Nurowski's conformal class of a maximally symmetric (2,3,5)-distribution and its Ricci-flat representatives

TL;DR

This paper demonstrates how solutions to specific differential equations related to the generalized Chazy equation naturally appear in the process of conformally rescaling metrics within Nurowski's class, leading to Ricci-flat representatives.

Contribution

It establishes a connection between solutions of the generalized Chazy equation and Ricci-flat metrics in Nurowski's conformal class for maximally symmetric (2,3,5)-distributions.

Findings

Solutions to the generalized Chazy equation appear in conformal rescaling.

Conformal rescaling can produce Ricci-flat metrics within the class.

The work links differential equations to geometric structures in conformal geometry.

Abstract

We show that the solutions to the second-order differential equation associated to the generalised Chazy equation with parameters $k=2$ and $k=3$ naturally show up in the conformal rescaling that takes a representative metric in Nurowski's conformal class associated to a maximally symmetric $(2,3,5)$-distribution (described locally by a certain function $\varphi(x,q)=\frac{q^2}{H''(x)}$) to a Ricci-flat one.

Equations

$$\mathcal{D}\subset [\mathcal{D},\mathcal{D}]\subset [\mathcal{D},[\mathcal{D},\mathcal{D}]]\cong TM.$$

$${\omega }_1=\mathrm{d}y-p\mathrm{d}x,{\omega }_2=\mathrm{d}p-q\mathrm{d}x,{\omega }_3=\mathrm{d}z-\phi (x,y,z,p,q)\mathrm{d}x.$$

$$\mathrm{d}{\theta }_1$$

$$\mathrm{d}{\theta }_2$$

$$\mathrm{d}{\theta }_3$$

$$\mathrm{d}{\theta }_4$$

$$\mathrm{d}{\theta }_5$$

$$g=2{\theta }_1{\theta }_5-2{\theta }_2{\theta }_4+\frac{4}{3}{\theta }_3{\theta }_3.$$

$${\omega }_1=$$

$${\omega }_2=$$

$${\omega }_3=$$

$${\omega }_4=\mathrm{d}q-\frac{H^{(3)}}{H^{\prime \prime }}q\mathrm{d}x,{\omega }_5=-\frac{H^{\prime \prime }}{2}\mathrm{d}x.$$

$${\theta }_1=$$

$${\theta }_4={\left(\frac{2}{H^{\prime \prime }}\right)}^{\frac{2}{3}}{\omega }_4+a_{41}{\theta }_1+a_{42}{\theta }_2+a_{43}{\theta }_3$$

$${\theta }_5={\left(\frac{2}{H^{\prime \prime }}\right)}^{\frac{2}{3}}{\omega }_5+a_{51}{\theta }_1+a_{52}{\theta }_2+a_{53}{\theta }_3.$$

$$10(H^{\prime \prime }{)}^3{H}^{(6)}-70(H^{\prime \prime }{)}^2{H}^{(3)}{H}^{(5)}-49(H^{\prime \prime }{)}^2(H^{(4)}{)}^2+280H^{\prime \prime }(H^{(3)}{)}^2{H}^{(4)}-175(H^{(3)}{)}^4=0.$$

$$P^{\prime \prime \prime }-2P{P}^{\prime \prime }+3P^{\prime 2}-\frac{4}{36-(\frac{3}{2}{)}^2}(6P^{\prime }-P^2{)}^2=0,$$

$$\tilde{g}=-\frac{2}{15}(P^{\prime }-\frac{4}{9}{P}^2){\omega }_1{\omega }_1+\frac{4}{9}P{\omega }_1{\omega }_2+\frac{4}{3}{\omega }_2{\omega }_2+2{\omega }_3{\omega }_5-2{\omega }_1{\omega }_4-4q{e}^{\int -\frac{2}{3}P\mathrm{d}x}{\omega }_2{\omega }_5.$$

