathbb {D}}_A\right] V_{CB\cdots }+ {\mathbb {D}}_A(X^CV_{CB\cdots }) = - V_{AB\cdots }+ {\mathbb {D}}_A(X^CV_{CB\cdots }). \end{aligned}$$
(58)

This can be used to commute \(X^E\) and \(X^D\) past the \({\mathbb {D}}\)’s until \(X^EK_{EA}=0\) can be applied. \(\square \)

Now we study the projection \(P:=P_{(2,2)} \) from \(\otimes ^4{\mathcal {T}}^*\) to \({\mathcal {T}}_{(2,2)}\) defined in Eq. (28). If \(S_{BCDE}\) is an element in \(\otimes ^4{\mathcal {T}}^*\) that is symmetric in D and E, i.e. \(S_{BCDE}=S_{BC(DE)}\), then a straightforward computation shows that for \(S_{BCDE}\in \otimes ^2{\mathcal {T}}^* \otimes {\mathcal {T}}_{(2)}\) we have

$$\begin{aligned} (PS)_{BCDE} = \tfrac{1}{4}\left( S_{(BC)DE}+ S_{(DE)BC}\right) -\tfrac{1}{8}\left( S_{(DC)BE} + S_{(EB)CD} + S_{(DB)CE}+ S_{(EC)BD}\right) .\nonumber \\ \end{aligned}$$
(59)

This implies indeed that

$$\begin{aligned} (S_{(ijk)}PS)_{BCDE}=0, \end{aligned}$$

i.e. the symmetrisation of PS over any three indices \(1\le i<j<k \le 4\) vanishes.

Next, for a section \(K_{DE}\in \Gamma ({\mathcal {T}}_{(2)} (2) )\) we set \(S_{BCDE}:={\mathbb {D}}_B{\mathbb {D}}_CK_{DE}\). Note that the differential splitting operator \({\mathcal {L}}\) is given by \({\mathcal {L}}(k_{bd})=(P{\mathbb {D}}^2K)_{BCDE}\).

We obtain the following statement, which was already observed in the proof of Theorem 11 and Proposition 15 for general rank:

Lemma 20

If \(K_{DE}\in \Gamma ({\mathcal {T}}_{(2)} (2) )\), then

$$\begin{aligned} X^EX^D{\mathbb {D}}_A(P{\mathbb {D}}^2K)_{BCDE}= \tfrac{1}{4} X^EX^D{\mathbb {D}}_A{\mathbb {D}}_B{\mathbb {D}}_CK_{DE} \end{aligned}$$

Proof

We use the formula Eq. (59) for \(S_{BCDE}:={\mathbb {D}}_B{\mathbb {D}}_CK_{DE}\) and apply \({\mathbb {D}}_A\) to it. Using relation Eq. (58) as well as \(X^DK_{DB}=0\) and Eq. (16), a direct computation shows that each of the last eight terms in the right-hand side of Eq. (59) vanishes when contracted with \(X^D\) and \(X^E\). For example,

$$\begin{aligned} X^EX^D{\mathbb {D}}_A{\mathbb {D}}_D{\mathbb {D}}_CK_{BE}= & {} -\,{\mathbb {D}}_CK_{BA}+X^D{\mathbb {D}}_A{\mathbb {D}}_CK_{BD}-X^D{\mathbb {D}}_A{\mathbb {D}}_CK_{BE}\\= & {} -\,{\mathbb {D}}_AX^D{\mathbb {D}}_CK_{BD}-X^D{\mathbb {D}}_A{\mathbb {D}}_DK_{BC}\\= & {} 2 {\mathbb {D}}_AK_{BC}-{\mathbb {D}}_AX^D{\mathbb {D}}_DK_{BC}\\= & {} 0. \end{aligned}$$

A similar computation shows that

$$\begin{aligned} X^EX^D{\mathbb {D}}_A{\mathbb {D}}_D{\mathbb {D}}_EK_{BC} = {\mathbb {D}}_AK_{BC} +[X^E,{\mathbb {D}}_A]{\mathbb {D}}_EK_{BC} =0. \end{aligned}$$

Hence, Eq. (59) implies that

$$\begin{aligned} X^EX^D{\mathbb {D}}_A(P{\mathbb {D}}^2K)_{BCDE}= \tfrac{1}{4} X^EX^D{\mathbb {D}}_A{\mathbb {D}}_{(B}{\mathbb {D}}_C)K_{DE}= \tfrac{1}{4} X^EX^D{\mathbb {D}}_A{\mathbb {D}}_B{\mathbb {D}}_CK_{DE}, \end{aligned}$$

where the second equality follows from Lemma 19. \(\square \)

The following lemma will give a formula for the projection P, when restricted to \({\mathcal {T}}\otimes {\mathcal {T}}_{(2,1)}\), i.e. applied to \(S_{BCDC}\in {\mathcal {T}}^*\otimes {\mathcal {T}}_{(2,1)}\).

Lemma 21

Let \(P:=P_{(2,2)} \) be the projection of \(\otimes ^4{\mathcal {T}}^*\) onto \({\mathcal {T}}_{(2,2)}\) defined above and \(S_{BCDC}\in {\mathcal {T}}^*\otimes {\mathcal {T}}_{(2,1)}\). Then

$$\begin{aligned} (PS)_{BCDE} =\tfrac{3}{4}\left( S_{BCDE} - S_{[BC]DE}\right) -\tfrac{3}{8}\left( S_{[DC]BE}+ S_{[EB]CD}+S_{[DB]CE}+ S_{[EC]BD}\right) .\nonumber \\ \end{aligned}$$
(60)

Proof

We use Eq. (59) under the additional assumption that \(S_{BCDC}\in {\mathcal {T}}^*\otimes {\mathcal {T}}_{(2,1)}\), i.e.

$$\begin{aligned} S_{B(CDE)}=0. \end{aligned}$$
(61)

For the third term on the right-hand side in Eq. (59) we compute

$$\begin{aligned} S_{(DC)BE} \ =\ S_{CDBE}+ S_{[DC]BE} \ =\ - S_{CBDE}- S_{CEBD} +S_{[DC]BE}, \end{aligned}$$

where the last equation uses Eq. (61). This allows to compute the sum of the last four terms in Eq. (59) as

$$\begin{aligned}&{S_{(DC)BE} + S_{(EB)CD} + S_{(DB)CE}+ S_{(EC)BD}=} \nonumber \\&\quad = -\,4 S_{(CB)DE} -S_{CEDB}- S_{CDBE} -S_{BEDC}-S_{BDEC} \nonumber \\&\qquad + \,S_{[DC]BE}+S_{[EB]CD}+S_{[DB]CE}+S_{[EC]BD} \nonumber \\&\quad = -\,2 S_{(CB)DE} + S_{[DC]BE}+S_{[EB]CD}+S_{[DB]CE}+S_{[EC]BD}, \end{aligned}$$
(62)

where the last equation again follows from Eq. (61).

