1. DERIVING THE EQUATION FOR RICCI-FLATNESS

We consider the rank 2 distribution 𝒟φ(x,q) on M_xyzpq associated to the underdetermined differential equation z′ = φ(x, y″) where φ(x,y″)=(y″)2H″(x) and H″(x) is a non-zero function of x. This is to say that the distribution 𝒟φ(x,q) is annihilated by the three 1-forms

where φ(x,q)=q2H″(x) . These three 1-forms are completed to a coframing on M_xyzpq by the additional 1-forms

Taking appropriate linear combinations, we let

with and

Imposing Cartan’s structure equations (0.1) on (θ₁, θ₂, θ₃, θ₄, θ₅) then gives the constraints a₅₁ = a₅₃ = 0 and a₄₁ = a₅₂, which we can set both to be zero, and we also find a42=130223(3H″H(4)-5(H(3))2)(H″)83 and a43=-213H(3)3(H″)43 . We obtain the 1-forms (θ₁, θ₂, θ₃, θ₄, θ₅) that give a coframing for a metric in Nurowski’s conformal class [11], related to the 1-forms (ω₁, ω₂, ω₃, ω₄, ω₅) as follows:

The metric g=2θ1θ5-2θ2θ4+43θ3θ3 is conformally flat, i.e. the metric g has vanishing Weyl tensor if and only if H(x) is a solution to the 6th-order nonlinear differential equation

This equation is called Noth’s equation [3]. In this case the distribution of the form 𝒟φ(x,q) is maximally symmetric and in the paper we will concern ourselves with the problem of finding Ricci-flat representatives in the conformal class of metrics associated to this distribution.

The explicit form of the metric given by the distribution 𝒟φ(x,q) is as follows. If we replace H″(x)=e∫23P(x)dx , then we find that equation (1.1) reduces to the k=32 generalised Chazy equation

and we find that the conformally rescaled metric g˜=2-23(H″)23g has the form

We can reexpress this metric as

By defining the new coframes

and making the further substitution Q = P² − 6P′, we get the following cosmetic improvement for g˜ :

From this we can rescale the metric g˜ further by a conformal factor Ω to obtain a Ricci-flat representative. When Ric(Ω2g˜)=0 , we say that Ω2g˜ is a Ricci-flat representative of Nurowski’s conformal class. We find that Ω2g˜ is Ricci-flat when Ω satisfies the second-order differential equation

We make the substitution Ω=1ρe-13∫Pdx to obtain

where ρ(x) is to be determined.

The function H(x) is related to another function F(x˜) by a Legendre transformation [3], [12]. We say that F(x˜) is the Legendre dual of H(x) determined by the relation H(x)+F(x˜)=xx˜ . This implies x˜=H′(x) with dx˜=H″dx and H″=1Fx˜x˜ . We can make use of this transformation to write dx=Fx˜x˜dx˜ . The Legendre dual of the distribution 𝒟φ(x,q) is therefore given by the annihilator of the three 1-forms

on the mixed jet space with local coordinates (x˜,y,z,p,q) . Relabelling x˜ with x, we have

Here F now becomes a function of x. These three 1-forms are completed to a coframing on M with local coordinates (x, y, z, p, q) by the additional 1-forms

(These are the Legendre transformed 1-forms ω₄ and ω₅). Similar as before, we consider the linear combinations with and

Imposing Cartan’s structure equations (0.1) on (θ1,θ2,θ3,θ4,θ5) once again gives b₅₁ = b₅₃ = 0 and b₄₁ = b₅₂, which we set to be zero. We also have

We obtain the 1-forms (θ₁, θ₂, θ₃, θ₄, θ₅) that give a coframing for a metric in Nurowski’s conformal class [11], related to the 1-forms (ω₁, ω₂, ω₃, ω₄, ω₅) as follows:

A representative metric of Nurowski’s conformal class is again given by (0.2). The condition that the metric g is conformally flat, i.e. the metric g has vanishing Weyl tensor, occurs when F(x) is a solution to the nonlinear differential equation

If we replace F″(x)=e∫12P(x)dx , then we find that the conformally rescaled metric g˜=213(F″)-23g has the form

Here equations (1.3) is reduced to the generalised Chazy equation

for P(x) with parameter k=23 . From the form of the metric g˜ we can locally rescale the metric again by a conformal factor to obtain Ricci-flat representatives.