$$\tilde{g}$$

$${\tilde{\omega }}_3$$

$${\tilde{\omega }}_5$$

$$\tilde{g}$$

$${\Omega }^{\prime \prime }\Omega -2({\Omega }^{\prime }{)}^2-\frac{2}{3}P\Omega {\Omega }^{\prime }-\frac{1}{18}{P}^2{\Omega }^2-\frac{1}{30}Q{\Omega }^2=0.$$

$${\rho }^{\prime \prime }-\frac{1}{45}Q\rho =0,$$

$${\omega }_1=$$

$${\omega }_2=$$

$${\omega }_3=$$

$${\omega }_1=$$

$${\omega }_2=$$

$${\omega }_3=$$

$${\omega }_4=\mathrm{d}q+\frac{F^{\prime \prime \prime }}{F^{\prime \prime }}q\mathrm{d}x,{\omega }_5=-\frac{1}{2}\mathrm{d}x.$$

$${\theta }_1={\omega }_3-2F^{\prime \prime }q{\omega }_2,{\theta }_2={\omega }_1,{\theta }_3=(2F^{\prime \prime }{)}^{\frac{1}{3}}{\omega }_2,$$

$${\theta }_4=(2F^{\prime \prime }{)}^{\frac{2}{3}}{\omega }_4+b_{41}{\theta }_1+b_{42}{\theta }_2+b_{43}{\theta }_3$$

$${\theta }_5=(2F^{\prime \prime }{)}^{\frac{2}{3}}{\omega }_5+b_{51}{\theta }_1+b_{52}{\theta }_2+b_{53}{\theta }_3.$$

$$10(F^{\prime \prime }{)}^3{F}^{(6)}-80(F^{\prime \prime }{)}^2{F}^{(3)}{F}^{(5)}-51(F^{\prime \prime }{)}^2(F^{(4)}{)}^2+336F^{\prime \prime }(F^{(3)}{)}^2{F}^{(4)}-224(F^{(3)}{)}^4=0.$$

$$\tilde{g}=\frac{1}{30}(6P^{\prime }-P^2)e^{\int -P\mathrm{d}x}{\omega }_1{\omega }_1-\frac{2}{3}P{e}^{-\int \frac{1}{2}P\mathrm{d}x}{\omega }_1{\omega }_2+\frac{8}{3}{\omega }_2{\omega }_2+4{\omega }_3{\omega }_5-4{\omega }_1{\omega }_4-8q{e}^{\int \frac{1}{2}P\mathrm{d}x}{\omega }_2{\omega }_5.$$

Full text

The article concerns the occurrence of the $k=2$ and $k=3$ generalised Chazy equation in a geometric setting, closely connected to the occurrence of the solutions of the generalised Chazy equation with parameters $k=\frac{2}{3}$ and $k=\frac{3}{2}$ respectively. We first discuss the set-up in which the differential equations will appear. This concerns the theory of maximally non-integrable rank 2 distribution $\mathcal{D}$ on a 5-manifold $M$. The maximally non-integrable condition of $\mathcal{D}$ determines a filtration of the tangent bundle $TM$ given by

[TABLE]

The distribution $[\mathcal{D},\mathcal{D}]$ has rank 3 while the full tangent space $TM$ has rank 5, hence such a geometry is also known as a $(2,3,5)$-distribution. Let $M_{xyzpq}$ denote the 5-dimensional mixed order jet space $J^{2,0}(\mathbb{R},\mathbb{R}^{2})\cong J^{2}(\mathbb{R},\mathbb{R})\times\mathbb{R}$ with local coordinates given by $(x,y,z,p,q)=(x,y,z,y^{\prime},y^{\prime\prime})$ (see also [14], [15]). Let $\mathcal{D}_{\varphi(x,y,z,y^{\prime},y^{\prime\prime})}$ denote the maximally non-integrable rank 2 distribution on $M_{xyzpq}$ associated to the underdetermined differential equation $z^{\prime}=\varphi(x,y,z,y^{\prime},y^{\prime\prime})$. This means that the distribution is annihilated by the following three 1-forms

[TABLE]