Now we look at the second term on the right-hand side of Eq. (59): using Eq. (61) we get that

$$\begin{aligned} \begin{array}{rcl} S_{(DE)BC} &{}=&{} -\,\tfrac{1}{2} \left( S_{DBCE}+ S_{DCEB}+ S_{EBCD}+ S_{ECDB}\right) \\ &{}=&{} -\,\tfrac{1}{2} \left( S_{BDCE}+ S_{CDEB}+ S_{BECD}+ S_{CEDB}\right) \\ &{}&{}{}-\, \left( S_{[DB]CE}+ S_{[DC]EB}+ S_{[EB]CD}+ S_{[EC]DB}\right) \\ &{}=&{} S_{(BC)DE}- \big ( S_{[DB]CE}+ S_{[DC]EB}+ S_{[EB]CD}+ S_{[EC]DB}\big ). \end{array} \end{aligned}$$

Hence, Eq. (27) from the flat case generalises to

$$\begin{aligned} S_{(BC)DE}= S_{(DE)BC} + \big ( S_{[DB]CE}+ S_{[DC]EB}+ S_{[EB]CD}+ S_{[EC]DB}\big ). \end{aligned}$$
(63)

Then, putting Eq. (62) and Eq. (63) together, for \(S_{BCDC}\in {\mathcal {T}}^*\otimes {\mathcal {T}}_{(2,1)}\), finishes the proof. \(\square \)

Now assume that \({\mathbb {D}}_C\) is the Thomas \({\mathbb {D}}\)-operator and \(K_{DE}\) is symmetric such that

$$\begin{aligned} {\mathbb {D}}_{(C}K_{DE)}=0. \end{aligned}$$
(64)

Then, set \(S_{BCDE}:={\mathbb {D}}_B {\mathbb {D}}_CK_{DE}\) in the above equations. Observe that

$$\begin{aligned} S_{[BC]DE}= & {} {\mathbb {D}}_{[B}{\mathbb {D}}_{C]}K_{DE}\ =\ \tfrac{1}{2}\left( {\mathbb {D}}_{B}{\mathbb {D}}_{C}K_{DE}-{\mathbb {D}}_{C}{\mathbb {D}}_{B}K_{DE}\right) \\= & {} \tfrac{1}{2} W_{BC}\ \sharp K_{DE} \ =\ -W_{BC}{}^F{}_{(D}K_{E)F}. \end{aligned}$$

Then, from Lemma 21 we get an explicit version of the curvature terms in Proposition 12:

Proposition 22

Let \({\mathbb {D}}\) be the Thomas \({\mathbb {D}}\)-operator for a projective structure with curvature \(W_{AB}{}^C{}_D\) and let P be the projection from \({\mathcal {T}}^*\otimes {\mathcal {T}}_{(2,1)}\) to \({\mathcal {T}}_{(2,2)}\). Then, \(K\in \Gamma ({\mathcal {T}}_{(2)})\) satisfies \({\mathbb {D}}_{(A}K_{BC)}=0\), i.e. \({\mathbb {D}}_{A}K_{BC}\in {\mathcal {T}}_{(2,1)}\), if and only if

$$\begin{aligned} (P{\mathbb {D}}^2K)_{BCDE} =\tfrac{3}{4} {\mathbb {D}}_B{\mathbb {D}}_CK_{DE} -\tfrac{3}{8}\left( W_{BC}\sharp K_{DE} + W_{D(B}\sharp K_{C)E} + W_{E(B} \sharp K_{C)D}\right) \end{aligned}$$
(65)

that is

$$\begin{aligned} {\mathbb {D}}_B{\mathbb {D}}_CK_{DE} +\tfrac{1}{2}\left( W_{BC}\sharp K_{DE} + W_{D(B}\sharp K_{C)E} + W_{E(B} \sharp K_{C)D}\right) \in {\mathcal {T}}_{(2,2)}. \end{aligned}$$

Proof

One direction immediately follows from Lemma 21 applied to \(S_{BCDE}:={\mathbb {D}}_B{\mathbb {D}}_CK_{ED}\).

For the other direction assume that Eq. (65) holds. Contracting with \(X^B\) and noting that \(X^BW_{B\cdots }=0\) as well as \(X^BK_{BC}=0\) implies that

$$\begin{aligned} X^B(P{\mathbb {D}}^2K)_{BCDE} = \tfrac{3}{4} X^B {\mathbb {D}}_B{\mathbb {D}}_CK_{DE} = \tfrac{3}{4} {\mathbb {D}}_CK_{DE} \end{aligned}$$
(66)

from the definition of \({\mathbb {D}}_B\). Hence, since \(P{\mathbb {D}}^2K\in \Gamma ({\mathcal {T}}_{(2,2)})\), the symmetrisation over CDE vanishes. \(\square \)

Note that, from Eq. (65) we obtain that

$$\begin{aligned} X^BX^C(P{\mathbb {D}}^2K)_{BCDE} =\tfrac{3}{4} X^BX^C {\mathbb {D}}_B{\mathbb {D}}_CK_{DE} = \tfrac{3}{4} X^C {\mathbb {D}}_CK_{DE} = \tfrac{3}{2} K_{DE}, \end{aligned}$$
(67)

because of Eq. (16) and Eq. (22).

Next we determine the connection for which \((P{\mathbb {D}}^2K)_{BCDE}\) is going to be parallel, i.e. we determine explicitly the curvature terms in Proposition 13, Theorem 15, and Corollary 17. To get a formula for its covariant derivative with respect to the projective tractor connection, we apply \({\mathbb {D}}\) to the equality in Proposition 22 to get

$$\begin{aligned} 4{\mathbb {D}}_C(P{\mathbb {D}}^2K)_{DEAB} =3 {\mathbb {D}}_C{\mathbb {D}}_D{\mathbb {D}}_EK_{AB} -\tfrac{3}{2}{\mathbb {D}}_C\left( W_{DE}\sharp K_{AB} + W_{A(D}\sharp K_{E)B} + W_{B(D}\sharp K_{E)A}\right) .\nonumber \\ \end{aligned}$$
(68)

We are now going to obtain a formula for \(T_{CDEAB}= {\mathbb {D}}_C {\mathbb {D}}_D{\mathbb {D}}_E K_{AB}\in \otimes ^5{\mathcal {T}}^*\). This is achieved by the following lemmas.

