Conformal Patterson-Walker metrics

CONFORMAL PATTERSON–WALKER METRICS

arXiv:1604.08471v2 [math.DG] 11 Jan 2017

ˇ MATTHIAS HAMMERL, KATJA SAGERSCHNIG, JOSEF SILHAN, ARMAN ˇ ˇ ´ TAGHAVI-CHABERT AND VOJTECH ZADNÍK Abstract. The classical Patterson–Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We achieve a complete description of Einstein metrics in the conformal class formed by the Patterson–Walker metric. Finally, we describe all symmetries of the conformal Patterson–Walker metric. In both cases we obtain descriptions in terms of geometric data on the original structure.

1. Introduction Given a torsion-free affine connection D on a smooth n-dimensional manifold M , the classical Patterson–Walker construction [27] yields a splitsignature (n, n) pseudo-Riemannian metric g on the total space of the cotangent bundle T ∗ M . The metric g is determined by the natural pairing of the vertical distribution V of T ∗ M and the horizontal distribution H ∼ = T M on T ∗ M . In particular, V and H (as determined by D) are totally isotropic with respect to g. Such metrics are endowed with a parallel pure spinor and a homothety, and satisfy an integrability condition on the Riemann curvature tensor. We shall show in section 2 that Patterson–Walker metrics are locally characterized by these data. When n = 2, this construction is generalised in [12, 11] where a conformal class of Patterson–Walker metrics is assigned to a projective class of volume-preserving torsion-free affine connections. As we shall see, this extends to any dimension. In order to accommodate projective invariance in this construction, we must replace T ∗ M by the density-valued cotangent bundle T ∗ M (2). Recall that the projective class p containing D is formed by all torsion-free affine connections which share the same geodesics (as unparametrized curves) as D. We shall suppose in addition that D preserves a volume form on M , and as such will be referred to as special. Then special b ∈ p give rise to Patterson–Walker metrics g, gˆ on T ∗ M (2) connections D, D which are conformally related, i.e. gˆ = e2f g for some smooth function f on M . In other words, the projective structure (M, p) induces a split-signature conformal structure (T ∗ M (2), c), see section 3 for details. Notice that certain geometrical data are to be expected on the conforf, c) induced from a projective class (M, p). Firstly, there mal manifold (M Date: January 12, 2017. 2000 Mathematics Subject Classification. 53A20, 53A30, 53B30, 53C07. Key words and phrases. Differential geometry, Parabolic geometry, Projective structure, Conformal structure, Einstein metrics, Conformal Killing field, Twistor spinors. 1

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ˇ ˇ ADN ´ ÍK HAMMERL, SAGERSCHNIG, SILHAN, TAGHAVI-CHABERT, Z

is a distinguished vector field k corresponding to the Euler vector field on T ∗ M (2). Secondly, there is an n-dimensional integrable distribution V on f corresponding to the vertical subbundle of T ∗ M (2). In fact, this distriM bution can be conveniently defined via a distinguished pure spinor field χ annihilating V . Here purity of χ corresponds to V = ker χ being maximally isotropic. Further, one expects an integrability condition imposed on the curvature of metrics in c and this we shall formulate in terms of the (conforfabcd of c. Our characterization result, proved mally invariant) Weyl tensor W in section 4, is then

Theorem 1. A conformal spin structure c of split signature (n, n) on a f is locally induced by an n-dimensional projective structure as manifold M a conformal Patterson–Walker metric if and only if the following properties are satisfied: f, c) admits a pure spinor χ with (maximally isotropic, n-dimensional) (a) (M integrable kernel ker χ satisfying the twistor spinor equation e a χ + 1 γaD / χ = 0, (1) D 2n e c is the Dirac operator and γ denotes the Clifford multiwhere D / = γcD plication. f, c) admits a (light-like) conformal Killing field k with k ∈ ker χ. (b) (M (c) The Lie derivative of χ with respect to the conformal Killing field k is 1 (2) Lk χ = − (n + 1)χ . 2 (d) The following integrability condition is satisfied for all v r , ws ∈ ker χ: fabcd v a wd = 0 . W

(3)

In section 5, we achieve a complete description of Einstein metrics within the induced conformal class in terms of the underlying geometric objects. In what follows, RDAC B is the curvature tensor of a torsion-free affine connection DA and WDA CB is the (projectively invariant) totally trace-free part of f on M f and E A ∼ RDAC B . That is, we use abstract indices Eea ∼ = T M on = TM M . Let us emphasize that the theorems below involve certain projectively invariant differential operators, and to formulate the invariance precisely will require the use of density-valued tensor fields. Leaving these details aside for the time being, the results can be stated as follows: Theorem 2. (a) If the affine connection D is Ricci-flat, then the induced Patterson– Walker metric g is Ricci-flat. (b) If the affine connection D admits an Euler-type vector field ξ satisfying the projectively invariant equation 1 A DP ξ P (4) DC ξ A = δC n and the integrability condition ξ D WDA CB = 0, then the induced Patterson– Walker metric g is conformal to a Ricci-flat metric σξ−2 g off the zero-set of a rescaling function σξ .


3

In fact, any Einstein metric in the conformal class c can be uniquely decomposed into two Einstein metrics of such types. Part (a) is a well-known fact for Patterson–Walker metrics that was already observed in [27, 9], and which we recover. To our knowledge, the construction of Ricci–flat Einstein metrics of part (b) is new, as is the decomposition result for general Einstein metrics. The decomposition of general Einstein metrics in c can be understood explicitly: if the Patterson–Walker metric g is conformal to an Einstein metric σ −2 g, then there is a canonical decomposition σ = σ+ + σ− −2 −2 g are Ricci-flat off the respective g and g− = σ− such that both g+ = σ+ zero-sets of σ± . Further, there is a Ricci-flat affine connection D− projectively related to D, which induces the Ricci-flat Patterson–Walker metric g− , and an Euler-type vector field ξ for D satisfying (4) and the integrability condition ξ D WDA CB = 0 such that g+ = σξ−2 g. Finally, in section 6 we study the Riemannian and conformal symmetries of the induced Patterson–Walker metric and present their complete description in terms of affine and projective properties of D and of p, respectively. f = T ∗ M (2) is Since the construction of the conformal structure c on M natural, symmetries of the projective structure p give rise to conformal symmetries (i.e. conformal Killing fields) of c. In fact, we can completely and explicitly understand the space of conformal Killing fields of c in terms of solutions to projectively invariant equations:

Theorem 3. (a) Any infinitesimal symmetry v A of the projective structure p induces a conformal Killing field ve0a of c. (b) Any skew-symmetric bivector wAB satisfying the projectively invariant equation 2 [A (5) δ DP wB]P DC wAB = − n−1 C D)

and the integrability condition wB(A WB(C E) = 0 induces a conformal a of c. Killing field ve− (c) Any Killing 1-form, i.e. a 1-form αA satisfying D(A αB) = 0, induces a a of c. conformal Killing field ve− In fact, any conformal Killing field of c can be uniquely decomposed as a a + c k a of components which correspond to solutions a +v e0a + ve− direct sum ve+ to the respective projective equations and a constant multiple of k.

Likewise, the construction of the Patterson–Walker metric g from a torsionfree affine connection D is natural, hence any symmetry of D gives a symmetry of g, i.e. a Killing field. In fact, we obtain a complete description of the space of Killing fields of g in terms of affine data: Theorem 4. (a) Any infinitesimal symmetry v A of the affine connection D induces a Killing field ve0a of g.

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(b) Any parallel bivector wAB for the affine connection D, DC wAB = 0, D) which satisfies the integrability condition wB(A RB(C E) = 0 induces a a of g. Killing field ve+ a of g. (c) Any Killing 1-form αA , D(A αB) = 0, induces a Killing field ve− In fact, any Killing field of g can be uniquely decomposed as a direct sum a of components which correspond to solutions to the respective a +v e0a + ve− ve+ affine equations.

The approach of the present paper is based on an extension of the twospinor calculus of [29] to higher dimensions, already used in [22], and developed more fully in [31, 32]. We shall set up this spinor calculus in section 3 and employ it to directly derive relationships between the original projective geometry and the induced conformal structure. A major step, which is particularly tailored for this approach, is our parallelizability result for pure twistor spinors with integrable distributions, Proposition 4.2, upon which Theorem 1 hinges. Projective and conformal geometries are instances of Cartan geometries, or more specifically, parabolic geometries. The geometric relationship studied in this article fits into the larger framework of so called Fefferman-type constructions for this class of structures. This is worked out in [20], which includes a characterisation result closely related to Theorem 1. The relation with the present treatment is briefly described in section 7.4. The present spinor-theoretic approach allows a succinct treatment, gives a shorter statement for the characterization of the induced structures than the one presented in [20], and allows us to give explicit descriptions of the Einstein metrics in the induced conformal class of metrics. Acknowledgments. The authors express special thanks to Maciej Dunajski for motivating the study of this construction and for a number of enlightening discussions on this and adjacent topics. KS thanks the INdAM (Istituto Nazionale di Alta Matematica) via project FIR 2013 - Geometria delle equazioni differenziali. She was supported by grant J3071-N13 of the ˇ was supported by the Czech science founAustrian Science Fund (FWF). JS ˇ ˇ dation (GACR) under grant P201/12/G028. AT-C was funded by GACR post-doctoral grant GP14-27885P and thanks the University of Turin for ˇ was supported by GACR ˇ grant GA201/08/0397. support. VZ 2. Patterson–Walker metrics Riemann extensions of affine connected spaces were first described in [27]. They are pseudo-Riemannian metrics on the total space of the cotangent bundle π : T ∗ M → M associated to torsion-free affine connections on M as follows: An affine connection D determines a horizontal distribution H ⊂ T (T ∗ M ) complementary to the vertical distribution V of the bundle projection π. Via the tangent map of π, the bundle H is isomorphic to T M , whilst V is canonically isomorphic to T ∗ M . Definition 2.1. The Riemann extension or the Patterson–Walker metric associated to a torsion-free affine connection D on M is the split-signature f := T ∗ M fully determined by the following conditions: metric g on M