In [10], it is shown how to associate canonically to such a (2,3,5)-distribution a conformal class of metrics of split signature $(2,3)$ (henceforth known as Nurowski’s conformal structure or Nurowski’s conformal metrics) such that the rank 2 distribution is isotropic with respect to any metric in the conformal class. The method of equivalence [3], [5], [10], [13], [9] gives the coframing for Nurowski’s metric. The 1-forms in the coframe satisfy the structure equations

[TABLE]

A representative metric in Nurowski’s conformal class [10] is given by

[TABLE]

When $g$ has vanishing Weyl tensor, the distribution is called maximally symmetric and has split $G_{2}$ as its group of local symmetries.

The historic example is the case where $\varphi$ is given by $\varphi(x,y,z,p,q)=q^{2}$. When $\varphi(x,y,z,p,q)=q^{2}$, we obtain the Hilbert-Cartan distribution associated to the Hilbert-Cartan equation $z^{\prime}=(y^{\prime\prime})^{2}$. When $\varphi(x,y,z,p,q)=q^{m}$, we obtain the distribution associated to the equation $z^{\prime}=(y^{\prime\prime})^{m}$. For such distributions, Nurowski’s metric [10] given by (0.2) has vanishing Weyl tensor precisely when $m\in\{-1,\frac{1}{3},\frac{2}{3},2\}$. In these cases the maximally symmetric distributions are all locally diffeomorphic to the Hilbert-Cartan or flat model obtained when $m=2$.

In this article, we consider distributions here of the form $\varphi(x,y,z,p,q)=\frac{q^{2}}{H^{\prime\prime}(x)}$. The Weyl tensor vanishes in the case where $H(x)$ satisfies the 6th-order ordinary differential equation (ODE) known as Noth’s equation [3]. For such maximally symmetric distributions we find the corresponding Ricci-flat representatives in Nurowki’s conformal class. This involves solving a second-order differential equation (see Proposition 35 of [14]) to find the conformal scale in which the Ricci tensor of the conformally rescaled metric vanishes, which turns out to be related to the solutions of Noth’s equation. The 6th-order ODE can be reduced to the generalised Chazy equation with parameter $k=\frac{3}{2}$ and its Legendre dual is another 6th-order ODE that can be reduced to the generalised Chazy equation with parameter $k=\frac{2}{3}$. We find the second-order differential equation that determines the conformal scale for Ricci-flatness involves solutions of the generalised Chazy equation with parameter $k=3$ and in the dual case $k=2$. This is the content of Theorems 3.1 and 3.2. We also give few remarks concerning the case for other parameters of $k$.

The aim of finding Ricci-flat representatives is motivated by the consideration that in the Ricci-flat, conformally flat case, we might be able to integrate the structure equations and reexpress them in Monge normal form to obtain the Hilbert-Cartan distribution. This is possible for the distributions of the form $\varphi(x,y,z,p,q)=q^{m}$, with $m\in\{-1,\frac{1}{3},\frac{2}{3},2\}$, but would require further investigations in the general setting.

The computations here are done using the indispensable DifferentialGeometry package in Maple 2018.

1. Deriving the equation for Ricci-flatness

We shall consider the rank 2 distribution $\mathcal{D}_{\varphi(x,q)}$ on $M_{xyzpq}$ associated to the underdetermined differential equation $z^{\prime}=\varphi(x,y^{\prime\prime})$ where $\varphi(x,y^{\prime\prime})=\frac{(y^{\prime\prime})^{2}}{H^{\prime\prime}(x)}$ and $H^{\prime\prime}(x)$ is a non-zero function of $x$. This is to say that the distribution $\mathcal{D}_{\varphi(x,q)}$ is annihilated by the three 1-forms

[TABLE]

where $\varphi(x,q)=\frac{q^{2}}{H^{\prime\prime}(x)}$. These three 1-forms are completed to a coframing on $M_{xyzpq}$ by the additional 1-forms

[TABLE]