Lemma 23

For every \(T\in \otimes ^5{\mathcal {T}}^*\) it holds

$$\begin{aligned}&{ T_{C(DE)AB}+T_{D(EC)AB}+T_{E(CD)AB} }\\&\quad =3T_{CDEAB} +3T_{C[ED]AB} +T_{D[EC]AB} +T_{E[DC]AB} +2T_{[EC]DAB} +2T_{[DC]EAB}. \end{aligned}$$

Proof

The proof is by inspection.\(\square \)

Lemma 24

Let \(T_{ABCDE}\in \otimes ^2 {\mathcal {T}}^*\otimes {\mathcal {T}}_{(2,1)}\), i.e. \(T_{AB(CDE)}=0\). Then

$$\begin{aligned} -3T_{CDEAB}= & {} 2T_{[EC]DAB} +2T_{[DC]EAB} +2T_{[AC]BDE} +2T_{[AD]BEC} +2T_{[AE]BCD}\\&+\,2T_{A[BC]DE} +2T_{A[BD]EC} +2T_{A[BE]CD}\\&+\,3T_{C[ED]AB} + T_{C[DA]BE}+ T_{C[DB]EA}+ T_{C[EA]BD}+ T_{C[EB]DA} +T_{C[AB]DE}\\&+\,T_{D[EC]AB} + T_{D[EA]BC}+ T_{D[EB]CA}+ T_{D[CA]BE}+ T_{D[CB]EA}+T_{D[AB]EC}\\&+\,T_{E[DC]AB} + T_{E[CA]BD}+ T_{E[CB]DA}+ T_{E[DA]BC}+ T_{E[DB]CA} +T_{E[AB]CD}. \end{aligned}$$

Proof

First we can swap the pair AB with DE by using Eq. (63) for the second equality in

$$\begin{aligned} T_{ABCDE}= & {} T_{C(AB)DE}+T_{C[AB]DE}+2T_{[AC]BDE}+2T_{A[BC]DE}\\= & {} T_{C(DE)AB} + T_{C[DA]BE}+ T_{C[DB]EA}+ T_{C[EA]BD}+ T_{C[EB]DA}\\&+\,T_{C[AB]DE}+2T_{[AC]BDE}+2T_{A[BC]DE}. \end{aligned}$$

In an analogous computation as in the flat case, this can be used to evaluate

$$\begin{aligned} 0=&\,3T_{AB(CDE)}\\ =&\, T_{C(DE)AB}+T_{D(EC)AB}+T_{E(CD)AB}\\&+\, T_{C[DA]BE}+ T_{C[DB]EA}+ T_{C[EA]BD}+ T_{C[EB]DA} +T_{C[AB]DE}+2T_{[AC]BDE}+2T_{A[BC]DE}\\&+\, T_{D[EA]BC}+ T_{D[EB]CA}+ T_{D[CA]BE}+ T_{D[CB]EA} +T_{D[AB]EC}+2T_{[AD]BEC}+2T_{A[BD]EC}\\&+\, T_{E[CA]BD}+ T_{E[CB]DA}+ T_{E[DA]BC}+ T_{E[DB]CA} +T_{E[AB]CD}+2T_{[AE]BCD}+2T_{A[BE]CD} \end{aligned}$$

Now we apply Lemma 23 to the terms \(T_{C(DE)AB}+T_{D(EC)AB}+T_{E(CD)AB}\) in this equation to get

$$\begin{aligned} 0 =&\, 3T_{CDEAB} +3T_{C[ED]AB} +T_{D[EC]AB} +T_{E[DC]AB} +2T_{[EC]DAB} +2T_{[DC]EAB}\\&+\, T_{C[DA]BE}+ T_{C[DB]EA}+ T_{C[EA]BD}+ T_{C[EB]DA} +T_{C[AB]DE}+2T_{[AC]BDE}+2T_{A[BC]DE}\\&+\, T_{D[EA]BC}+ T_{D[EB]CA}+ T_{D[CA]BE}+ T_{D[CB]EA} +T_{D[AB]EC}+2T_{[AD]BEC}+2T_{A[BD]EC}\\&+ \,T_{E[CA]BD}+ T_{E[CB]DA}+ T_{E[DA]BC}+ T_{E[DB]CA} +T_{E[AB]CD}+2T_{[AE]BCD}+2T_{A[BE]CD}, \end{aligned}$$

which implies the formula in the lemma. \(\square \)

By applying this lemma to \(T_{CDEAB}= {\mathbb {D}}_C {\mathbb {D}}_D{\mathbb {D}}_E K_{AB}\in \Gamma ( \otimes ^2{\mathcal {T}}^*\otimes {\mathcal {T}}_{(2,1)})\) for \(K_{AB}\in \Gamma ({\mathcal {T}}_{(2)})\) and by replacing skew-symmetrisations by curvature, for example,

$$\begin{aligned} T_{[EC]DAB} = \tfrac{1}{2}\left( {\mathbb {D}}_E{\mathbb {D}}_C{\mathbb {D}}_DK_{AB}-{\mathbb {D}}_C{\mathbb {D}}_E{\mathbb {D}}_DK_{AB}\right) = \tfrac{1}{2}W_{EC}\sharp {\mathbb {D}}_DK_{AB} \end{aligned}$$

and

$$\begin{aligned} T_{A[BC]DE} =\tfrac{1}{2}\left( {\mathbb {D}}_A{\mathbb {D}}_B{\mathbb {D}}_CK_{DE}- {\mathbb {D}}_A{\mathbb {D}}_C{\mathbb {D}}_BK_{DE}\right) = \tfrac{1}{2} {\mathbb {D}}_A (W_{BC}\sharp K_{DE}), \end{aligned}$$

we obtain the following result. Here and henceforth we use the following convention: the notation |B| or \(| A\cdots B|\) means that the index B, or the indices \(A\cdots B\), are excluded from any surrounding symmetrisation.