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(a) both V and H are isotropic with respect to g, (b) the value of g with one entry from V and another entry from H is given by the natural pairing between V ∼ = T ∗ M and H ∼ = T M. It follows that V is parallel with respect to the Levi-Civita connection of the constructed metric. Hence Riemann extensions are special cases of pseudo-Riemannian manifolds admitting a parallel isotropic distribution known as Walker manifolds or Walker metrics. We can give local coordinate expression for these Riemann extensions. Let us introduce local coordinates {xA } on M and fibre coordinates {pA } so f. Here, indices run from 1 to that θ = pA dxA is the tautological 1-form on M n, but we shall view them as abstract indices. Let further ΓAC B = Γ(AC B) be the Christoffel symbols of a torsion-free affine connection DA on M . The horizontal distribution H associated to the affine connection DA is spanned by ∂ ∂ + ΓAC B pC . A ∂x ∂pB

(6)

Defining α ⊙ β := 12 (α ⊗ β + β ⊗ α) for any 1-forms α and β, we can write the Patterson–Walker metric explicitly as g = 2 dxA ⊙ dpA − 2 ΓAC B pC dxA ⊙ dxB , (7) D E from which it is clear that both V = ∂p∂B and H spanned by (6) are indeed isotropic with respect to (7). Since V and H are totally isotropic and dual to each other via the metric, we can associate to them a pair of pure spinors defined up to scale. These f to V and H. With spinors will allow us to construct projections from T M a slight abuse of notation to be clarified subsequently, it will be convenient to employ abstract index notation on spinor fields (see [28]): sections of the irreducible spinor bundles Se+ and Se− will be adorned with primed and ′ unprimed upper-case Roman indices, i.e. αA ∈ Γ(Se+ ) and β A ∈ Γ(Se− ), ∗ ) and λ ∈ Γ(S ∗ ). In e− and similarly for dual spinor bundles, κA′ ∈ Γ(Se+ A f, g) is generated by the γ-matrices particular, the Clifford algebra of (T M ′ γa B A and γa BA′ , which satisfy A γ(a A C γb) CB ′ = −gab δeB ′ , ′

B γ(a AC ′ γb) C B = −gab δeA ,

′

′

A′ and δ eA are the identity elements on Se+ and Se− respectively. where δeB ′ B ′ Let χA ∈ Γ(Se+ ) be a spinor field annihilating V , and define a linear map B′ A f) → Γ(Se− ) . : Γ(T M χA a := γa B ′ χ

∗ ) be a spinor A′ is pure. Similarly, let η ˇA′ ∈ Γ(Se+ Then V = ker χA a since χ ′ A field annihilating H so that χ and ηˇA′ are dual, and chosen such that ′ ηˇA′ χA = − 21 . Defining

f) → Γ(Se∗ ) , ηˇaA := ηˇB ′ γa B A : Γ(T M − ′

we then have H = ker ηˇaA since ηˇA′ is pure.

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Therefore, we can identify H with the image of χA a , and V with the image of ηˇaA . In this situation the upper case Roman index refers to an ndimensional representation. Viewed as projections, the spinors satisfy [31] aB χA = 0, aχ

a ηˇA ηˇaB = 0 ,

a B χA ˇB = δA , aη

(8)

B is the identity on im χA . In sum, we have a splitting where δA a A ∼ f=V ⊕H ∼ TM ˇaA = im ηˇaA ⊕ im χA a = ker χa ⊕ ker η

where H ∼ ea ∈ Γ(H), we can write = V ∗ , and for any vea ∈ Γ(V ), w vea = α eA χaA ,

a w ea = βeA ηˇA ,

for some α eA ∈ im ηˇaA ,

for some βeA ∈ im χaA .

There is the freedom in rescaling both χA and ηˇA′ such that χA ηˇA′ = − 21 , which will be fixed by the following consideration. If the torsion-free affine connection D preserves in addition a volume form on M , then the connection D is said to be special, and all our affine connections will have this property. This means that we can always choose our coordinates {xA } such that the preserved volume form is given by dx1 ∧ . . . ∧ dxn , up to constant multiple, and thus, the Christoffel symbols satisfy ΓAC C = 0. Henceforth, we denote e a the Levi-Civita connection of the Patterson–Walker metric (7) on by D f M induced E by a special torsion-free affine connection ′ DA on M . Since D ∂ V = ∂pB and H is spanned by (6), we can choose χA and ηˇA′ such that ea = χaA D

∂ , ∂pA

′

′

∂ ∂ a e ηˇA + ΓAC B pC , Da = A ∂x ∂pB

(9)

and the non-trivial commutation relations b e a e b e e a , ηˇB ec , ec , [χaA D Db ] = ΓB AC χcC D [ˇ ηA Da , ηˇB Db ] = RAB C D pC χcD D

are satisfied. Here we use the convention RAB C D v D = 2 D[A DB] v C for the curvature tensor RBC DA of DA . We can immediately see that H is integrable e c of the if and only if DA is flat. We then obtain the Christoffel symbols Γ a b ea connection D D B A B C e = 2 χA Γ ˇ[bB χC aη abc c] ΓA C + χa χb χc RBC A pD .

In particular, using (8) and the fact ΓAB C is trace-free, we immediately see ′ a + that the spinor χA determined by (9) is parallel. Writing vea = veA ηˇA aA α eA χ , we have ∂ ∂ C a B b e + ΓA D p C veB + ΓAB C e vC , Da ve ηˇA χb = ∂xA ∂pD ∂ ∂ C b a e ae + ΓA D p C α e B − ΓAC B α eC − veC RCADB pD , D v ηˇA ηˇbB = ∂xA ∂pD e a veb χaA ηˇbB = ∂ α D eB , ∂pA e a veb χaA χB = ∂ veB . D b ∂pA


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In particular, if veB = v B (x) and α eB = αB (x) do not depend on pA , then bB b e a veb = DA v B χA . (10) D ˇB + DA αB − v C RCADB pD χA aχ aη Next, the Riemann tensor can be computed to be A B D A B D C e R = 2 χ χ η ˇ χ + χ χ η ˇ χ a b [cC d] c d [aC b] RAB D abcd

E B C D + 2 χA [a χb] χc χd DA RCD B pE , (11)

from which we deduce that eabcd v a wd = 0 R

for all v a , wa ∈ Γ(V ).

We have a distinguished vector field k and a 2-form µ, defined by ∂ , k := 2 pA ∂pA µ := 2 dpA ∧ dxA .

(12)

(13)

As a 1-form, ka is twice the tautological one-form θa on T ∗ M . As a skewsymmetric endomorphism, µab acts as the identity on H and as minus the identity on V : b a = ηˇB , µab ηˇB

µab χbB = −χaB .

(14)

It is then straightforward to check that k satisfies the conformal Killing field equation e a kb − µab − gab = 0, D (15)

and in particular ka is a light-like vertical homothety, Lk g = 2 g. Now, Patterson–Walker metrics can be locally characterized as follows:

f, g) be a spin structure of split signature (n, n) Proposition 2.2. Let (M admitting a parallel pure spinor χ with integrable associated distribution V , and a homothety k ∈ Γ(V ) such that (15). Suppose further that the Riemann tensor satisfies (12). f, there exist coordinates (xA , pA ) Then, in a neighborhood of any point of M such that the metric g takes the form (7) where ΓAC B are the Christoffel symbols for a special torsion-free affine connection D on the leaf space of V . f, g) is the Riemannian extension associated to D. In particular, (M f, there exist coordinates {xA , pA } Proof. In a neighborhood of any point of M such that the metric takes the form [4, 23] g = 2 dpA ⊙ dxA − 2 ΘAB dxB ⊙ dxA ,

(16)

where the distribution V is spanned by the vector fields ∂p∂A and {xA } are coordinates on the leaf space M , and the functions ΘAB = Θ(AB) (x, p) satisfy the differential conditions ∂ Θ = 0. (17) ∂pB BA Since k is a homothety tangent to V , we can write ∂ , g(k, −) = kA dxA , k = kA ∂pA

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for some functions kA . The exterior derivative of this 1-form is given by µ=

∂ ∂ kB dxA ∧ dxB + kB dpA ∧ dxB . A ∂x ∂pA

This gives µ

∂ ,− ∂pA

=

1 ∂ kB dxB . 2 ∂pA

Since χ is parallel, differentiating ka χA a = 0 yields µab γ b χ + γa χ = 0, f, acts according to (15). This means that µ, as an endomorphism of T M 1 ∂ B , i.e. k = 2 p + φ for by minus the identity on V . Hence 2 ∂pA kB = δA B B B A some functions φB of x . We can perform a change of the coordinates pA to eliminate the functions φB in kB while preserving the form of the metric. At this stage, we have the following local coordinate forms for the homothety ka , its associated 1-form ka , and its exterior derivative µab : ∂ , ∂pA µ = 2 dpA ∧ dxA . k = 2 pA

g(k, −) = 2 pA dxA

Now, k is a homothety satisfying Lk gab = 2 gab , and the equivalent condition on ΘAB is pC

∂ ΘAB = ΘAB . ∂pC

(18)

This says that ΘAB is homogeneous of degree 1 in pA . On the other hand, the curvature condition (12) is equivalent to ∂2 Θ = 0, ∂pB ∂pD AC

(19)

which tells us that ΘAC is linear in pA . Putting things together we see that, given the metric (16), the conditions (17), (18) and (19) are satisfied if and only if ΘAB takes the form ΘAB = ΓAC B pC ,

(20)

for some ΓAC B = Γ(AC B) (x), which is moreover trace-free by virtue of (17). The condition (12) is the obstruction for the Levi-Civita connection to descend to an affine connection on M , cf. [1, 9]. We have therefore recovered the Patterson–Walker metric (7), and ΓAC B can be identified with the Christoffel symbols of a special affine connection D on the leaf space of V. 3. Conformal Patterson–Walker metrics We now deal with a projective-to-conformal analog of the construction from previous section.