Taking appropriate linear combinations, we let

[TABLE]

with

[TABLE]

and

[TABLE]

Imposing Cartan’s structure equations (Nurowski’s conformal class of a maximally symmetric $(2,3,5)$-distribution and its Ricci-flat representatives) on $(\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5})$ then gives the constraints $a_{51}=a_{53}=0$ and $a_{41}=a_{52}$, which we can set both to be zero, and we also find $a_{42}=\frac{1}{30}\frac{2^{\frac{2}{3}}(3H^{\prime\prime}H^{(4)}-5(H^{(3)})^{2})}{(H^{\prime\prime})^{\frac{8}{3}}}$ and $a_{43}=-\frac{2^{\frac{1}{3}}H^{(3)}}{3(H^{\prime\prime})^{\frac{4}{3}}}$. The metric $g=2\theta_{1}\theta_{5}-2\theta_{2}\theta_{4}+\frac{4}{3}\theta_{3}\theta_{3}$ 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

[TABLE]

This equation is called Noth’s equation [3]. In this case the distribution of the form $\mathcal{D}_{\varphi(x,q)}$ 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 $\mathcal{D}_{\varphi(x,q)}$ is as follows. If we replace $H^{\prime\prime}(x)=e^{\int\frac{2}{3}P(x){\rm d}x}$, we find that equation (1.1) reduces to the $k=\frac{3}{2}$ generalised Chazy equation

[TABLE]

and we find that the conformally rescaled metric $\tilde{g}=2^{-\frac{2}{3}}(H^{\prime\prime})^{\frac{2}{3}}g$ has the form

[TABLE]

We can reexpress this metric as

[TABLE]

By defining the new coframes

[TABLE]

and making the further substitution $Q=P^{2}-6P^{\prime}$, we get the following cosmetic improvement for $\tilde{g}$:

[TABLE]

From this we can rescale the metric $\tilde{g}$ further by a conformal factor $\Omega$ to obtain a Ricci-flat representative. When ${\rm Ric}(\Omega^{2}\tilde{g})=0$, we say that $\Omega^{2}\tilde{g}$ is a Ricci-flat representative of Nurowski’s conformal class. We find that $\Omega^{2}\tilde{g}$ is Ricci-flat when $\Omega$ satisfies the second-order differential equation

[TABLE]

We make the substitution $\Omega=\frac{1}{\rho}e^{-\frac{1}{3}\int P{\rm d}x}$ to obtain

[TABLE]

where $\rho(x)$ is to be determined.

The function $H(x)$ is related to another function $F(\tilde{x})$ by a Legendre transformation [3], [11]. We say that $F(\tilde{x})$ is the Legendre dual of $H(x)$ determined by the relation $H(x)+F(\tilde{x})=x\tilde{x}$. This implies $\tilde{x}=H^{\prime}(x)$ with ${\rm d}\tilde{x}=H^{\prime\prime}{\rm d}x$ and $H^{\prime\prime}=\frac{1}{F_{\tilde{x}\tilde{x}}}$. We can make use of this transformation to write ${\rm d}x=F_{\tilde{x}\tilde{x}}{\rm d}\tilde{x}$. The Legendre dual of the distribution $\mathcal{D}_{\varphi(x,q)}$ is therefore given by the annihilator of the three 1-forms

[TABLE]

on the mixed jet space with local coordinates $(\tilde{x},y,z,p,q)$. Relabelling $\tilde{x}$ with $x$, we have

[TABLE]

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

[TABLE]

(These are the Legendre transformed 1-forms $\omega_{4}$ and $\omega_{5}$). Similar as before, we consider the linear combinations

[TABLE]

with

[TABLE]

and

[TABLE]