Proposition 25

Let \({\mathbb {D}}\) be the Thomas \({\mathbb {D}}\)-operator for a projective structure with curvature \(W_{AB}{}^C{}_D\) and let P be the map from \({\mathcal {T}}^*\otimes {\mathcal {T}}_{(2,1)}\) to \({\mathcal {T}}_{(2,2)}\) defined in Eq. (28). Then, \(K\in {\mathcal {T}}_{(2)}\) satisfies \({\mathbb {D}}_{(A}K_{BC)}=0\), i.e. \({\mathbb {D}}_{A}K_{BC}\in {\mathcal {T}}_{(2,1)}\), if and only if,

$$\begin{aligned} {\begin{array}{rcl} {\mathbb {D}}_C(P{\mathbb {D}}^2K)_{DEAB} &{}=&{} \tfrac{1}{2}W_{C(D}\sharp {\mathbb {D}}_{E)}K_{AB} -\tfrac{3}{4} W_{A(C}\sharp {\mathbb {D}}_{|B|}K_{DE)} -\tfrac{3}{4} {\mathbb {D}}_A (W_{B(C}\sharp K_{DE)})\\ &{}&{} -\,\tfrac{1}{8} {\mathbb {D}}_{C}\left( W_{AB}\sharp K_{DE} - W_{E(A}\sharp K_{B)D}-W_{D(A}\sharp K_{B)E}\right) \\ &{}&{}-\,\tfrac{1}{8}{\mathbb {D}}_D \left( W_{AB}\sharp K_{EC} + W_{EC}\sharp K_{AB} + 2 W_{E(A}\sharp K_{B)C}+ 2 W_{C(A}\sharp K_{B)E}\right) \\ &{}&{}-\,\tfrac{1}{8}{\mathbb {D}}_E \left( W_{AB}\sharp K_{DC} +W_{DC}\sharp K_{AB} + 2 W_{C(A}\sharp K_{B)D}+ 2 W_{D(A}\sharp K_{B)C}\right) . \end{array}}\end{aligned}$$
(69)

Proof

First assume that Eq. (69) holds. We contract this equation with \(X^A\) and \(X^B\). It is a direct computation to see that the right-hand side is zero: to see this, recall that \(X^AW_{A\cdots }=0\) and \(X^AK_{AC}=0\) and that Eq. (58) applied to \(V_{C\cdots }\) with \(X^CV_{C\cdots }=0\) gives

$$\begin{aligned} X^C{\mathbb {D}}_A V_{CB\cdots } =\left[ X^C,{\mathbb {D}}_A\right] V_{CB\cdots }+ {\mathbb {D}}_A(X^CV_{CB\cdots }) =- V_{AB\cdots }. \end{aligned}$$
(70)

Then, from the obtained \(X^AX^B{\mathbb {D}}_C(P{\mathbb {D}}^2K)_{DEAB}=0\) and from Lemmas 19 and 20 we obtain the required symmetry of \({\mathbb {D}}_CK_{ED}\).

For the other direction we apply Lemma 24 to \(T_{CDEAB}= {\mathbb {D}}_C {\mathbb {D}}_D{\mathbb {D}}_E K_{AB}\in \otimes ^2{\mathcal {T}}^*\otimes {\mathcal {T}}_{(2,1)}\). Equation in Lemma 24 then becomes

$$\begin{aligned} -3{\mathbb {D}}_C{\mathbb {D}}_D{\mathbb {D}}_EK_{AB}= & {} -\,2 W_{C(E}\sharp {\mathbb {D}}_{D)}K_{AB} + 3 W_{A(C}\sharp {\mathbb {D}}_{|B|}K_{DE)} +3{\mathbb {D}}_A (W_{B(C}\sharp K_{DE)})\\&+\, \tfrac{1}{2}{\mathbb {D}}_{C}\left( 3 W_{ED}\sharp K_{AB}+W_{AB}\sharp K_{DE} - 2 W_{A(E}\sharp K_{D)B}-2W_{B(D}\sharp K_{E)A}\right) \\&+\,\tfrac{1}{2}{\mathbb {D}}_D \left( W_{AB}\sharp K_{EC} + W_{EC}\sharp K_{AB} + 2 W_{E(A}\sharp K_{B)C}+ 2 W_{C(A}\sharp K_{B)E}\right) \\&+\,\tfrac{1}{2}{\mathbb {D}}_E \left( W_{AB}\sharp K_{DC} +W_{DC}\sharp K_{AB} + 2 W_{C(A}\sharp K_{B)D}+ 2 W_{D(A}\sharp K_{B)C}\right) . \end{aligned}$$

Now we plug this in for the term \({\mathbb {D}}_C{\mathbb {D}}_D{\mathbb {D}}_EK_{AB}\) in Eq. (69) that was obtained by differentiating the equality in Proposition 22:

$$\begin{aligned} 4{\mathbb {D}}_C(P{\mathbb {D}}^2K)_{BCDE}= & {} 3 {\mathbb {D}}_C{\mathbb {D}}_D{\mathbb {D}}_EK_{AB} -\tfrac{3}{2}{\mathbb {D}}_C\left( W_{DE}\sharp K_{AB} + W_{A(D}\sharp K_{E)B} + W_{B(D}\sharp K_{E)A}\right) \\= & {} 2 W_{C(E}\sharp {\mathbb {D}}_{D)}K_{AB} - 3 W_{A(C}\sharp {\mathbb {D}}_{|B|}K_{DE)} -3{\mathbb {D}}_A (W_{B(C}\sharp K_{DE)})\\&-\, \tfrac{1}{2}{\mathbb {D}}_{C}\left( W_{AB}\sharp K_{DE} + W_{A(E}\sharp K_{D)B}+ W_{B(D}\sharp K_{E)A}\right) \\&-\,\tfrac{1}{2}{\mathbb {D}}_D \left( W_{AB}\sharp K_{EC} + W_{EC}\sharp K_{AB} + 2 W_{E(A}\sharp K_{B)C}+ 2 W_{C(A}\sharp K_{B)E}\right) \\&-\,\tfrac{1}{2}{\mathbb {D}}_E \left( W_{AB}\sharp K_{DC} +W_{DC}\sharp K_{AB} + 2 W_{C(A}\sharp K_{B)D}+ 2 W_{D(A}\sharp K_{B)C}\right) . \end{aligned}$$

This finishes the proof. \(\square \)