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3.1. Calculus for projective geometry. As before, we shall use upper case Roman abstract indices as in [28] for tensors on M . For instance, αA ∈ EA denotes a 1-form on M , v AB ∈ E [AB] denotes a bivector on M . This convention should not be confused with unprimed spinor indices. By and large, we follow the treatment given in [15, 14, 2]. b A are in a given projective Two torsion-free affine connections DA and D A A class p if and only if for any ξ ∈ E , b A ξ B = DA ξ B + Q B ξ C , D Q C = 2 δC ΥB) , (21) AC

AB

(A

for some 1-form ΥA . Similar formulae can be obtained on 1-forms and tensors by means of the Leibniz rule. We shall assume M to be oriented. Let us fix a volume form εA1 ...An ∈ b A in p, E[A1 ...An ] . Then, by (21), for any two affine connections DA and D we have b A εB ...Bn = DA εB ...Bn − (n + 1) ΥA εB ...Bn , D (22) 1

1

1

for some 1-form ΥA . We can always choose εA1 ...An ∈ E [A1 ...An ] such that B1 Bn εA1 ...An εB1 ...Bn = n!δ[A . In general, DA does not preserve εA1 ...An . . . δA n] 1 so that if we set 1 (DA εB1 ...Bn ) εB1 ...Bn , (23) ΥA := (n + 1)!

b A given by (21) or (22) preserves εA ...An . Thus, we can the connection D 1 always find a special connection, i.e. a connection that preserves a given volume form, in the projective class p, and such a connection can be shown to be unique, cf. [6] and [13]. With no loss of generality, we shall henceforth restrict ourselves to special torsion-free affine connections. These enjoy nice properties. In particular, if RAB C D is the curvature tensor of a special torsion-free affine connection D with Ricci tensor RicAB := RP AP B , then the Schouten tensor 1 RicAB , n−1 is symmetric. Hence P vanishes if and only if D is Ricci-flat. The projective Weyl curvature and the Cotton tensor are defined respectively by PAB :=

C C WAB CD = RAB C D + PAD δB − PBD δA ,

YCAB = 2D[A PB]C .

(24)

The connection D is called projectively flat if it is projectively equivalent to a flat affine connection. For manifolds of dimension n = 2, the Weyl curvature vanishes identically and the only obstruction to projective flatness is the Cotton tensor Y . For n ≥ 3 projective flatness is equivalent to the vanishing of the Weyl curvature W . By (22), any two volume forms ε and εˆ related by εˆ = e(n+1)φ ε correspond b differing by the 1-form to two special torsion-free affine connections D and D ΥA = DA φ. We note that under such a projective change, the Rho tensor transforms according to b AB = PAB + ΥA ΥB − DA ΥB , P (25) b AB associated to D b A remains symmetric. so that the Schouten P

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We therefore have a special subclass of torsion-free affine connections of p, projectively related by exact 1-forms, and thus parametrized by smooth functions on M . We can conveniently define the density bundle of projective w weight w as E(w) := (∧n T M )− n+1 on M , where dim M = n. We will refer to everywhere positive sections of E(1) as projective scales. Any projective scale σ, say, determines a special torsion-free affine connection DA in p, which extends to an affine connection, also denoted DA , on E(w), and for which DA σ = 0. For any two torsion-free affine connections in p, we have b A f = DA f + wΥA f , D

f ∈ E(w) ,

(26)

An oriented projective structure determines a distinguished section εA1 ...An ∈ E[A1 ...An ] (n + 1), which we shall refer to as the projective volume form. Any choice of projective scale σ corresponds to a special connection D preserving b in the volume form ε = σ −(n+1) ε. Since, for any two connections D and D b p, we have Dε = Dε by (26) and (22), we conclude that Dε = 0 for any connection D in p.

3.2. Calculus for conformal geometry. As before, we shall use lower f, e.g. gab ∈ Ee(ab) denotes a symmetric case Roman indices for tensors on M f. The reader can refer to [2] for more details on conformal 2-tensor on M geometry and its calculus. − w e f 2n We define the density bundle of conformal weight w as E[w] := ∧2n T M

f, where dim M f = 2n. We will refer to everywhere positive sections of on M e as conformal scales. The Levi-Civita connection extends to an affine E[1] e connection on E[w]. The conformal structure can be equivalently seen as f ⊗ E[2] e referred to as the a density-valued metric gab ∈ Ee(ab) [2] = S 2 T ∗ M f. Any conformal scale τ ∈ E[1] e determines a metric conformal metric on M −2 e a preserves gab , gab = τ gab in c. The associated Levi-Civita connection D f f[2]. gab and τ . The conformal metric allows us to identify T M with T ∗ M ∗ ∗ [1] Similarly, one can identify Se± with Se± [1] when n is even, and with Se∓ when n is odd, by means of weighted spin bilinear forms. e is given by For a (pseudo-)Riemannian metric g, the Schouten tensor P ! e 1 Sc e gab − Pab = Ric gab , 2n − 2 2(2n − 1) g and Sc e is the Ricci and scalar curvature of g, respectively. Since where Ric e is a trace modification of Ric, g the Schouten tensor vanishes if and only if P g is Ricci-flat. The conformal Weyl curvature and the Cotton tensors of g are defined respectively by e e c f c =R e c − 2 δc P W ab d ab d [a b]d + 2gd[a Pb] ,

e b]c . e [a P Yecab = 2D

The metric g is called conformally flat if it can be (locally) rescaled to a flat metric. For manifolds of dimension 2n ≥ 4 conformal flatness is equivalent f . The transformation rules for to the vanishing of the Weyl curvature W Levi-Civita connections and Schouten tensors under conformal changes can be given explicitly, see e.g. [2].


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3.3. Conformal extensions of projective structures. The Riemann extension of an affine connected space can be adapted to weighted cotangent bundles T ∗ M (w) = T ∗ M ⊗ E(w). The only difference in the weighted case is that a choice of torsion-free affine connection D gives rise to a weighted metric. This means that the natural pairing between H ∼ = T M and ∗ M (w) defines a symmetric bilinear form on the tangent bundle of V ∼ T = T ∗ M (w) with values in π ∗ E(w), the pull-back of the line bundle over M with respect to the natural projection π : T ∗ M (w) → M . A special connection D yields a trivialization of E(w), and thus the pairing can be regarded as R-valued. In particular, D defines a Patterson–Walker metric on T ∗ M (w). We shall denote by θ the (weighted) tautological 1-form on T ∗ M (w). This bundle is trivialized by any choice of projective scales. Let σ and σ ˆ be two such scales related by σ ˆ = e−φ σ for some smooth function φ. Then, θ := σ −w θ and θˆ := σ ˆ −w θ are two (tautological) 1-forms related by wφ ˆ θ = e θ. In both cases, there exists canonical coordinates {xA , pA } and {xA , pÂ } in which θ = pA dxA and θˆ = pÂ dxA . Thus, a projective change induces the change of canonical fiber coordinates pA 7→ pÂ = ewφ pA . b A ∈ p be the special affine connections in p associated to Let DA and D b A differs from DA via (21) by ΥA = DA φ. σ and σ ˆ respectively, so that D b A are related by This means that the Christoffel symbols of DA and D b C = Γ C + δ C ΥB + δ C ΥA . Γ A B

A B

A

B

A straightforward computation then gives

bAC B pˆC dxB = ewφ dpA − ΓAC B pC dxB dˆ pA − Γ

+ ewφ ((w − 1)pA ΥB − pB ΥA ) dxB , (27)

so that using (7) yields

gˆ = ewφ (g + 2 (w − 2)pB ΥA dxB ⊙ dxA ) ,

As a consequence, we immediately conclude:

b be projectively equivalent special torsion-free Proposition 3.1. Let D and D affine connections on M and let g and gˆ be the associated Patterson–Walker metrics on T ∗ M (w). Then g and gˆ are conformally equivalent if and only if w = 2. f := T ∗ M (2) we have thus obtained the notion of the conformal Setting M f, c) of a projective structure (M, p): extension (M

Definition 3.2. The conformal extension or the conformal Patterson–Walker metric associated to an oriented projective structure p on M is the splitf = T ∗ M (2) represented by the Patterson– signature conformal structure c on M Walker metric of a special torsion-free affine connection D ∈ p.

Remark 3.3. A slightly different construction, which was first introduced in [12] when n = 2, involves the so-called Thomas projective parameters. In dimension n, these are defined by [16, 33] 2 δC Γ D , (28) ΠAC B := ΓAC B − n + 1 (A B) D

12


where ΓAC B are the Christoffel symbols of any affine connection in p with respect to some coordinate system {xA }. In fact, the ΠAC B do not depend on the choice of connection in p, and are thus a set of projectively invariant functions. However, the ΠAC B depend on the choice of coordinates {xA } in the sense that they do not transform as Christoffel symbols, let alone as a tensor in general. Consider a general coordinate transformation xA 7→ y A ∂y A 1 A on M with Jacobian JBA := ∂x B , and set φ := n+1 log det JB . Then, we have [16] ∂ C C ∂φ C ΠADB JD = Π′ D E JAD JBE + A JBC − 2J(A , (29) ∂x ∂xB) where Π′ AC B are the Thomas projective parameters defined by the Christoffel symbols with respect to {y A }. Each of the coordinate systems {xA } and {y A } defines volume forms ε := dx1 ∧ . . . ∧ dxn and εˆ := dy 1 ∧ . . . ∧ dy n , respectively, preserved by special b A in p respectively. These are projectively related connections DA and D by ΥA = DA φ. We therefore have an induced change of canonical fiber coordinates on T ∗ M (w) given by pA 7→ qA := ewφ pB (J −1 )B A . Define two ∗ metrics on the open subset of T M (w) over the overlap of the charts of {xA } and {y A } by g := 2 dxA ⊙ dpA − 2 ΠAC B pC dxA ⊙ dxB , gˆ := 2 dy A ⊙ dqA − 2 Π′ A

C

B qC

(30)

dy A ⊙ dy B .

Then, using (29), one can immediately check that ∂φ A B wφ . gˆ = e g + 2(w − 2)qA B dy ⊙ dy ∂y In particular, g and gˆ are conformally equivalent if and only if w = 2. We have therefore constructed a conformal class of metrics of the form (30) on f = T ∗ M (2) from the projective class p on M : a metric in the conforM mal class corresponds to the Thomas projective parameters representing p in a given coordinate system {xA }, up to coordinate transformations that preserve the volume form dx1 ∧ . . . ∧ dxn . Different Thomas projective parameters for different coordinate systems yield conformally related metrics. Finally, with no loss, we can take ΠAC B = ΓAC B in the definition (28), where ΓAC B are the Christoffel symbols of the special connection DA preserving the volume form ε, up to constant multiple, on (M, p). In this case, the metric (30) can be identified with the Patterson–Walker metric (7). As this identification holds for any choice of coordinate system, the conformal f is none other than the conformal class of metrics of the form (30) on M Patterson–Walker metric of Definition 3.2.