Imposing Cartan’s structure equations (Nurowski’s conformal class of a maximally symmetric $(2,3,5)$-distribution and its Ricci-flat representatives) on $(\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5})$ again gives $b_{51}=b_{53}=0$ and $b_{41}=b_{52}$, which we set to be zero. We also obtain $b_{42}=-\frac{1}{30}\frac{2^{\frac{2}{3}}(3F^{\prime\prime}F^{(4)}-4(F^{(3)})^{2})}{(F^{\prime\prime})^{\frac{10}{3}}}$ and $b_{43}=\frac{2^{\frac{1}{3}}F^{(3)}}{3(F^{\prime\prime})^{\frac{5}{3}}}$. 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

[TABLE]

If we replace $F^{\prime\prime}(x)=e^{\int\frac{1}{2}P(x){\rm d}x}$, we find that the conformally rescaled metric $\tilde{g}=2^{\frac{1}{3}}(F^{\prime\prime})^{-\frac{2}{3}}g$ has the form

[TABLE]

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

[TABLE]

for $P(x)$ with parameter $k=\frac{2}{3}$. From the form of the metric $\tilde{g}$ we can locally rescale the metric again by a conformal factor to obtain Ricci-flat representatives.

We find that the Ricci tensor of $\Omega^{2}\tilde{g}$ is zero when $\Omega$ satisfies

[TABLE]

If we make the substitution $\Omega=\frac{1}{\nu}$, then we obtain the differential equation

[TABLE]

where $Q=P^{2}-6P^{\prime}$ and $\nu$ is to be determined. From the form the metric $\tilde{g}$ in (1.4), we can also define new coframes by

[TABLE]

We have used that $e^{-\int\frac{P}{2}{\rm d}x}=\frac{1}{F^{\prime\prime}}$. Also replacing $6P^{\prime}-P^{2}=-Q$, this gives the cosmetic improvement for $\tilde{g}$:

[TABLE]

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

[TABLE]

and Chazy’s equation

[TABLE]

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 $k=\frac{2}{3}$, $\frac{3}{2}$, $2$ and $3$ was also further investigated in [12]. The solution to the generalised Chazy equation is given by the following (see also [4] and [12]). Let

[TABLE]

where $s=s(\alpha,\beta,\gamma,x)$ is a solution to the Schwarzian differential equation

[TABLE]

and

[TABLE]

is the Schwarzian derivative with the potential $V$ given by

[TABLE]

The combination $y=-2w_{1}-2w_{2}-2w_{3}$ solves the generalised Chazy equation when

[TABLE]

The combination $y=-w_{1}-2w_{2}-3w_{3}$ solves the generalised Chazy equation when

[TABLE]

with permutations of $w_{1}$, $w_{2}$ and $w_{3}$ in $y$ corresponding to permutations of the values $\alpha$, $\beta$ and $\gamma$ in $(\alpha,\beta,\gamma)$. The combination $y=-w_{1}-w_{2}-4w_{3}$ solves the generalised Chazy equation whenever

[TABLE]

again permuting $w_{1}$, $w_{2}$ and $w_{3}$ in $y$ corresponds to permuting the values $\alpha$, $\beta$, $\gamma$ in $(\alpha,\beta,\gamma)$. Following [1], the functions $w_{1}$, $w_{2}$ and $w_{3}$ satisfy the following system of differential equations:

[TABLE]

where

[TABLE]

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

[TABLE]

with the same potential $V$ as given in (2.2). This corresponds to the general solution of the Schwarzian differential equation (2.1) after exchanging dependent and independent variables [8]. In this case $x=\frac{u_{2}}{u_{1}}$ where $u_{1}$ and $u_{2}$ are linearly independent solutions to (2.4). Using the further substitution $u(s)=(s-1)^{\frac{1-\gamma}{2}}s^{\frac{1-\beta}{2}}z(s)$, the equation (2.4) can be brought to the hypergeometric differential equation

[TABLE]

with

[TABLE]

From the differential equations (2), we can recover $s$ by $s=\frac{w_{1}-w_{3}}{w_{2}-w_{3}}$. From this we deduce $s^{\prime}=2(w_{1}-w_{2})s$ and we also obtain the relation ${\rm d}s=2(w_{1}-w_{2})s{\rm d}x$.