Now we are going to expand the terms in Eq. (69) using the Leibniz rule

$$\begin{aligned} {\mathbb {D}}_A (W_{BC}\sharp K_{DE}) = ({\mathbb {D}}_A W_{BC})\sharp K_{DE} +W_{BC}\sharp ({\mathbb {D}}_A K_{DE)})+ W_{BC}{}^{H}{}_{A} {\mathbb {D}}_{H}K_{DE}, \end{aligned}$$
(71)

and then substituting \(K_{DE}\) and \({\mathbb {D}}_AK_{DE}\) terms by contractions of \(X^F\) with \(L_{FADE}=(P{\mathbb {D}}^2K)_{FADE}\) using relations Eq. (67) and Eq. (66):

$$\begin{aligned} K_{DE}=\tfrac{2}{3}X^FX^GL_{FGDE},\qquad {\mathbb {D}}_AK_{DE}=\tfrac{4}{3}X^FL_{FADE}. \end{aligned}$$

To this end, first one checks that \(X^FW_{FBCD}=0\) and \({\mathbb {D}}_AX^F=\delta _A{}^F\) imply that

$$\begin{aligned} W_{BC}\sharp (X^FQ_{F\cdots })= X^F W_{BC}\sharp Q_{F\cdots } , \end{aligned}$$

and

$$\begin{aligned} {\mathbb {D}}_A W_{BC}\sharp (X^FQ_{F\cdots })= X^F {\mathbb {D}}_AW_{BC}\sharp Q_{F\cdots } -W_{BC}{}^{H}{}_{A}Q_{H\cdots }, \end{aligned}$$

for any tensor \(Q_{F\cdots }\). For \(Q=L\) and \(Q=X^FL_{F\cdots }\) this implies

$$\begin{aligned} W_{BC}\sharp ({\mathbb {D}}_A K_{DE)}) = \tfrac{4}{3}W_{BC}\sharp (X^FL_{FADE}) = \tfrac{4}{3} X^F W_{BC}\sharp L_{FADE} \end{aligned}$$

and

$$\begin{aligned} ({\mathbb {D}}_A W_{BC})\sharp K_{DE}= & {} \tfrac{2}{3} ({\mathbb {D}}_A W_{BC})\sharp (X^FX^GL_{FGDE})\\= & {} \tfrac{2}{3} X^FX^G{\mathbb {D}}_A W_{BC}\sharp L_{FGDE} - \tfrac{4}{3} X^FW_{BC}{}^{H}{}_{A}L_{FHDE} . \end{aligned}$$

Substituting this into Eq. (71), the terms \(W_{BC}{}^{H}{}_{A} {\mathbb {D}}_{H}K_{DE}\) are cancelled and we get

$$\begin{aligned} {\mathbb {D}}_A (W_{BC}\sharp K_{DE}) = \tfrac{2}{3} X^FX^G{\mathbb {D}}_A W_{BC}\sharp L_{FGDE} + \tfrac{4}{3} X^F W_{BC}\sharp L_{FADE}. \end{aligned}$$
(72)

Then, we compute step by step the terms in the right-hand side of Eq. (69):

$$\begin{aligned}&{ W_{C(D}\sharp {\mathbb {D}}_{E)}K_{AB} -\tfrac{1}{4}\left( {\mathbb {D}}_D \left( W_{EC}\sharp K_{AB} \right) +{\mathbb {D}}_E \left( W_{DC}\sharp K_{AB} \right) \right) =}\\&\quad = 2 X^F W_{C(D}\sharp L_{E)FAB} -\tfrac{1}{3}X^FX^G {\mathbb {D}}_{(D}W_{E)C}\sharp L_{FGAB}. \end{aligned}$$

Next we consider the terms that are not evidently symmetric in A and B: using \(L_{A(CDE)}=0\) as well as the second Bianchi identity for the Weyl tensor we compute

$$\begin{aligned}&{ -\tfrac{3}{4}\left( {\mathbb {D}}_A (W_{B(C}\sharp K_{DE)}) + W_{A(C}\sharp {\mathbb {D}}_{|B|}K_{DE)}\right) }\\&\qquad \quad {-\,\tfrac{1}{8}\left( {\mathbb {D}}_{C}\left( W_{AB}\sharp K_{DE}\right) + {\mathbb {D}}_{D}\left( W_{AB}\sharp K_{EC}\right) + {\mathbb {D}}_{E}\left( W_{AB}\sharp K_{CD}\right) \right) }\\&\qquad = -\,2 X^F W_{(B|(C}\sharp L_{DE)|A)F}-\tfrac{1}{2} X^FX^G{\mathbb {D}}_{(A}W_{B)(C}\sharp L_{DE)FG}\\&\qquad = \tfrac{2}{3} X^F\left( W_{C(A}\sharp L_{B)FED} - 2 W_{(A|(D}\sharp L_{E)C|B)F} \right) \\&\qquad \quad - \,\tfrac{1}{6} X^FX^G \left( {\mathbb {D}}_{(A}W_{B)C}\sharp L_{DEFG} +2 {\mathbb {D}}_{(A}W_{B)(D}\sharp L_{E)CFG}\right) , \end{aligned}$$

and

$$\begin{aligned}&{ \tfrac{1}{2} {\mathbb {D}}_{C}\left( W_{E(A}\sharp K_{B)D}+W_{D(A}\sharp K_{B)E}\right) }\\&\qquad \quad { -{\mathbb {D}}_D \left( W_{E(A}\sharp K_{B)C}+ W_{C(A}\sharp K_{B)E}\right) -{\mathbb {D}}_E \left( W_{C(A}\sharp K_{B)D}+ W_{D(A}\sharp K_{B)C}\right) }\\&\qquad = \tfrac{4}{3}X^F\left( W_{C(A}\sharp L_{B)FED} + 2W_{(D|(A}\sharp L_{B)F|E)C}+ 3W_{(D|(A}\sharp L_{B)|E)FC}\right) \\&\qquad \quad -\tfrac{2}{3}X^FX^G\left( 2{\mathbb {D}}_{(D}W_{E)(A}\sharp L_{B)CFG} + {\mathbb {D}}_{(A}W_{|C(D}\sharp L_{E)|B)FG} + {\mathbb {D}}_{(D}W_{|C(A}\sharp L_{B)|E)FG}\right) . \end{aligned}$$

Now note that because of the pairwise symmetry of L and the skew symmetry of W, we have

$$\begin{aligned} W_{(A|(D}\sharp L_{E)C|B)F}=-W_{(D|(A}\sharp L_{B)F|E)C}. \end{aligned}$$