To deal with the conformal class of Patterson–Walker metrics of Definition 3.2, rather than a metric, we shall henceforth view the quantities ′ introduced in section 2 as being weighted. In particular, γa AB ′ and γa A B have conformal weight 1. By definition, the conformal Killing field ka has weight 0, so that the 1-form ka is twice the weighted tautological 1-form θa ′ f, i.e. ka = 2 θa ∈ E[2]. e on M Next, requiring that the spinor χA remain parallel with respect to the Levi-Civita connection of any Patterson–Walker


13

metric, restricts its possible conformal weight. Following the conventions of ′ [28, 31], and for convenience, χA will have weight 0, from which it follows that ηˇA′ has weight 0. Lemma 3.4. Any projective scale σ ∈ E(1) lifts to a conformal scale σ e∈ e e E[1], and thus by extension any section of E(w) lifts to a section of E[w]. e eaσ Conversely, any section σ e of E[w] such that χaA D e = 0, with respect to any Patterson–Walker metric in c, descends to a section of E(w). ...Bℓ B1 ...Bℓ (w) gives rise to a section of ∈ EAB11...A Further, any section σA 1 ...Ak k B ...B 1 ℓ EeA1 ...Ak [w − k + ℓ]. For contravariant tensors, the lifts depend on the choice of special torsion-free affine connection on p.

Proof. Proposition 3.1 assigns to a special affine connection on M , i.e. a f, i.e. a section of section of σ ∈ E(1), a Patterson–Walker metric on M e f σ e ∈ E[1]. This can also be verified by noting that the volume form on M −2 induced by gab = σ e gab takes the form εe = εA1 ...An dxA1 . . . dxA1 ∧ εB1 ...Bn dpB1 . . . dpBn .

where εA1 ...An is the volume form determined by σ, and εA1 ...An its inverse. Since a special projective change induces a change pÂ = e2φ pA for some function φ, the volume form εe transforms to b εe = e2nφ εe as expected. The e a , for any converse statement follows from the fact that the vectors χaA D Patterson–Walker metric in c, span the vertical distribution. According to our conventions, we obtain weighted projectors and injectors a ∈ E ea [−1]. Now choosing an affine connection D ∈ p, eA ˇA χA a ∈ Ea [1] and η A any section v A ∈ E A (w) can be canonically lifted vea ∈ Eea [w]. This means in particular that as a spinor field, v A = vea χA a gives rise to a section of A e E [w + 1]. Similarly (but independently of the choice of D ∈ p), any section αA ∈ EA (w) gives rise to a section of EeA [w − 1]. This generalizes to tensor fields of higher valence. Now let D be a special torsion-free affine connection on M and g its f. We can decompose (11) further so as to Patterson–Walker metric on M e e f express the conformal Weyl, Schouten and Cotton tensors W abcd , Pab , Ycab of the Patterson–Walker metric g in terms of the projective Weyl, Schouten respectively Cotton tensors WAB DC , PAB and YABC : D A B C A B C f χ η ˇ χ η ˇ + χ χ W = 2 χ χ c d [a b]D WAB C a b [c d]D abcd B C D E + 2 χA [a χb] χ[c χd] DA WCD B pE + pC YDAB , (31) e = χA χB P , P ab a b AB

Yecab =

A B χC c χa χb YCAB

(32) .

(33)

Remark 3.5. By direct inspection, we find: (a) By (31), the induced Patterson–Walker metric is conformally flat if and only if the original affine connection is projectively flat.

14


(b) By (32), the induced Patterson–Walker metric is Ricci-flat if and only if the original affine connection is Ricci-flat. (c) By (7) and (32), a Patterson–Walker metric is Einstein if and only if it is already Ricci-flat. 4. Characterization of conformal Patterson–Walker metrics We shall now prove our characterization Theorem 1 which exactly specifies those split-signature conformal spin structures that are associated to a projective structure via the conformal extension in the sense of definition 3.2. For this purpose we start by collecting properties of the induced conformal structures: f, c) associated to an oriented Proposition 4.1. The conformal extension (M projective structure (M, p) satisfies all the properties (a)–(d) of Theorem 1. e it trivially satisfies the twistor Proof. Since χ is parallel with respect to D, spinor equation (1). We have already observed in (15) that k ∈ Γ(V ) is a (light-like) conformal Killing field. The general formula for the Lie derivative of χ with respect to the conformal Killing field k is

e k )γ a γ b χ − 1 (D e p kp )χ. e a χ − 1 (D (34) Lk χ = k a D 4 [a b] 4n e a χ = 0, D e a kb = µab + gab and µab χbB = −χB Hence it is immediate that D a (according to (14) and (15)) imply (2). The integrability condition (3) follows immediately from (31). For the converse direction we begin with two technical results which will provide a normal form for structures satisfying the above conformal properties.1 Proposition 4.2. Let χ be a pure real twistor spinor on a conformal pseudof, c) of signature (n, n) with associated totally isotropic Riemannian manifold (M n-plane distribution V . Suppose V is integrable. Then locally, there is a conformal subclass of metrics in c for which χ is parallel, i.e. if g is any such e Dχ e = 0. Any two such metrics are metric with Levi-Civita connection D, related by a conformal factor constant along the leaves of V . ′

ˇA := Proof. In abstract index notation for spinors we write χA and χ

√1 (D / χ)A . 2n

The key idea is to use the transformation rule for χ ˇA under a conformal ∞ f) and gˆ = e2φ g a change of metric: For a smooth function φ ∈ C (M rescaled metric, the spinor χ ˇA transforms according to (see e.g. [3, 19]) 1 e aA χ ˇA 7→ χ ˇA + √ (D a φ)χ . 2

(35)

1AT-C thanks Andree Lischewski for pointing out an unnecessary curvature condition in the statement of Proposition 4.2, which appeared in an earlier version of [31] (preprint arXiv:1212.3595). See also his analogous result in [24].


15

′

Thus, to find a conformal scaling for which χA is parallel, we must first show that χ ˇA can be expressed as 1 eaφ , χ ˇA = √ χaA D 2

(36)

for some smooth function φ. We assume of course that χ ˇA is non-vanishing, for otherwise our spinor was already parallel. ′ e a χB ′ = − √1 χ ˇB , and contracting We write the twistor equation on χ as D 2 a

with χaA this gives

′ e a χB ′ = − √1 χaA χ χaA D ˇB a . 2

(37)

The condition that V is integrable can be re-expressed as [22, 31] e a χB = αA χB , χaA D ′

′

(38)

for some spinor αA , which necessarily lies in the image of χaA . The equations (37) and (38) together imply 1 ′ A B′ ˇB − √ χaA χ a =α χ . 2 It is shown in [31], that since χ is pure, the last formula implies that √ A ˇ . (39) αA = 2χ In particular, this implies that χ ˇA also lies in the image of χaA and thus ′

′

χaA χ ˇB ˇ A χB . a = −2 χ

(40)

By differentiating (36) one obtains the integrability conditions eaχ χa[A D ˇB] = α[A χ ˇB]

(41)

for the existence of φ (see e.g. [22, 31]). By (39), the right-hand-side of (41) vanishes. On the other hand, the prolongation of the twistor equae ab χaB leads to the vanishing of the left-hand-side of eaχ tion D ˇA = − √12 P (41). Hence both sides of (41) are zero, and the integrability conditions are therefore satisfied and we can find a local solution φ of (36). Finally, by (35), adding to φ a smooth function constant along V , yields a metric in c conformal related to gˆ, for which χ is also parallel. This produces the required conformal subclass of c. Remark 4.3. (a) The relation between pure twistor spinors and the integrability of their associated distributions is already given in [31]. Similar results are obtained in odd dimensions in [32]. (b) A similar argument is employed in [10] in the four-dimensional case to show the existence of a suitable parallelizing scale. / χ)A being pure with (c) Formula (40) is in fact equivalent to χ ˇA = √12n (D ′

associated n-plane distribution intersecting that of χA maximally in an (n − 1)-dimensional distribution [31].

16


Lemma 4.4. Let χ be a parallel pure spinor with associated distribution V and ka a conformal Killing field tangent to V such that Lk χ = − 21 (n + 1)χ. Then ka is a homothety satisfying (15). e a kb −µab +gab ϕ = 0 Proof. We write the conformal Killing field equation as D e [a kb] and ϕ = − 1 D e p kp . Since χ is parallel, differentiating with µab = D 2n a A k χa = 0 yields µab γ b χ − ϕγa χ = 0

(42)

µab γ a γ b χ + 2nϕχ = 0 .

(43)

so that On the other hand, Lk χ = − 12 (n + 1)χ now reads as

1 1 1 − µab γ a γ b χ + ϕχ = − (n + 1)χ , (44) 4 2 2 where we have used (34) and the fact that χ is parallel. Combining (43) and (44) yields ϕ = −1, hence (15) follows. f, c) be a conformal spin structure of split signaProposition 4.5. Let (M ture (n, n) satisfying the properties (a)–(d) of Theorem 1. Then the local leaf space of the integrable distribution associated to the pure twistor spinor f, c) is the conformal extension admits a projective structure p such that (M associated to p.

Proof. From Proposition 4.2 we know that, locally, we can find metrics g and gˆ in c such that the twistor spinor χ is parallel with respect to the b e and D, e and gˆ = e2φ g for some corresponding Levi-Civita connections D f which is constant on the leaves of V . From Lemma smooth function φ on M a 4.4 we know that k is a homothety satisfying (15). Since χ is parallel with e ab is annihilated by V , and thus the e the Schouten tensor P respect to D, integrability condition (3) is equivalent to the condition (12) on the Riemann b eabcd . The same argument applies also to D e and the corresponding tensor R Riemann tensor. We can therefore apply Proposition 2.2 to each of the metrics g and gˆ with respective special torsion-free affine connections D and b on M . D b are projectively related. Now, the LeviWe shall show that D and D b e and D e are related by Civita connections D b e aξb = D e a ξ b + Υ a ξ b + Υ c ξ c δ b − ξa Υ b , D a

e a φ. Since φ is constant along the leaves of V , the correspondwhere Υa = D ing 1-form Υa is strictly horizontal. Hence we consider both φ and Υa as the pull-back of a smooth function φ and a 1-form ΥA on the leaf space M , f and ξ A respectively. Therefore, for ξ a being a projectable vector field on M denoting its projection to M , the two underlying affine connections differ by b A ξ B = DA ξ B + ΥA ξ B + ΥC ξ C δB . D A

b are projectively equivalent, cf. (21). That is why D and D


17

f is locally identified with Finally, from Proposition 3.1 it follows that M

T ∗ M (2).