3. Main results: Solving the equations for Ricci-flatness

In this section we give the general solution to the differential equation (1.2) where $Q=P^{2}-6P^{\prime}$ and $P$ is a solution of the $k=\frac{3}{2}$ generalised Chazy equation in Theorem 3.1 and the general solution to the differential equation (1.5) where again $Q=P^{2}-6P^{\prime}$ and $P$ is a solution of the $k=\frac{2}{3}$ generalised Chazy equation in Theorem 3.2. We first prove the following theorem

Theorem 3.1**.**

The solution to the differential equation

[TABLE]

where $Q=P^{2}-6P^{\prime}$ and $P$ is a solution to the $k=\frac{3}{2}$ generalised Chazy equation, is given by $\rho=\frac{u}{v}$ where $v$ is the solution to the second-order differential equation associated to the $k=\frac{3}{2}$ generalised Chazy equation and $u$ is a solution to the second-order differential equation associated to the $k=3$ generalised Chazy equation.

Proof.

To prove the claim, we consider the second-order differential equation of the form $v_{ss}+\frac{1}{4}Vv=0$ associated to the generalised Chazy equation with parameter $k=\frac{3}{2}$, where $V$ is the function given by

[TABLE]

We find that $v=v(s(x))$ as a function of $x$ satisfies

[TABLE]

We have used that

[TABLE]

and the differential equations (2). Furthermore, the Wronskian $W=v_{1}(v_{2})_{s}-v_{2}(v_{1})_{s}$ of the solutions to this differential equation satisfies $W_{s}=0$, so $W=c_{0}$ and we have

[TABLE]

from the consideration that $s^{\prime}=2(w_{1}-w_{2})s=\frac{v_{1}^{2}}{W}$, and furthermore we obtain from the differential equation that the Wronskian $W=\frac{v(s(x))^{2}}{2(w_{1}-w_{2})s(x)}$ satisfies, that

[TABLE]

This equation implies the differential equation for $v$ above, by using the fact that $w_{i}$ satisfy the differential equations (2).

Upon making the substitution $\rho=\frac{u(x)}{v(x)}$ into equation (1.2), and using equation (3.1), we obtain a differential equation for $u(x)$ remaining. This differential equation for $u(x)$ turns out to be of the form

[TABLE]

which is the same differential equation for $v$ with different constants $\tilde{\alpha}$, $\tilde{\beta}$, $\tilde{\gamma}$. This is the differential equation associated to the generalised Chazy equation with parameter $k=3$. We see this automatically when we compute the values with $Q=P^{2}-6P^{\prime}$ where $P$ is the solution of the generalised Chazy equation with parameter $k=\frac{3}{2}$. Specialising to the case where $k=\frac{3}{2}$, we obtain the following:

For the solutions given by $P=-2w_{1}-2w_{2}-2w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{4}{3},\frac{4}{3},\frac{4}{3})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{2}{3},\frac{2}{3},\frac{2}{3})$. When $(\alpha,\beta,\gamma)=(\frac{4}{3},\frac{1}{3},\frac{1}{3})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{2}{3},\frac{1}{3},\frac{1}{3})$.

For the solutions given by $P=-w_{1}-2w_{2}-3w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{1}{3},\frac{1}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{3},\frac{1}{3},\frac{1}{2})$. When $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{4}{3},\frac{1}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{3},\frac{2}{3},\frac{1}{2})$. When $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{1}{3},2)$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{3},\frac{1}{3},1)$.

Finally for the solutions given by $P=-4w_{1}-w_{2}-w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{2}{3},\frac{2}{3})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{2}{3},\frac{1}{3},\frac{1}{3})$. When $(\alpha,\beta,\gamma)=(\frac{8}{3},\frac{2}{3},\frac{2}{3})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{4}{3},\frac{1}{3},\frac{1}{3})$.

The values $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ are precisely the ones that show up in the solutions of the $k=3$ generalised Chazy equation. See [12] for the list of $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ when $k=3$.