This allows us to collect some of the terms above as

$$\begin{aligned}&{ \tfrac{2}{3}W_{(D|(A}\sharp L_{B)F|E)C} +W_{(D|(A}\sharp L_{B)|E)FC} - \tfrac{4}{3} W_{(A|(D}\sharp L_{E)C|B)F}}\\&\quad = 2 W_{(D|(A}\sharp L_{B)F|E)C} + W_{(D|(A}\sharp L_{B)|E)FC}\\&\quad = W_{(D|(A}\sharp L_{B)F|E)C} + W_{(A|(D}\sharp L_{E)F|B)C}, \end{aligned}$$

where the last equality follows from \(L_{ECBF}=L_{BFEC}\) and \(L_{B(FEC)}=0\). Hence, we get the following formula for \( {\mathbb {D}}_CL_{DEAB}\) for \(L:=P({\mathbb {D}}^2K)\):

$$\begin{aligned} \begin{array}{rcl} {\mathbb {D}}_CL_{DEAB} &{}=&{} X^F \left( W_{C(D}\sharp L_{E)FAB} + W_{C(A}\sharp L_{B)FED}\right) \\ &{}&{}+\,X^F \left( W_{(D|(A}\sharp L_{B)F|E)C}+ W_{(A|(D}\sharp L_{E)F|B)C} \right) \\ &{}&{} - \,\tfrac{1}{6} X^FX^G \left( {\mathbb {D}}_{(D}W_{E)C}\sharp L_{ABFG}+ {\mathbb {D}}_{(A}W_{B)C}\sharp L_{DEFG}\right) \\ &{}&{} -\,\tfrac{1}{3}X^FX^G\left( {\mathbb {D}}_{(D}W_{E)(A}\sharp L_{B)CFG} + {\mathbb {D}}_{(A}W_{B)(D}\sharp L_{E)CFG}\right) \\ &{}&{} -\,\tfrac{1}{6}X^FX^G\left( {\mathbb {D}}_{(A}W_{|C(D}\sharp L_{E)|B)FG} + {\mathbb {D}}_{(D}W_{|C(A}\sharp L_{B)|E)FG}\right) . \end{array} \end{aligned}$$
(73)

Having this formula, we can formulate the following result:

Theorem 26

Let \((M,{\varvec{p}})\) be an arbitrary projective manifold. Then, the splitting operator \({\mathcal {L}}: S^2T^*M(4)\rightarrow {\mathcal {T}}_{(2,2)}\) gives an isomorphism between weighted Killing tensors of rank 2 and sections \(L_{DEAB}\) of the tractor bundle \({\mathcal {T}}_{(2,2)}\) of weight zero that satisfy Eq. (73).

Proof

Given a rank 2 tensor \(k_{ab}\) we define \(L_{DEAB}={\mathbb {D}}_D{\mathbb {D}}_EK_{AB}\) and \(L_{DEAB}:=(P{\mathbb {D}}^2K)_{DEAB}\). Then, if \(k_{ab} \) is Killing, it follows from Proposition 25 and the above computations that \(L_{DEAB}\) satisfies Eq. (73).

On the other hand, let \(L_{DEAB}\) be a section of \(\mathcal T_{(2,2)}\) of weight zero that satisfies Eq. (73). Contracting Eq. (73) with \(X^D\) and \(X^E\), one can easily check, using the same arguments as before and that \(L_{(DEF)B}=0\), that the right-hand side vanishes and thus

$$\begin{aligned} 0= X^DX^E{\mathbb {D}}_CL_{DEAB} \end{aligned}$$

Then, from Proposition 8 it follows that \(L_{DEAB}\) defines a Killing tensor \(k_{ab}\). Moreover, we see that \(L_{DEAB}={\mathcal {L}}(k)_{DEAB}\) unless the map

$$\begin{aligned} L_{DEAB}\mapsto K_{AB}=X^DX^EL_{DEAB}\in {\mathcal {T}}_{(2)}(2). \end{aligned}$$

has a kernel. So let us assume there is a section \(L_{DEAB} \) of \({\mathcal {T}}_{(2,2)}\) that satisfies Eq. (73) and such that

$$\begin{aligned} X^DX^EL_{DEAB}=0. \end{aligned}$$
(74)

Applying \({\mathbb {D}}_C\) to this and using \(0= X^DX^E{\mathbb {D}}_CL_{DEAB}\) implies that \(0=X^DL_{DEAB}\). Applying \({\mathbb {D}}_C\) to this gives

$$\begin{aligned} 0=L_{CEAB}+X^D{\mathbb {D}}_CL_{DEAB}= L_{CEAB}. \end{aligned}$$

Here the second equality uses Eq. (73), which allows us to compute

$$\begin{aligned} X^D{\mathbb {D}}_CL_{DEAB}= & {} X^DX^F\left( W_{C(A}\sharp L_{B)FED} +\tfrac{1}{2} W_{E(A}\sharp L_{B)FDC}\right) . \end{aligned}$$

But now \(L_{B(FED)}=0 \) and Eq. (74) imply that

$$\begin{aligned} X^DX^F W_{C(A}\sharp L_{B)FED}=-X^DX^F W_{C(A}\sharp L_{B)DEF}=0, \end{aligned}$$

which proves that \(X^D{\mathbb {D}}_CL_{DEAB}=0\) and finishes the proof. \(\square \)

Note that the right-hand side of Eq. (73) indeed defines a section \({\mathcal {R}}_C\sharp \) of \({\mathcal {T}}^*\otimes {\mathcal {T}}_{(2,2)}\) as claimed in the proof of Theorem 15.

In order to extract a covariant derivative from this, we have to contract it with \(W^{C}{}_{c}\). In general this contraction is not projectively invariant. However, since \(L_{DEAB}\) has weight zero, applying \({\mathbb {D}}_C\) to it and contracting with \(X^C\) gives zero, \(X^C {\mathbb {D}}_C L_{DEAB}=0\). Hence, the contraction \(W^{C}{}_{c} {\mathbb {D}}_C L_{DEAB}\) is also projectively invariant for sections \(L_{DEAB}\) that satisfy Eq. (73). However, we need that the curvature term in right-hand side of Eq. (73) is projectively invariant as claimed in the proof of Theorem 15, i.e. that the right-hand side of Eq. (73) is projectively invariant for any\(L_{DEAB}\in {\mathcal {T}}_{(2,2)}\) not only for solutions of Eq. (73). This is the statement of the following lemma.

Lemma 27

For any \(L_{ABDE}\in {\mathcal {T}}_{(2,2)}\) the right-hand side in Eq. (73) gives zero when contracted with \(X^C\). In particular, the section of \({\mathcal {T}}^*\otimes \mathrm {End}({\mathcal {T}}_{(2,2)})\) defined by the right-hand side in Eq. (73) is projectively invariant.