Combining propositions 4.1 and 4.5 we immediately obtain our characterization Theorem 1. The conformal Patterson–Walker metric constructed above is also equipped with another distinguished spinor as explained below. f, c) admits a pure spinor Proposition 4.6. The conformal extension (M e field ηA ∈ EA [1], which, for any choice of Patterson–Walker metric, takes the form 1 (45) ηA = √ kb ηˇbA , 2 2 and satisfies a ηA ˇaB = −2 ηB ηˇA′ , ′η

(46)

a := η γ a B , i.e. the totally isotropic n-plane distribution U := where ηA ′ B A′ ker ηˇaA′ intersects the horizontal distribution H maximally and intersects the vertical√distribution V in the line distribution spanned by k a . In particular, ka = 2 2ηA χaA . Further, ηA satisfies the conformally invariant equation e a ηA − √1 γ B ′ ηˇB ′ = 1 kd W fdabc (γ b γ c )B ηB . D (47) a A A 8 2 f, c) is conformally flat, In particular, ηA is a twistor spinor if and only if (M i.e. (M, p) is projectively flat.

Proof. That ηA is pure follows from the fact that it lies in the image of ηˇaA since ηˇA′ is a pure spinor. That it satisfies (46) follows from a direct computation and commuting γ-matrices. Since ka and ηˇaA have conformal weights 0 and 1 respectively, ηA has conformal weight 1 by (45). We now check that (45) is independent of the choice of connection in p. b A in p Consider any two projectively related special connections DA and D b on M f annihilated by pure corresponding to horizontal distributions H and H b spinors ηˇA′ and ηˇA′ respectively. Note that with a choice of trivialization, (13) allows us to make the identification 1 (48) ηA = √ pA , 2 b are related as and similarly for b ηˇA′ . Since the 1-forms annihilating H and H b in (27) with w = 2, we can then readily check that ηˇA′ and ηˇA′ are related a where Υ = Υ χA , or equivalently, by by b ηˇA′ = ηˇA′ − √12 Υa ηA ′ a A a √ b (49) ηˇaA = ηˇaA + 2(ηA ΥB − ηB ΥA )χB a . ′

Since ka annihilates χA , the result follows immediately. The final part of the proposition follows from a direct, albeit lengthy, computation. The identification (48) will prove to be very convenient in explicit computations, and will be used ubiquitously in sections 5 and 6.

18


Remark 4.7. We can investigate the geometric properties of the distributions V = ker χA ˇaA′ and V ∩ U = hka i viewed as G-structures on a , U = ker η f with structure group taken to be the stabilizer of hχA′ i, hηA i or hka i M in Spin(n, n) at a point. These can be expressed in terms of differential ′ conditions on the fields χA , ηA or ka defined up to scale, and are related to the notion of intrinsic torsion of the G-structure. For pure spinor fields, this is the topic of the articles [31, 32], to which we refer for details. ′

(a) For χA parallel, the intrinsic torsion is trivial. This implies in particular bB aA e χC χ Da χ b = 0, i.e. V , as any integrable totally isotropic n-plane distribution on (M, c), is totally geodetic [22, 30, 31]. a b fdabc η a ′ η b ′ η c ′ , which by e (b) From (47), we deduce ηA′ Da ηB ′ ηbC ′ = kd W A B C a b e the Bianchi identity implies that η[A′ Da ηB ′ ηbC ′ ] = 0. The distribu a D b f e tion U is integrable, i.e. ηA η ′ a B ′ ηbC ′ = 0, if and only if (M , c) is conformally flat. (c) Being light-like and conformal Killing, ka generates a shear-free con e c k[a kb] = 0 and gruence of null geodesics tangent to U ∩ V , i.e. kc D Lk gab = f gab + t(a kb) for some function f and 1-form ta . e b kc] does not vanish. (d) Moreover, this congruence is also twisting, i.e. k[a D Since ka annihilates the rank-(2n − 1) distribution U + V , this means that U + V is not integrable. f with Remark 4.8. In four dimensions, i.e. n = 2, we can identify T M e e S+ ⊗ S− , and use the two-spinor calculus of [28]. We can choose a spin invariant skew-symmetric bilinear form εAB on Se− , with inverse εAB , to be e a of a Patterson–Walker metric preserved by the Levi-Civita connection D in c, and identify εAB as the volume form on M preserved by the corresponding special connection DA ∈ p. It can be shown [5, 10] that the function ΘAB can be expressed in terms of a single function Θ = Θ(x, p), 2 i.e. ΘAB = εAC εBD ∂pC∂∂pD Θ. Then equations (18) and (19) tell us that the function Θ must be a polynomial of degree 3 in the coordinates pA , i.e. where ΓAB C are the Christoffel symbols for an affine connection on the projective surface M . The Weyl tensor can be expressed as e ABCD εA′ B ′ εC ′ D′ , fabcd = Ψ e A′ B ′ C ′ D′ εAB εCD + Ψ W

e ABCD are the self-dual and anti-self-dual parts of e A′ B ′ C ′ D′ and Ψ where Ψ ′ ′ the Weyl tensor. Writing v a = χA v A and wa = χA wA for two arbitrary elements of V for some spinors v A and wA , we see that (3) is equivalent to ′ ′ e ABCD χB ′ χD′ = 0 . e A′ B ′ C ′ D′ vB wD + v A wC Ψ χA χC Ψ

fabcd γ c γ d χ = 0 for the existence of a twistor The integrability condition W ′ ′ e A′ B ′ C ′ D′ χA = 0, i.e. the self-dual Weyl tensor is of Petrov spinor χA is Ψ e ABCD = 0, i.e. the type N. Combined with (3), we immediately conclude Ψ Weyl tensor is self-dual.


19

Generalizations of Patterson–Walker lifts in dimension four were recently considered in [11]. 5. Einstein metrics e is an almost Einstein scale if We say that a non-trivial density σ e ∈ E[1] −2 gab = λ gab the metric gab = σ e gab is Einstein off the zero-set of σ e, i.e. Ric for some constant λ. One can show that this is equivalent to σ e satisfying the conformally invariant equation e ab σ e (a D e b) + P e = 0. (50) D 0 f, c) gives rise to solutions to We now show that any Einstein scale on (M overdetermined projectively invariant differential equations. One of these is a projective analogue of equation (50), to be precise, a solution σ ∈ E(1) to (51) D(A DB) + PAB σ = 0 .

Away from their singularity sets, solutions to this equation determine Ricciflat affine connections DA in p. Thus, they are sometimes referred to as almost Ricci-flat scales. We shall also consider a generalization of Euler vector fields to weighted vector fields: i.e. a solution ξ A ∈ E A (−1) satisfying 1 B DA ξ B − δA (DC ξ C ) = 0 , (52) n Equation (52) implies C D(A DB) ξ C + δ(A PB)D ξ D = 0 ,

(53)

WAB CD ξ D = 0 .

(54)

With reference to Lemma 3.4 and the fact that ηA has conformal weight 1 we prove: Lemma 5.1. Let σ ∈ E(1) and ξ A ∈ E A (−1). Then √ σ e− := π ∗ σ , σ e+ := 2 ξ A ηA

(55)

e f to M , and ξ A is are sections of E[1]. Here, π is the projection from M viewed as a section of EeA .

e and σ a ∈ Eea [w], Before we proceed, we note that for any ka ∈ Eea , σ ∈ E[w] we have w e a eaσ ebσ e b kb . e b ka − w σ Lk σ e = ka D e− σ eDa k , Lk σ ea = kb D ea D ea − σ eb D 2n 2n Choosing a Patterson–Walker metric, these simplify to e aσ Lk σ e = ka D e − we σ,

ebσ Lk σ ea = kb D ea − σ eb µba − (w + 1)e σa ,

(56)

where we have made use of (15). Similar formulae for the Lie derivative on weighted forms can be obtained using the Leibniz rule or the fact that ka is a conformal Killing field. Lemma 5.2. The lifts satisfy Lk σ e± = ±e σ± .

20


eaσ eaσ Proof. By (13), we have ka D e+ = 2 σ e+ and ka D e− = 0. Applying (56) with w = 1 completes the proof. Proposition 5.3. (a) Suppose σ ∈ E(1) satisfies (51). Then its lift σ e− given by (55) is an almost Einstein scale, i.e. a solution to (50). (b) Suppose ξ A ∈ E A (−1) satisfies (52) together with the integrability condition ξ D WDA CB = 0 .

(57)

Then its lift σ e+ given by (55) is an almost Einstein scale. In both cases, the rescaled metrics they define are Ricci-flat off the singular sets of σ e± .

Proof. (a) Let σ be a Ricci-flat scale with associated torsion-free affine connection DA in p on M that is Ricci-flat, i.e. PAB = 0. Then DA is special and determines a Patterson–Walker metric g with corresponding conformal scale σ e− as given by (55). Reading off (32), we see that e ab = 0, i.e. g is Ricci-flat. P a k . Then, using the Leibniz rule, (b) Let us rewrite (55) as σ e+ = 21 ξ A ηˇA a (10), with v A = ξ A and αA = 0, (14) and (15), we obtain eaσ D e+ = DA ξ B pB χA + ξ B ηˇaB . (58) a

Similarly, A B B eaD ebσ D e+ = DA DB ξ C − ξ D RDAC B χA χ(a ηˇb)B . a χa pC + 2 DA ξ

(59)

Finally, using (11), (24) and (32), we find 1 B C B e ab σ e (a D e b) σ χA ˇb)B D e+ + P e+ = 2 DA ξ − δA DC ξ (a η n 0 C B + DA DB ξ C + δA PBD ξ D − ξ D WDA CB pC χA a χb . (60)

That σ e+ is an almost Einstein scale follows immediately from (52), (53) and (57). To show that the rescaled metric is Ricci-flat, we compute the b e of the Rho tensor of the rescaled metric via the transformation trace P b e = P− e D e := P e a and Υa := −σ −1 D e a σ+ . e aΥa +(1−n)ΥaΥa , where P rule P a + e Using (58), (59) and the fact P = 0 for a Patterson–Walker metric, one b e = 0 as required. easily verifies P

e with Lk σ Lemma 5.4. Let σ e ∈ E[1] e = rσ e for some real constant r. Then in p . In particular, σ e+ is homogeneous of σ e is homogeneous of degree r+1 A 2 degree 1 and σ e− of degree 0. Proof. This follows from (56) with w = 1 and (13).

e be an almost Einstein scale. Then Proposition 5.5. Let σ e ∈ E[1] σ e=σ e+ + σ e−

where Lk σ e± = ±e σ± . Further, for any choice of Patterson–Walker metric, σ e± can be expressed as the lifts (55), where


21

(a) σ = σ e− (x) is an almost Ricci-flat scale on (M, p). A eaσ (b) ξ = χaA D e+ satisfies (52) together with the integrability condition (57).