∎

The determination of solutions to equation (1.5) is similar to that of Theorem 3.1. We prove the following

Theorem 3.2**.**

The solution to the differential equation

[TABLE]

where $Q=P^{2}-6P^{\prime}$ and $P$ is a solution of the $k=\frac{2}{3}$ generalised Chazy equation, is given by $\nu=\frac{u}{v}$, where $v$ is a solution to the second-order differential equation associated to the $k=\frac{2}{3}$ generalised Chazy equation and $u$ is a solution to the second-order differential equation associated to the $k=2$ generalised Chazy equation.

Proof.

The proof of the claim is similar to the proof of the previous theorem. From the differential equation of the form $v_{ss}+\frac{1}{4}Vv=0$ associated to the $k=\frac{2}{3}$ generalised Chazy equation, where $V$ is the function given by

[TABLE]

we find that $v=v(s(x))$ as a function of $x$ satisfies

[TABLE]

Like in the proof of Theorem 3.1, it can also be deduced that (3.1) holds for $v$, i.e.

[TABLE]

which again implies the differential equation for $v$ above, by using the fact that $w_{i}$ satisfy the differential equations (2).

Upon making the substitution $\nu=\frac{u(x)}{v(x)}$ into equation (3.2), and using equation (3.3), we obtain a differential equation for $u(x)$ remaining. The differential equation for $u(x)$ is again

[TABLE]

which is the same differential equation for $v$ but with different constants $\tilde{\alpha}$, $\tilde{\beta}$, $\tilde{\gamma}$. The equation (3.4) corresponds to the second-order differential equation associated to the $k=2$ generalised Chazy equation. To see this, we shall compute these constants when $Q=P^{2}-6P^{\prime}$ and $P$ is the solution of the generalised Chazy equation with parameter $k=\frac{2}{3}$. Specialising to the case where $k=\frac{2}{3}$, we obtain the following:

For the solutions given by $P=-2w_{1}-2w_{2}-2w_{3}$, when $(\alpha,\beta,\gamma)=(3,3,3)$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(1,1,1)$. When $(\alpha,\beta,\gamma)=(3,\frac{1}{3},\frac{1}{3})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(1,\frac{1}{3},\frac{1}{3})$.

For the solutions given by $P=-w_{1}-2w_{2}-3w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{3}{2},\frac{1}{3},\frac{1}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{2},\frac{1}{3},\frac{1}{2})$. When $(\alpha,\beta,\gamma)=(\frac{3}{2},3,\frac{1}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{2},1,\frac{1}{2})$. When $(\alpha,\beta,\gamma)=(\frac{3}{2},\frac{1}{3},\frac{9}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{2},\frac{1}{3},\frac{3}{2})$.

Finally for the solutions given by $P=-4w_{1}-w_{2}-w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{3}{2},\frac{3}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{2}{3},\frac{1}{2},\frac{1}{2})$. When $(\alpha,\beta,\gamma)=(6,\frac{3}{2},\frac{3}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(2,\frac{1}{2},\frac{1}{2})$.

The values $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ are precisely the ones that show up in the solutions of the $k=2$ generalised Chazy equation. See also [12] for the list of $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ 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 $\nu=\frac{u}{v}$ 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 $\tilde{k}$ with

[TABLE]

The values $(\alpha,\beta,\gamma)$ appearing in $V$ in the differential equation $v_{ss}+\frac{1}{4}Vv=0$ are related to the values $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ appearing in $V$ in the differential equation $u_{ss}+\frac{1}{4}Vu=0$ by the following. For the solutions given by $P=-2w_{1}-2w_{2}-2w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{2}{k},\frac{2}{k},\frac{2}{k})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

When $(\alpha,\beta,\gamma)=(\frac{2}{k},\frac{1}{3},\frac{1}{3})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

and $\tilde{\beta}=\frac{1}{3}$, $\tilde{\gamma}=\frac{1}{3}$. Here and subsequently, we shall consider the positive square root that gives positive $\tilde{\alpha}$, $\tilde{\beta}$ and $\tilde{\gamma}$.