We find that the Ricci tensor of Ω2g˜ is zero when Ω satisfies

If we make the substitution Ω=1η , then we obtain the differential equation

where Q = P² − 6P′ and η is to be determined. From the form of the metric g˜ in (1.4), we can also define new coframes by

We have used that e-∫P2dx=1F″ . Also replacing 6P′ − P² = −Q, this gives the cosmetic improvement for g˜ :

We now investigate the solutions to (1.2) and (1.5). They are given by Theorems 3.1 and 3.2. We first review some results about the solutions to the generalised Chazy equation.

2. GENERALISED CHAZY EQUATION

The generalised Chazy equation with parameter k is given by

and Chazy’s equation is obtained in the limit as k tends to infinity. The generalised Chazy equation was introduced in [6], [7] and studied more recently in [8], [1], [2] and [4]. The generalised Chazy equation with parameters k=23, 32, 2 and 3 was also further investigated in [13]. The solution to the generalised Chazy equation is given as follows (see also Table 2 in Section 3.3 of [4] and Proposition 2.2 of [13]). Let where s = s (α, β, γ, x) is a solution to the Schwarzian differential equation and is the Schwarzian derivative with the potential V given by

The combination y = −2w₁ − 2w₂ − 2w₃ solves the generalised Chazy equation when

This combination corresponds to cases 1(b) and 3(b) of Table 2 in [4]. The combination y = −w₁ − 2w₂ − 3w₃ solves the generalised Chazy equation when

with permutations of w₁, w₂ and w₃ in y corresponding to permutations of the values α, β and γ in (α, β, γ ). This combination corresponds to cases 1(a), 2(a) and 2(b) of Table 2 in [4]. The combination y = −w₁ − w₂ − 4w₃ solves the generalised Chazy equation whenever again permuting w₁, w₂ and w₃ in y corresponds to permuting the values α, β, γ in (α, β, γ ). This combination corresponds to cases 2(c) and 3(a) of Table 2 in [4]. Following [1], the functions w₁, w₂ and w₃ satisfy the following system of differential equations: where

The second-order differential equation associated to the generalised Chazy equation with parameter k is given by

with the same potential V as given in (2.2) and (α, β, γ) is one of the triples in (2.3), (2.4) or (2.5). The equation (2.7) corresponds to the general solution of the Schwarzian differential equation (2.1) after interchanging dependent and independent variables [8]. In this case x=u2u1 where u₁ and u₂ are linearly independent solutions to (2.7). Using the further substitution u(s)=(s-1)1-γ2s1-β2z(s) , the equation (2.7) can be brought to the hypergeometric differential equation with

From the differential equations (2.6), we can recover s by s=w1-w3w2-w3 . From this we deduce s′ = 2(w₁ − w₂)s and we also obtain the relation ds = 2(w₁−w₂)sdx.

3. MAIN RESULTS: SOLVING THE EQUATIONS FOR RICCI-FLATNESS

In this section we give the general solution to the differential equation (1.2) where Q = P² – 6P′ and P is a solution of the k=32 generalised Chazy equation in Theorem 3.1 and the general solution to the differential equation (1.5) where again Q = P² – 6P′ and P is a solution of the k=23 generalised Chazy equation in Theorem 3.2. We first prove the following theorem.

Upon making the substitution ρ=u(x)v(x) into equation (1.2), and using equation (3.3), we obtain a differential equation for u(x) of the form

which is the same differential equation for v with different constants α˜ , β˜ , γ˜ . We claim that this is the differential equation uss+14V˜u=0 associated to the generalised Chazy equation with parameter k = 3, with and (α˜,β˜,γ˜) is one of the triples in (2.3), (2.4) or (2.5) with k = 3. We compute the triples (α˜,β˜,γ˜) when Q = P² − 6P′ and P is the solution of the generalised Chazy equation with parameter k=32 . Fixing (α, β, γ ) to be one of the triples in (2.3), (2.4) or (2.5) determines the values α˜ , β˜ , γ˜ up to sign. Specialising to the case where k=32 , we obtain the following:

For the solutions given by P = −2w₁−2w₂−2w₃, when (α,β,γ)=(43,43,43) , we find (α˜,β˜,γ˜)=(23,23,23) . When (α,β,γ)=(43,13,13) , we find (α˜,β˜,γ˜)=(23,13,13) .