Proof. We use a Patterson–Walker metric throughout. Using (56) with w = 1, together with the Leibniz rule and the fact that µab kb = −ka , we compute

eaD ebσ e = ka kb D L2k σ e+σ e. e ab σ e (a D e b) + P eaD ebσ Since σ e is an almost Einstein scale, ka kb D e = ka kb D e= 0

e ab kb = 0, where we have used the fact that, for a Patterson–Walker metric, P 2 e=σ e, i.e. (Lk − 1)(Lk + 1)e σ = 0. This equation is the 0 by (32). Hence Lk σ characteristic polynomial for Lk viewed as a linear operator acting on the finite-dimensional space of Einstein scales, and the decomposition of this space follows immediately. Details and generalizations are given in [18]. Next, assume that σ e± are almost Einstein scales with Lk σ e± = ±e σ± , so eaD e bσ that χaA χbB D e± = 0. In coordinates, this condition reads ∂2 σ e± = 0 . ∂pA pB This means that σ e± are polynomials of degree 1 in pA with coefficients A depending on x only, i.e. σ e± = ξ A pA + σ, where ξ A = ξ A (x) and σ = σ(x). Now, using (56) with w = 1, Lk σ e± = ±e σ can be recast as ae e aσ k Da σ e+ = 2 σ e+ , ka D e− = 0 .

Using (13), these conditions tell us that σ e+ is homogeneous of degree 1 in pA and σ e− homogeneous of degree 0 in pA . Since they are also polynomials in pA , we conclude that σ e± take the form (55). For the last part of the proposition, we assume σ e± are almost Einstein scales with Lk σ e± = ±e σ± so that σ e± are given by (55). We proceed as follows. (a) The almost Einstein scale σ e− defines a conformally related Patterson– −2 e ab = 0. By (32), we conclude immediately Walker metric σ e− gab with P PAB = 0, i.e. the corresponding affine connection on M is Ricci-flat. (b) Equation (50) with σ e=σ e+ implies that the left-hand side of (60) vanishes, and in particular, each term of the right-hand side must vanish separately, i.e. ξ A satisfies (52) and C D(A DB) ξ C + δ(A PB)D ξ D − ξ D WD(A CB) = 0 .

(61)

But with reference to (54) and (53), together with the Bianchi identity, equation (61) implies (57), i.e. ξ D WDA CB = 0. Combining Proposition 5.3 and Proposition 5.5 now gives Theorem 2. 6. Symmetries f, c), i.e. a soluWe now show that any conformal Killing vector vea on (M tion of e a veb = φeab − ψe gab , D (62)


22

e [a veb] and ψe = − 1 gab D e a veb , gives rise to solutions of overdewhere φeab = D 2n termined projectively invariant differential equations on (M, p) as claimed by Theorem 3. Before we proceed, we recall the prolongation equations for (62): e ab veb − βea , e a ψe = P D

(63)

e a[b vec] + e a φebc = −2 ga[b βec] − 2 P D e a βeb = D

e c φecb P a

Here, βea is defined by (63).

fdabc ved W

,

e ab − ve Yabd . − ψe P d

(64) (65)

6.1. Projectively invariant equations. An infinitesimal projective symmetry is a vector field v A that preserves the projective structure, i.e. for any DA in p and vector field X A , Lv DA X B − DA Lv X B = QAC B X C ,

where

C QAB C = 2 δ(A ΥB) ,

(66)

for some 1-form ΥA . It can be shown that v A is an infinitesimal projective symmetry if and only if it satisfies the following projectively invariant overdetermined system of partial differential equations [15] (67) D(A DB) v C + PAB v C + v D WD(A CB) = 0 . 0

Define

B φB A := DA v −

1 B δ DC v C n A

ψ :=

1 DC v C , n

1 βA := − DA DB v B − PAB v B . n+1

(68)

Then, under a projective transformation, using (21), the fields transform as 1 A C A A φÂ B = φB − ΥC v δB + ΥB v , n n+1 B ψˆ = ψ + ΥC v C , βÂ = βA − ΥB φB A − ΥA ψ − ΥA ΥB v . n Equation (67) can be written in prolonged form as vÂ = v A ,

(69)

B DA v B − φB A − δA ψ = 0 , n+1 βA + PAB v B = 0 , DA ψ + n

1 C δ(A PB)D v D − (n−1)βB) = 0 , (70) n C DA βB − PAB ψ − PAC φC − v Y ABC = 0 . B

C D C D(A φC B) + PAB v + v WD(A B) −

The first two equations immediately follow from (68), the third one from (67), and the last one from the divergence of the latter equation. Next, we shall consider the following projectively invariant equation DC wAB +

2 [A δ DD wB]D = 0 , n−1 C

(71)


23

where wAB ∈ E [AB] (−2). Defining ν A :=

1 DC wCA , n−1

(72)

one can easily verify the transformation rules under a projective change w ˆ AB = wAB ,

νÂ = ν A − wAB ΥB .

(73)

Differentiating (72), one can show that equation (71) is equivalent to the system [A

DC wAB − 2 δC ν B] = 0 , DA ν B + PAC wCB +

1 wCD WCD BA = 0 . 2(n − 2)

(74)

Finally, we shall consider a weighted 1-form αA ∈ EA (2) that satisfies the Killing equation D(A αB) = 0 .

(75)

6.2. Projectively invariant lifts. Let v A ∈ E A , wAB ∈ E [AB] (−2) and A αA ∈ EA (2), and φB A , ψ, and ν are given by (68) and (72). At this stage, we A AB do not assume that v , w and αA satisfy (67), (71) and (75) respectively. Lemma 6.1. Choosing a special torsion-free affine connection D ∈ p, we define the vector fields √ n−1 aB a + ψk a , (76) ve0a := v A ηˇA − 2φA B ηA χ 2(n + 1) √ 1 a a − √ (ν B ηB )ka , ve+ := 2wAB ηA ηˇB (77) 2 a ve− := αA χaA , (78)

f. Then the forms of these vectors are independent of the choice of on M D ∈ p.

Proof. We first check the conformal weight of each expression using Lemma 3.4. For instance, we view wAB and ν A as sections of EeAB and EeA [1] respeca have weight 1 and −1 respectively, we see that the tively. Since ηA and ηˇB a , have weight 0 as required. both terms in (77), and thus ve+ Next, under a projective change of affine connections in p, ηA , χA a and a a is invariant. The invariance of v k are invariant. In particular, ve− e0a and a can be verified by observing that the change of horizontal distribution as ve+ given (49) induced by a projective change, and using (48), counterbalances the transformation rules (69) and (73). Lemma 6.2. The vector fields in Lemma 6.1 satisfy the following properties: a a and L v a = ±2 v e± (a) Lk ve± k e0 = 0; a are tangent to U = ker η ′ and V = ker χA respectively, i.e. a and v e− (b) ve+ aA a a χA = 0; a ve+ ηaA′ = 0 and ve− a (c) ve0a is not tangent to U + V = ker ka , i.e. ve0a ka is not identically zero.

24


Proof. (a) First observe that [2pA ∂p∂A , ∂p∂B ] = −2 ∂p∂B which, using (9) is equivalent to the first relation in the display b e e a , χbA D e b ] = −2 χbA D e b , [ka D e a , ηˇA [ka D Db ] = 0 , Lk pA = 2 pA . (79)

The second relation follows similarly using (9) and the last one is obvious. Further, Lk v A = 0 and similarly for all sections depending only on xA . Using (48), these relations and the Leibniz rule, it is a straightforward computation to verify part (a). a χA = 0 follows from (8). Further recall η a η ˇA′ from (b) Here ve− a A′ ˇaB = −2 ηB η a inserts (46). Since wAB is skew-symmetric, the first summand of ve+ trivially into ηaA′ . The second summand inserts trivially using (45) since k is null. e0a ka = 2 v A pA (c) It follows from (48), and the properties of ka and χA a that v A A which is zero if and only if v is zero. But if v is zero then φB A and ψ are zero since they are defined by (68). Proposition 6.3. (a) Suppose v A ∈ E A is an infinitesimal projective symmetry, i.e. satisfies (67). Then its lift ve0a given by (76) is a conformal Killing field. (b) Suppose wAB ∈ E [AB] (−2) satisfies (71) together with the integrability condition wB(A WB(C

D) E)

= 0.

(80)

a given by (77) is a conformal Killing field. Then its lift ve+ a (c) Suppose αA ∈ EA (1) satisfies the Killing equation (75). Then its lift ve− given by (78) is a (conformal) Killing field.

Proof. In the following we work with a choice of Patterson–Walker metric g ∈ c. (a) Suppose v A satisfies (67), so that φB A , ψ and βA are given by (68), and lift v A to vea := ve0a as given by (76). Then, using (10), n−1 1 1 1 C B B C A B e DC v + ψ gab D(a veb) = DA v − DC v δA − φA χ(a ηˇb)B + n 2 n n+1 C D C − DA φC B + PAB v + v WDA B 1 C B − δA PBD v D − (n − 1)βB pC χA (a χb) . (81) n Since (67) is equivalent to (70), it is clear that the first and third terms of (81) vanish, and so (81) is proportional to the metric, i.e e v a is a conformal Killing field. a as given by (77). (b) Suppose wAB satisfies (71), and lift wAB to vea := ve+ Then, using (10), [B C] BC e χA D(a veb) = DA w − 2δA ν ˇb)B pC − (ν C pC )gab (a η D B − χA DA ν C + PAE wEC δB − wEC WEA DB pC pD . (82) (a χb) Since (71) is equivalent to (74), and we assume in addition (80), we e (a veb) = −(ν C pC )gab , i.e. vea is conformal Killing. immediately conclude D


25

a as given by (c) Suppose αA is a solution to (75), and lift αA to vea := ve− (78). Then, using (10),

e (a veb) = (DA αB ) χA χB . D (a b)

By (75), we now conclude that vea is a (conformal) Killing field.