For the solutions given by $P=-w_{1}-2w_{2}-3w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{1}{3},\frac{1}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

When $(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{2}{k},\frac{1}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

When $(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{1}{3},\frac{3}{k})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

Finally for the solutions given by $P=-4w_{1}-w_{2}-w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{4}{k},\frac{1}{k},\frac{1}{k})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

When $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{1}{k},\frac{1}{k})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

In all cases the appropriate substitution of $\tilde{\alpha}$, $\tilde{\beta}$ and $\tilde{\gamma}$ in terms of the Chazy parameter $\tilde{k}$ 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 $\tilde{k}$, related to $k$ by (4.1). The further substitution $k=\frac{3}{m}$ and $\tilde{k}=\frac{3}{\tilde{m}}$ into (4.1) gives

[TABLE]

which has integer solutions when considered as a negative Pell equation. For integer solutions $m$ and $\tilde{m}$ we obtain

[TABLE]

They take on values $(m,\tilde{m})=(2,1)$, $(38,17)$, $(682,305)$, $(12238,5473)$ and so on for $n\in\mathbb{N}\cup\{0\}$. They also give the corresponding pairs of Chazy parameters $(k,\tilde{k})=(\frac{3}{2},3)$, $(\frac{3}{38},\frac{3}{17})$ 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 $\nu=\frac{u}{v}$ 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 $\tilde{k}$ with

[TABLE]

In this case we obtain the relationship between the values $(\alpha,\beta,\gamma)$ and $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ as follows. For $P=-2w_{1}-2w_{2}-2w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{2}{k},\frac{2}{k},\frac{2}{k})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

Considering integer solutions $\alpha$ and $\tilde{\alpha}$ to the negative Pell equation $10\tilde{\alpha}^{2}-\alpha^{2}=1$ (and also $\beta$, $\tilde{\beta}$ and $\gamma$, $\tilde{\gamma}$ respectively), we find

[TABLE]

where $n\in\mathbb{Z}$. Positive integer solutions are given by $(\alpha,\tilde{\alpha})=(3,1)$, $(117,37)$, $(4443,1405)$, $(168717,53353)$ and so on for $n\in\mathbb{N}\cup\{0\}$. They give the relationship between the pairs of Chazy parameters $k=\frac{2}{\alpha}$ and $\tilde{k}=\frac{2}{\tilde{\alpha}}$, with $(k,\tilde{k})=(\frac{2}{3},2)$, $(\frac{2}{117},\frac{2}{37})$ and so on for $n\in\mathbb{N}\cup\{0\}$. 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 $(\alpha,\beta,\gamma)$ and $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ are as follows. For the same $P$, when $(\alpha,\beta,\gamma)=(\frac{2}{k},\frac{1}{3},\frac{1}{3})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

and $\tilde{\beta}=\frac{1}{3}$, $\tilde{\gamma}=\frac{1}{3}$.

For the solutions given by $P=-w_{1}-2w_{2}-3w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{1}{3},\frac{1}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

When $(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{2}{k},\frac{1}{2})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

When $(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{1}{3},\frac{3}{k})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

Finally for the solutions given by $P=-4w_{1}-w_{2}-w_{3}$, when $(\alpha,\beta,\gamma)=(\frac{4}{k},\frac{1}{k},\frac{1}{k})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

When $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{1}{k},\frac{1}{k})$, we find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

[TABLE]

In all cases the appropriate substitution of $\tilde{\alpha}$, $\tilde{\beta}$ and $\tilde{\gamma}$ in terms of the Chazy parameter $\tilde{k}$ gives equation (4.2), and therefore the equation for $u$ is the second-order differential equation associated to the generalised Chazy equation with parameter $\tilde{k}$, related to $k$ by (4.2). Altogether, with the exception of the parameters $k=\frac{3}{2}$ and $k=\frac{2}{3}$ as mentioned above, they give Ricci-flat but non-conformally flat examples of Nurowski’s metric.