Proof

Clearly both of the terms of the form \(X^CW_{C(D}\sharp L_{E)FAB}\) in the first line of Eq. (73) vanish separately because \(X^CW_{C ABC}=0\). Also both terms of the form \(X^C X^F X^G {\mathbb {D}}_{(D}W_{E)(A}\sharp L_{B)CFG}\) in the fourth line of Eq. (73) vanish separately because \(L_{B(CFG)}=0\). Similarly both terms of the form \(X^CX^FX^G {\mathbb {D}}_{(D}W_{E)C}\sharp L_{ABFG}\) in the third line of Eq. (73) vanish separately because \(X^CW_{C ABC}=0\) and

$$\begin{aligned} X^C {\mathbb {D}}_{(D}W_{E)C}\sharp L_{ABFG} = - \delta ^C {}_{(D} W_{E)C}\sharp L_{ABFG} = W_{(EC)}\sharp L_{ABFG} =0. \end{aligned}$$

All the other terms in the second and fifth line of Eq. (73) do not vanish separately but cancel against each other when contracted with \(X^C\). In fact we have

$$\begin{aligned} X^C\left( {\mathbb {D}}_{(A}W_{|C(D}\sharp L_{E)|B)FG} + {\mathbb {D}}_{(D}W_{|C(A}\sharp L_{B)|E)FG}\right) =- W_{(A|(D}\sharp L_{E)|B)FG}-W_{(D|(A}\sharp L_{B)|E)FG}=0, \end{aligned}$$

and for the terms in the second line

$$\begin{aligned} X^C X^F \left( W_{(D|(A}\sharp L_{B)F|E)C}+ W_{(A|(D}\sharp L_{E)F|B)C} \right) =0, \end{aligned}$$

because of the skew symmetry of \(W_{DA}\). \(\square \)

In order to obtain from Eq. (73) an equation involving the tractor derivative \(\nabla _c\), we have to contract it with \(W^{C}{}_{c}\). First we look at terms that for which the contracted index C appears on the curvature (or its derivative) \(W_{AC}\). These will turn out to be manifestly invariant as we can eliminate \(W^{C}{}_{c}\): first we observe that

$$\begin{aligned} W^{C}{}_{c} W_{CA}\sharp L_{BFED} = Z_{A}{}^{a}\kappa _{ca}\sharp L_{BFED}, \end{aligned}$$

where \(\kappa _{ca}{}^H{}_G\) is the tractor curvature defined in Eq. (13). Hence, for the terms in the first line in Eq. (73) we get

$$\begin{aligned} X^F \left( W_{C(D}\sharp L_{E)FAB} + W_{C(A}\sharp L_{B)FED}\right) = X^F\left( Z_{(A}{}^{a}\kappa _{|ca|}\sharp L_{B)FED} + Z_{(D}{}^{a}\kappa _{|ca|}\sharp L_{E)FAB} \right) , \end{aligned}$$

which is manifestly invariant. Next we compute, using formulae Eq. (12) and that the weight of \(W_{CD}{}^{H}{}_{F}\) is \(-2\), that

$$\begin{aligned} W^{C}{}_{c}{\mathbb {D}}_AW_{BC}= & {} -\,2 Y_AZ_{B}{}^{b}\kappa _{bc} + Z_{A}{}^{a} W^{C}{}_{c}\nabla _aW_{BC}\\= & {} -\,2 Y_AZ_{B}{}^{b}\kappa _{bc} + Z_{A}{}^{a}\left( \nabla _a (W^{C}{}_{c}W_{BC})-\nabla _aW^{C}{}_{c} W_{BC}\right) \\= & {} -\,2 Y_AZ_{B}{}^{b}\kappa _{bc} + Z_{A}{}^{a}\nabla _a (Z_{B}{}^{b}\kappa _{bc})\\= & {} -\,(2 Y_AZ_{B}{}^{b}+Y_B Z_{A}{}^{b})\kappa _{bc} + Z_{A}{}^{a}Z_{B}{}^{b}\nabla _a \kappa _{bc}, \end{aligned}$$

because \(\nabla _aW^{C}{}_{c} W_{BC} =-P_{ac}X^CW_{BC} =0\) and \(\nabla _a Z_{B}{}^{b}=-\delta _{a}^bY_B\). Hence, for the expressions in the third line of Eq. (73) we get,

$$\begin{aligned} X^FX^G {\mathbb {D}}_{(A}W_{B)C}\sharp L_{DEFG} = X^FX^G \left( -3Y_{(A}Z_{B)}{}^{b}\kappa _{bc}+ Z_{(A}{}^{a}Z_{B)}{}^{b}\nabla _a \kappa _{bc}\right) \sharp L_{DEFG} \end{aligned}$$

and

$$\begin{aligned} X^FX^G {\mathbb {D}}_{(D}W_{E)C}\sharp L_{ABFG} = X^FX^G \left( -3Y_{(D}Z_{E)}{}^{b}\kappa _{bc} + Z_{(D}{}^{a}Z_{E)}{}^{b}\nabla _{a} \kappa _{bc}\right) \sharp L_{ABFG}. \end{aligned}$$

Similarly we get for the expressions in the fifth line of Eq. (73),

$$\begin{aligned}&{\left( {\mathbb {D}}_{(A}W_{|C(D}\sharp L_{E)|B)FG} + {\mathbb {D}}_{(D}W_{|C(A}\sharp L_{B)|E)FG}\right) }\\&\quad = -\,\tfrac{1}{2}\left( {\mathbb {D}}_{(A}W_{D)C}\sharp L_{BEFG} + {\mathbb {D}}_{(A}W_{E)C}\sharp L_{DBFG} + {\mathbb {D}}_{(B}W_{D)C}\sharp L_{AEFG} + {\mathbb {D}}_{(B}W_{E)C}\sharp L_{DAFG} \right) \\&\quad = \tfrac{1}{2}\left( 3Y_{(A}Z_{D)}{}^{b}\kappa _{bc}- Z_{(A}{}^{a}Z_{D)}{}^{b}\nabla _a \kappa _{bc}\right) \sharp L_{BEFG}\\&\qquad +\,\tfrac{1}{2}\left( 3Y_{(A}Z_{E)}{}^{b}\kappa _{bc}- Z_{(A}{}^{a}Z_{E)}{}^{b}\nabla _a \kappa _{bc}\right) \sharp L_{BDFG}\\&\qquad +\,\tfrac{1}{2}\left( 3Y_{(B}Z_{D)}{}^{b}\kappa _{bc}- Z_{(B}{}^{a}Z_{D)}{}^{b}\nabla _a \kappa _{bc}\right) \sharp L_{AEFG}\\&\qquad +\,\tfrac{1}{2}\left( 3Y_{(B}Z_{E)}{}^{b}\kappa _{bc}- Z_{(B}{}^{a}Z_{E)}{}^{b}\nabla _a \kappa _{bc}\right) \sharp L_{ADFG}. \end{aligned}$$