For the solutions given by P = −w₁−2w₂−3w₃, when (α,β,γ)=(23,13,12) , we find (α˜,β˜,γ˜)=(13,13,12) . When (α,β,γ)=(23,43,12) , we find (α˜,β˜,γ˜)=(13,23,12) . When (α,β,γ)=(23,13,2) , we find (α˜,β˜,γ˜)=(13,13,1) .

Finally, for the solutions given by P = −4w₁ − w₂ − w₃, when (α,β,γ)=(23,23,23) , we find (α˜,β˜,γ˜)=(23,13,13) . When (α,β,γ)=(83,23,23) , we find (α˜,β˜,γ˜)=(43,13,13) .

The values of (α˜,β˜,γ˜) are precisely the triples (2.3), (2.4) or (2.5) that show up in the solutions of the k = 3 generalised Chazy equation. See [13] for the list of (α˜,β˜,γ˜) when k = 3.

The determination of solutions to equation (1.5) is similar to that of Theorem 3.1.

Upon making the substitution η=u(x)v(x) into equation (3.4), and using equation (3.6), we obtain a differential equation for u(x) again given by

which is the same differential equation for v but with different constants α˜ , β˜ , γ˜ . Equation (3.7) corresponds to the second-order differential equation uss+14V˜u=0 associated to the k = 2 generalised Chazy equation, with and (α˜,β˜,γ˜) is one of the triples in (2.3), (2.4) or (2.5) with k = 2. To see this, we shall compute these constants when Q = P²−6P′ and P is the solution of the generalised Chazy equation with parameter k=23 . Specialising to the case where k=23 , we obtain the following:

For the solutions given by P = −2w₁−2w₂−2w₃, when (α, β, γ ) = (3, 3, 3), we find (α˜,β˜,γ˜)=(1,1,1) . When (α,β,γ)=(3,13,13) , we find (α˜,β˜,γ˜)=(1,13,13) .

For the solutions given by P=-w1-2w2-3w3 , when (α,β,γ)=(32,13,12) , we find (α˜,β˜,γ˜)=(12,13,12) . When (α,β,γ)=(32,3,12) , we find (α˜,β˜,γ˜)=(12,1,12) . When (α,β,γ)=(32,13,92) , we find (α˜,β˜,γ˜)=(12,13,32) .

Finally, for the solutions given by P = −4w₁−w₂−w₃, when (α,β,γ)=(23,32,32) , we find (α˜,β˜,γ˜)=(23,12,12) . When (α,β,γ)=(6,32,32) , we find (α˜,β˜,γ˜)=(2,12,12) .

The values of (α˜,β˜,γ˜) are again precisely the triples (2.3), (2.4) or (2.5) that show up in the solutions of the k = 2 generalised Chazy equation. See also [13] for the list of (α˜,β˜,γ˜) when k = 2.

4. SOLUTION TO THE EQUATION FOR RICCI-FLATNESS FOR GENERAL CHAZY PARAMETER

More generally, when P is a solution to the generalised Chazy equation with parameter k, the metric g is no longer conformally flat but we can still find the conformal scale for which the Ricci tensor vanishes.

In the case of (1.2) with solutions given by ρ=uv where v is the second-order differential equation associated to the generalised Chazy equation with parameter k, we find that u is a solution to the second-order differential equation associated to the generalised Chazy equation with parameter k˜ with

The values (α, β, γ) appearing in V in the differential equation vss+14Vv=0 are related to the values (α˜,β˜,γ˜) appearing in V˜ in the differential equation uss+14V˜u=0 by the following. For the solutions given by P = −2w₁−2w₂−2w₃, when (α,β,γ)=(2k,2k,2k) , we find (α˜,β˜,γ˜) with

When (α,β,γ)=(2k,13,13) , we find (α˜,β˜,γ˜) with

and β˜=13 , γ˜=13 . Here and subsequently, we consider the positive square root that gives positive α˜ , β˜ and γ˜ .