(83)

6.3. Decomposition of conformal Killing fields. Before we proceed, we record the following technical lemma. f. Choose a Patterson– Lemma 6.4. Let vea ∈ Eea be a vector field on M a A a aA Walker metric so that e v = ve ηˇA + α eA χ for some veA and α eA . Then a a Lk ve = 2 r ve for some real constant r if and only if veA and α eA are homogeneous of degree r and r + 1 in pA respectively. In particular,

(a) Lk vea = 0 if and only if e v A and α eA are homogeneous of degree 0 and 1 in pA respectively. (b) Lk vea = 2 vea if and only if veA and α eA are homogeneous of degree 1 and 2 in pA respectively; (c) Lk vea = −2 vea if and only if veA and α eA are homogeneous of degree −1 and 0 in pA respectively;

Proof. This follows from (56), or (79), and (13).

Proposition 6.5. A conformal Killing field vea ∈ Eea can be uniquely decomposed as a a vea = ve+ + ve0a + ve− + c ka

(84)

a e eb − 1 D a a, L v a = ±2 v e ec = e± where Lk ve± k e0 = 0, c is some constant, and µ b Da v 0 0 n cv e [a kb] , with respect to any Patterson–Walker metric. Further, 0 with µab = D a and v a can be expressed as the lifts (76), (77) and (78) respectively, e− ve0a , ve+ where e a kb veb is an infinitesimal projective symmetry, i.e. satis(a) v A = 12 χaA D 0 fies (67). b satisfies (71) together with the integrability condie e+ (b) wAB = 12 χaA χB b Da v tion (80). a satisfies the Killing equation (75). (c) αA = ηˇaA ve−

Proof. We work with a Patterson–Walker metric gab and the relation (48) throughout. Following the argument given in the proof of Proposition 5.5, we first show that for any conformal Killing field vea , Lk (Lk − 2) (Lk + 2) vea = 0 .

(85)

Differentiating (62) once and substituting (64) and (63) yield eaD e b vec = −ga[b βec] − 2Pa[b vec] + ved W fdabc − gbc Pad ved − gbc vea . D

Now, using (56) with w = 0 gives

e b vea + veb µ b − vea . Lk vea = kb D a

(86)

26


eaD e b vec = 0, where we have We note that Lk µab = 0 and using (86), ka kb D a d f made use of the fact that k Wabcd k = 0, and, for a Patterson–Walker e ab kb = 0 — see (31) and (32). Then we compute metric, P e d vec µ c − 2 vec µ c + 2 vea , L3k vea = 4 Lk vea , L2k vea = 2 kd D a a

which is equivalent to (85). The result follows immediately. a +α Next, we write vea = veA ηˇA eA χaA . Then contracting (86) with three vertical fields yields e aD e b vec = χaA χbB χcC D

∂2 veC = 0 , ∂pA pB

i.e.

where wAB and ψ A only depend on xA . Similarly,

veA = wAB pB + ψ A ,

∂3 α eD = 0 , ∂pA pB pC BC α eA = ψA pB pC + ϕB A pB + αA ,

d e e e χaA χbB χcC ηˇD Da Db Dc ved =

i.e.

(BC)

A BC = ψ , ϕB where ψA A and αA only depend on x . A Now, applying Lemma 6.4 gives the following conditions: n−1 B (a) if Lk vea = 0, then veA = ψ A and α eA = ϕB A pB = −φA pB + n+1 ψpA where n+1 C φC C = 0 and ψ = n(n−1) ϕC with factors chosen for later convenience; BC p p ; (b) if Lk vea = 2 vea , then veA = wAB pB and α eA = ψA B C a a A (c) if Lk ve = −2 ve , then ve = 0 and α eA = αA (x); In case (a), we immediately conclude that vea takes the form (76), while in (c) that vea takes the form (78). For case (b), we return to the conformal Killing equation and equation (86), and find e a veb = w(AB) = 0 , χa(A χbB) D i.e. wAB = w[AB] , (A c e e AB χa(A χbB) ηˇC Da Db vec = ψC = −2 δC βea χaB) ,

i.e.

(A

AB ψC = δC ν B) ,

for some ν A , from which it follows that vea takes the form (77). At this stage, we do not know that v A , wAB and αA satisfy (67), (71) and (75) AB and ν A respectively, nor that v A , φB A and ψ are related by (68), and w by (72). a are tangent to the distribua and v e− Next, we note that by Lemma 6.2, ve+ ′ tions U and V annihilated of ηA and χA respectively. Since ka is tangent to a could potentially be of the form f k a for some smooth function both then ve± a = f k a . Then L v a a tells us that f must be f . So suppose that ve± e± k e± = ±2 v a non-constant. But since k is a conformal Killing field, f must necessarily a cannot be proportional to k a . be constant. Hence, ve± a Finally, suppose ve0 = c k a for some constant c. Then Lk ve0a = 0. But e ed = −2 c (n + 1) leads to a contradiction. Hence, e a veb − 1 D computing µab D 0 0 n dv a ve0 cannot be proportional to ka . To conclude the proof, we show that v A , φB A and ψ are related by (68), and wAB and ν A by (72), and that v A , wAB and αA satisfy (67), (71) and (75) respectively. e ec = 0 so that vea = vea given by e a veb − 1 D (a) Suppose Lk vea = 0 and µab D 0 n cv e a veb gives (81) again. Taking the trace-free part of (76). Computing D


27

1 B B C and (70). Now, substituting φB (81) yields φB A = DA v − n δA DC v A into (70) precisely yields (67). Finally,

e a veb − µab D

1e c n−1 Dc ve = DC v C − nψ . n n

Since, by assumption the left-hand side vanishes, we have ψ = n1 DC v C . a given by (77). Computing D e a veb (b) Suppose Lk vea = 2 vea so that vea = ve+ gives (82) again. The trace-free part of (82) vanishes, which is equivalent to DA wBC −

1 1 C B C C DD wBD δA − δA ν + ν B δA = 0, n n

D) D) D(A ν (C + P(A|E wE(C δ|B) − wE(C WE(A B) = 0 .

(87) (88)

1 The symmetric part of (87) yields ν A = n−1 DC wCA and the skewsymmetric part reduces to (71). In particular, wAB satisfies (71). This in turn implies (74), which, substituted into (88), yields 1 (C D) D) EF w WEF (A δB) + wE(C WE(A B) = 0 . 2(n − 2)

Taking the trace shows that both terms vanish separately, as required. a given by (78). Computing D e a veb (c) Suppose Lk vea = −2 vea so that vea = ve− gives (83) again, from which we immediately conclude that αA is Killing.

We end the section with a geometric interpretation of a light-like conformal Killing field vea with Lk vea = 0.

f such that Lk vea = Proposition 6.6. Let vea be a conformal Killing field on M 0 with associated infinitesimal projective symmetry v A as in Proposition 6.5. Then vea is light-like if and only if v B DB v A =

2 (DC v C )v A , n+1

(89)

for any affine connection DA in the projective class on M . In particular, if vea is light-like then v A is geodetic. Proof. We compute the norm of the lift vea = ve0a as defined in Lemma 6.1: n−1 A a A B v vea = 2 e ψ v − φB v pA , n+1

B = where we have used (48) as before. So, vea vea = 0 if and only if φA Bv n−1 A A n+1 ψv . Since v is an infinitesimal projective symmetry, we know that φB ea is light-like if and only if v A satisfies A and ψ are given by (68) so that v (89). This condition in particular implies that v A is geodetic with respect to DA , and thus with respect to the projective structure.

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6.4. Decomposition of Killing fields of Patterson–Walker metrics. We now consider the Patterson–Walker metric g induced by a given affine connection D on M . Let vea be an infinitesimal symmetry of g, i.e. Lveg = 0, which is well-known to be equivalent to the overdetermined equation e (a veb) = 0 . D (90)

Such a field is also known as a Killing field. We want to understand how ve decomposes in terms of objects on the affine structure (M, D) in analogy to Proposition 6.5 and Theorem 3. Before we proceed, we recall the definition of an infinitesimal affine symmetry as a vector field v A that preserves the affine structure, i.e. it satisfies (66) with ΥA = 0. Following [34, 15], one can check that such a vector field satisfies the overdetermined second order equation DA DB v C + v D RDAC B = 0 .

(91)

One can show that (91) is equivalent to the system B DA v B − φB A − δA ψ = 0 ,

D C DA φC B + v RDA B = 0 ,

DA ψ = 0 ,

1 1 C B C B where we have set φB A := DA v − n DC v δA and ψ := n DC v . f: Let us define the following vector fields on M √ 1 a aB ve0a := v A ηˇA − 2 φA − ψka , B ηA χ 2 √ a a , ve+ := 2 wAB ηA ηˇB a ve− := αA χaA ,

(92)

(93) (94) (95)

AB and α are tensor fields on M , with w AB = w [AB] where v A , φB A A , ψ, w C and φC = 0. One then easily checks that an infinitesimal affine symmetry v A , a parallel bivector wAB and a Killing 1-form αA give rise to Killing fields via the lifts (93), (94) and (95) respectively.

Remark 6.7. Had we lifted an infinitesimal affine symmetry v A by means of a is a homothety with D a = 2n2 ψ. e a ve+ (76), we would have discovered that ve+ n+1 n Since ka is a homothety, we can modify (76) by adding the term − n+1 ka to it and thus obtain the Killing field (93). Proposition 6.8. A Killing field vea ∈ Eea can be uniquely decomposed as a a vea = e v0a + ve− + ve+ ,

(96)

a can be expressed a and v a a, L v a = ±2 v e− e0a , ve+ e± where Lk ve± k e0 = 0. Further, v as the lifts (93), (94) and (95) respectively, where e a kb veb is an infinitesimal affine symmetry, i.e. satisfies (a) v A = 21 χaA D 0 (91). b is parallel, i.e. D w AB = 0, and satisfies the e e+ (b) wAB = 12 χaA χB C b Da v D)

integrability condition wB(A RB(C E) = 0. a satisfies the Killing equation (75). (c) αA = ηˇaA ve−

Proof. Since every Killing field of g is in particular a conformal Killing field with respect to the conformal Patterson–Walker metric [g] = c, we can recycle the proof of Proposition 6.5. In particular, we obtain the decomposition


29

(96). Note that unlike in decomposition (84), the homothety ka does not occur in (96) since ka is not a Killing field. Next, following the same reaa and v a take the forms (93), (94) and (95). e− soning, we deduce that ve0a , ve+ The only difference here is the choice of factors in (93). Finally, we compute e (a veb) = 0. When vea = vea , we find D 0 1 1 1 B C B B C A e D(a veb) = DA v − DC v δA − φA χ(a ηˇb)B + DC v − ψ gab n 2 n D C C A B − DA φC B + v RDA B + δB DA ψ pC χ(a χb) , (97)

which tells us that v A is an infinitesimal affine symmetry, as can be checked a , (82) with directly from the defining equations (90) and (91). When vea = e v+ A AB a a ν = 0 implies that w is parallel. When ve = ve− , (83) gives us that αA is Killing. Taken together, we thus obtain Theorem 4.