Finally, we compute

$$\begin{aligned} W_{(D|(A}\sharp L_{B)F|E)C}+ W_{(A|(D}\sharp L_{E)F|B)C} = Z_{(A}{}^{a}Z_{|(D}{}^{d}\kappa _{|ad|}\sharp \left( L_{E)F|B)C} - L_{E)C|B)F} \right) \end{aligned}$$

and

$$\begin{aligned} {\mathbb {D}}_{(A}W_{B)(D}\sharp L_{E)CFG} = -\left( 3 Y_{(A}Z_{B)}{}^{b} Z_{(D}{}^{d}\kappa _{|bd|} - Z_{(A}{}^{a}Z_{B)}{}^{b} Z_{(D}{}^{d}\nabla _{|a}\kappa _{bd|} \right) \sharp L_{E)CFG}, \end{aligned}$$

to rewrite Eq. (73) in terms of the tractor connection as

$$\begin{aligned} \begin{array}{rcl} \nabla _cL_{DEAB} &{}=&{} X^F\left( Z_{(A}{}^{a}\kappa _{|ca|}\sharp L_{B)FED} + Z_{(D}{}^{a}\kappa _{|ca|}\sharp L_{E)FAB} \right) \\ &{}&{} +\, X^F W^{C}{}_{c} Z_{(A}{}^{a}Z_{|(D}{}^{d}\kappa _{|ad|}\sharp \left( L_{E)F|B)C} - L_{E)C|B)F} \right) \\ &{}&{} - \,\tfrac{1}{12}X^FX^G \left( 3Y_{(A}Z_{D)}{}^{b}\kappa _{bc}- Z_{(A}{}^{a}Z_{D)}{}^{b}\nabla _a \kappa _{bc}\right) \sharp L_{BEFG}\\ &{}&{} - \,\tfrac{1}{12}X^FX^G \left( 3Y_{(A}Z_{E)}{}^{b}\kappa _{bc}- Z_{(A}{}^{a}Z_{E)}{}^{b}\nabla _a \kappa _{bc}\right) \sharp L_{BDFG}\\ &{}&{} - \,\tfrac{1}{12}X^FX^G \left( 3Y_{(B}Z_{D)}{}^{b}\kappa _{bc}- Z_{(B}{}^{a}Z_{D)}{}^{b}\nabla _a \kappa _{bc}\right) \sharp L_{AEFG}\\ &{}&{} - \,\tfrac{1}{12}X^FX^G \left( 3Y_{(B}Z_{E)}{}^{b}\kappa _{bc}- Z_{(B}{}^{a}Z_{E)}{}^{b}\nabla _a \kappa _{bc}\right) \sharp L_{ADFG}\\ &{}&{} +\, \tfrac{1}{6} X^FX^G \left( 3Y_{(A}Z_{B)}{}^{b}\kappa _{bc}- Z_{(A}{}^{a}Z_{B)}{}^{b}\nabla _a \kappa _{bc}\right) \sharp L_{DEFG}\\ &{}&{}+\, \tfrac{1}{6} X^FX^G \left( 3Y_{(D}Z_{E)}{}^{b}\kappa _{bc} - Z_{(D}{}^{a}Z_{E)}{}^{b}\nabla _{a} \kappa _{bc}\right) \sharp L_{ABFG}\\ &{}&{} -\,\tfrac{1}{3} X^FX^G W^{C}{}_{c}\left( 3 Y_{(A}Z_{B)}{}^{b} Z_{(D}{}^{d}\kappa _{|bd|} - Z_{(A}{}^{a}Z_{B)}{}^{b} Z_{(D}{}^{d}\nabla _{|a}\kappa _{bd|} \right) \sharp L_{E)CFG}\\ &{}&{} -\,\tfrac{1}{3}X^FX^GW^{C}{}_{c}\left( 3 Y_{(D}Z_{E)}{}^{b} Z_{(A}{}^{d}\kappa _{|bd|} - Z_{(D}{}^{a}Z_{E)}{}^{b} Z_{(A}{}^{d}\nabla _{|a}\kappa _{bd|} \right) \sharp L_{B)CFG}, \end{array} \end{aligned}$$
(75)

where \(\nabla _c\) is the projective tractor connection and \(\kappa _{bc}\) its curvature. The right-hand side of this equation defines the section \({\mathcal {Q}}_a\sharp \in \Gamma (T^*M\otimes {\mathcal {T}}_{(2,2)})\) in Corollary 17. Hence we arrive at:

Theorem 28

Let \((M,{\varvec{p}})\) be an arbitrary projective manifold. Then, the splitting operator \({\mathcal {L}}: S^2T^*M(4)\rightarrow {\mathcal {T}}_{(2,2)}\) gives an isomorphism between weighted Killing tensors of rank 2 and sections \(L_{DEAB}\) of the tractor bundle \({\mathcal {T}}_{(2,2)}\) of weight zero that satisfy Eq. (75) for the projective tractor connection \(\nabla _a\), or equivalently, parallel sections of the connection \(\nabla _a-{\mathcal {Q}}_a\sharp \). Moreover, the right-hand side of Eq. (75) is projectively invariant.

Proof

The proof follows immediately from Theorem 26 and Lemma 27 and from the computations above. \(\square \)

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Correspondence to A. Rod Gover.

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ARG gratefully acknowledges support from the Royal Society of New Zealand via Marsden Grant 16-UOA-051. TL was partially supported by the Grant 346300 for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund.

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Gover, A.R., Leistner, T. Invariant prolongation of the Killing tensor equation. Annali di Matematica 198, 307–334 (2019). https://doi.org/10.1007/s10231-018-0775-3

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Keywords

  • Integrability
  • Hidden symmetries
  • Projective differential geometry
  • Riemannian manifolds
  • Affine manifolds

Mathematics Subject Classification

  • Primary 53B10
  • Secondary 53A20