For the solutions given by P = −w₁−2w₂−3w₃, when (α,β,γ)=(1k,13,12) , we find (α˜,β˜,γ˜) with

When (α,β,γ)=(1k,2k,12) , we find (α˜,β˜,γ˜) with

When (α,β,γ)=(1k,13,3k) , we find (α˜,β˜,γ˜) with

Finally for the solutions given by P = −4w₁−w₂−w₃, when (α,β,γ)=(4k,1k,1k) , we find (α˜,β˜,γ˜) with

When (α,β,γ)=(23,1k,1k) , we find (α˜,β˜,γ˜) with

In all cases the appropriate substitution of α˜ , β˜ and γ˜ in terms of the Chazy parameter k˜ gives equation (4.1), so it can be seen that the equation for u is the second-order differential equation associated to the generalised Chazy equation with parameter k˜ , related to k by (4.1). The further substitution k=3m and k˜=3m˜ into (4.1) gives

which has integer solutions when considered as a negative Pell equation. For integer solutions m and m˜ we obtain

They take on values (m,m˜) = (2, 1), (38, 17), (682, 305), (12238, 5473) and so on for n∈𝕅∪{0} . They also give the corresponding pairs of Chazy parameters (k,k˜)=(32,3), (338,317) and so on, with the fundamental solution (n = 0) agreeing with the result of Theorem 3.1 in the conformally flat case.

In the case of (1.5) with solutions given by η=uv where v is the second-order differential equation associated to the generalised Chazy equation with parameter k, we find that u is a solution to the second-order differential equation associated to the generalised Chazy equation with parameter k˜ with

In this case we obtain the relationship between the values (α, β, γ ) and (α˜,β˜,γ˜) as follows. For P = −2w₁−2w₂−2w₃, when (α,β,γ)=(2k,2k,2k) , we find (α˜,β˜,γ˜) with

Considering integer solutions α and α˜ to the negative Pell equation 10α˜2-α2=1 (and also β, β˜ and γ, γ˜ respectively), we find

where n∈𝕑 . Positive integer solutions are given by (α,α˜) = (3, 1), (117, 37), (4443, 1405), (168717, 53353) and so on for n∈𝕅∪{0} . They give the relationship between the pairs of Chazy parameters k=2α and k˜=2α˜ , with (k,k˜)=(23,2), (2117,237) and so on for n∈𝕅∪{0} . For these parameters, the associated hypergeometric functions are algebraic. Again the fundamental solution (n = 0) agrees with the result of Theorem 3.2 in the conformally flat case.

The determination of the other values of (α, β, γ) and (α˜,β˜,γ˜) is as follows. For the same P, when (α,β,γ)=(2k,13,13) , we find (α˜,β˜,γ˜) with

and β˜=13 , γ˜=13 .

For the solutions given by P = −w₁−2w₂−3w₃, when (α,β,γ)=(1k,13,12) , we find (α˜,β˜,γ˜) with

When (α,β,γ)=(1k,2k,12) , we find (α˜,β˜,γ˜) with When (α,β,γ)=(1k,13,3k) , we find (α˜,β˜,γ˜) with

Finally, for the solutions given by P = −4w₁ − w₂ − w₃, when (α,β,γ)=(4k,1k,1k) , we find (α˜,β˜,γ˜) with

When (α,β,γ)=(23,1k,1k) , we find (α˜,β˜,γ˜) with

In all cases the appropriate substitution of α˜ , β˜ and γ˜ in terms of the Chazy parameter k˜ gives equation (4.2), and therefore the equation for u is the second-order differential equation associated to the generalised Chazy equation with parameter k˜ , related to k by (4.2). Altogether, with the exception of the parameters k=32 and k=23 as mentioned above, they give Ricci-flat but non-conformally flat examples of Nurowski’s metric.