Remark 6.9. The fact that ka does not occur in (96) allows us to dispense e ec = 0 given in Proposition e a veb − 1 D with the additional requirement µab D 0 0 n cv e a veb − 1 D e ec = 2n 6.5. In fact, if ve0a is given by (93), then µab D 0 0 n cv Finally, we give the analogue of Proposition 6.6.

f such that Lk vea = 0 with Proposition 6.10. Let vea be a Killing field on M A associated infinitesimal affine symmetry v as in Proposition 6.8. Then vea is light-like if and only if v B DB v A = 0, i.e. v A is tangent to affinely parametrised geodesics on M .

Proof. The proof is completely analogous to that of Proposition 6.6: for a B p . The Killing field vea given by (93), we find vea vea = −2 ψ v A + φA A Bv B result follows from the definitions of φA and ψ — see (92). 7. Special cases and further remarks

7.1. Case n = 2. In the special case n = 2, the projective volume form εAB ∈ E[AB] (3) on (M, p), with inverse εAB ∈ E [AB] (−3), allows us to identify E A (−1) with EA (2), and E AB (−2) with E(1). In particular, it is straightforward to check that ξ A ∈ E A (−1) is a solution of the Euler-type equation (52) if and only if αA := ξ B εBA ∈ EA (2) satisfies the Killing equation (75). Similarly, wAB ∈ E [AB] (−2) is a solution of (71) if and only if σ := 12 wAB εAB ∈ E(1) is a Ricci-flat scale, i.e. if it satisfies (51). f, c): any conformal Killing vector This is also reflected at the level of (M a a a field v˜± with Lk v˜± = ±2 v˜± gives rise to an almost Einstein scale σ ˜∓ with Lk σ ˜∓ = ∓˜ σ∓ . Conversely, any such Einstein scale arises in this way. Remark 7.1. Let us assume that M is a two-dimensional surface equipped with Riemannian metric gAB and Levi-Civita covariant derivative DA , and endowed with a Killing field αA . Then DA αB = λ εAB for some λ ∈ C∞ (M ). B and therefore constitutes Then ξ A := (∗α)A = αB εAB satisfies DA ξ B = λ δA a (non-trivial) Euler-type field on the projective surface M with projective class p spanned by D. Clearly, ξ A and αA are orthogonal to each other.

30


This remark applies in particular to any surface of revolution in R3 in which case αA represents the infinitesimal generator of the rotation. 7.2. Case n = 3. In the special case n = 3, the projective volume form εABC ∈ E[ABC] (4) on (M, p), with inverse εABC ∈ E [ABC] (−4), allows us to identify E AB (−2) with EA (2). One can then easily check that wAB ∈ E [AB] (−2) is a solution of (71) satisfying the integrability condition (80) if and only if αA := 12 wBC εBCA ∈ EA (2) satisfies the Killing equation (75), D)

together with the integrability condition αF εF B(A WB(C E) = 0. a with L v a a gives Correspondingly, any conformal Killing vector ve+ e+ k e+ = 2 v a a a rise to a conformal Killing vector ve− with Lk ve− = −2 ve− . The explicit a is a conformal Killing field. form of this relation is as follows. Assume ve+ Since DA εBCD = 0, for any affine connection D ∈ p, then it is clear that B C e ebcd = 0 with respect to the pullback εeabc := χA a χb χc εABC satisfies Da ε a := any Patterson–Walker metric. A short computation then shows that v˜− 1 a eb c e bc D v˜+ is indeed a conformal Killing field. 2ε

7.3. Contact projective structures in odd dimensions. There is a specific class of (odd-dimensional) projective structures on M allowing a compatible contact structure. According to [17], these are the projective structures subordinate to the so-called contact projective structures. It follows that under a curvature condition imposed on the contact projective structure (known as the vanishing of the contact torsion) one obtains a projective structure p on M admitting a Killing 1-form αA . In particular, every projective structure (M, p) determined by a contact projective structure with vanishing contact torsion gives rise to an infinitesimal conformal f, c). symmetry of (M

7.4. Relation to Cartan geometry and tractor calculus. The original oriented projective structure (M, p) can be equivalently described as a Cartan geometry of type (SL(n+1), P ) with P a parabolic subgroup of SL(n+1), f, c) can be equivalently described as a and the conformal spin structure (M Cartan geometry of type (Spin(n+1, n+1), Pe), with Pe a parabolic subgroup, see [8]. This viewpoint was used in [20] to relate the respective geometries (see also [26, 25] for similar Cartan geometric approaches). The formulation in [20] follows the so-called Fefferman-type construction, which is based on a group inclusion SL(n + 1) ֒→ Spin(n + 1, n + 1) of the underlying (Cartan) structure groups. f, c) can also be underThe decomposition of conformal Killing fields of (M f, c) are equivalent to stood in this framework: Conformal Killing fields of (M eω infinitesimal symmetries of the equivalent Cartan geometry (G, e ) and according to [7] those infinitesimal symmetries can be described equivalently f associated to the by sections of the conformal adjoint tractor bundle AM adjoint representation of Spin(n + 1, n + 1) on so(n + 1, n + 1), parallel with respect to a certain connection referred to as the prolongation connection. Likewise, projective infinitesimal symmetries are described as suitable parf, c) is allel sections of the projective adjoint tractor bundle AM . Since (M (locally) induced in a natural way from the projective structure (M, p), the


31

adjoint tractor bundle decomposes naturally according to the decomposition of so(n + 1, n + 1) into its SL(n + 1)-irreducible components R ⊕ sl(n + 1) ⊕ Λ2 Rn+1 ⊕ Λ2 (Rn+1 )∗ . Decomposing an infinitesimal symmetry into its constituents with respect to this decomposition and reinterpreting the resulting sections on the original projective structure (M, p) gives an alternative (algebraic) approach to derive Theorem 3. Let us illustrate this formalism within the general approach of the present f∼ article. A choice of metric g in c splits the adjoint tractor bundle as AM = e e e e Ea [2]⊕ Eab [2]⊕ E ⊕ Ea . Similarly, a choice of torsion-free affine connection D in p splits the projective adjoint tractor bundle, which is associated to sl(n+1) A ⊕ E A ⊕ E ⊕ E . A conformal Killing field v E ea can then be as AM ∼ = B B e βea ), where φeab , ψe and βea were defined e = (e expressed as a section Σ va , φeab , ψ, at the beginning of section 6. The defining equation (62) together with its e prolongation (63), (64) and (65) then can be understood equivalently as Σ f. Similarly, being parallel with respect to the prolongation connection on AM A an infinitesimal projective symmetry v can be expressed as a section Σ = A (v A , φA B , ψ, βA ), where φB , ψ and βA were defined at the beginning of section 6.1. The defining equation (67) together with its prolongation (70) can be interpreted as Σ being parallel with respect to the prolongation connection e and Σ is given in terms of the lift vã of on AM . The relation between Σ 0 Lemma 6.1. An analogous approach can be employed to describe almost f, c) in terms of parallel sections of the standard tractor Einstein scales on (M bundle and relate them to projective data. This Cartan geometric approach to relate particular overdetermined equations on the respective projective and conformal structures will be the subject of forthcoming work [21]. References [1] Z. Afifi. Riemann extensions of affine connected spaces. Quart. J. Math., Oxford Ser. (2), 5:312–320, 1954. [2] T.N. Bailey, M. Eastwood, and A.R. Gover. Thomas’s structure bundle for conformal, projective and related structures. Rocky Mountain J. Math., 24(4):1191–1217, 1994. [3] T. Branson. Conformal structure and spin geometry. In Dirac operators: yesterday and today, pages 163–191. Int. Press, Somerville, MA, 2005. [4] R. L. Bryant. Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor. In Global analysis and harmonic analysis (Marseille-Luminy, 1999), volume 4 of Sémin. Congr., pages 53–94. Soc. Math. France, Paris, 2000. [5] R.L. Bryant. Recent advances in the theory of holonomy. Astérisque, (266):Exp. No. 861, 5, 351–374, 2000. Séminaire Bourbaki, Vol. 1998/99. [6] R. Bryant, M. Dunajski. M. Eastwood, Metrisability of two-dimensional projective structures. J. Differential Geom., 83(3):465–499, 2009. ˇ [7] A. Cap. Infinitesimal automorphisms and deformations of parabolic geometries. J. Eur. Math. Soc. (JEMS), 10(2):415–437, 2008. ˇ [8] A. Cap and J. Slov´ ak. Parabolic Geometries I: Background and General Theory. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2009. [9] A. Derdzinski. Noncompactness and maximum mobility of type III Ricci-flat self-dual neutral Walker four-manifolds. Q. J. Math., 62(2):363–395, 2011.

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M. H.: University of Greifswald, Department of Mathematics and Informatics, Walther-Rathenau-Str. 47, 17489 Greifswald, Germany K. S.: INdAM-Politecnico di Torino, Dipartimento di Scienze Matematiche, corso Duca degli Abruzzi 24, 10129 Torino, Italy ˇ Masaryk University, Faculty of Science, Kotla ´r ˇska ´ 2, 61137 Brno, Czech J. S.: Republic ` di Torino, Dipartimento di Matematica “G. Peano”, Via A. T.-C.: Universita Carlo Alberto, 10 - 10123, Torino, Italy ˇ Masaryk University, Faculty of Education, Por ˇ´ıc ˇ´ı 31, 60300 Brno, Czech V. Z: Republic E-mail address: [email protected], [email protected], [email protected], [email protected], [email protected]