# Recurrent Lorentzian Weyl spaces

Andrei Dikarev  ,  Anton S. Galaev  and  Eivind Schneider

###### Abstract.

We find the local form of all non-closed Lorentzian Weyl manifolds
$(M,c,\nabla)$ with recurrent curvature tensor. The recurrent curvature tensor
turns out to be weighted parallel, i.e., the obtained spaces provide certain
generalization of locally symmetric affine spaces for the Weyl geometry. If
the dimension of the manifold is greater than 3, then the conformal structure
is flat, and the recurrent Weyl structure is locally determined by a single
function of one variable. Two local structures are equivalent if and only if
the corresponding functions are related by a transformation from
$\mathrm{SAff}_{1}(\mathbb{R})\times\mathrm{PSL}_{2}(\mathbb{R})\times\mathbb{Z}_{2}$.
We find generators for the field of rational scalar differential invariants of
this Lie group action. The global structure of the manifold $M$ may be
described in terms of a foliation with a transversal projective structure. It
is shown that all locally homogeneous structures are locally equivalent, and
there is only one simply connected homogeneous non-closed recurrent Lorentzian
Weyl manifold. Moreover, there are 5 classes of cohomogeneity-one spaces, and
all other spaces are of cohomogeneity-two. If $\dim M=3$, the non-closed
recurrent Lorentzian Weyl structures are locally determined by one function of
two variables or two functions of one variables, depending on whether its
holonomy algebra is 1- or 2-dimensional. In this case, two structures with the
same holonomy algebra are locally equivalent if and only if they are related,
respectively, by a transformation from an infinite-dimensional Lie pseudogroup
or a 4-dimensional subgroup of $\mathrm{Aff}(\mathbb{R}^{3})$. Again we
provide generators for the field of rational differential invariants. We find
a local expression for the locally homogeneous non-closed recurrent Lorentzian
Weyl manifolds of dimension 3, and also of those of cohomogeneity one and two.
In the end we give a local description of the non-closed recurrent Lorentzian
Weyl manifolds that are also Einstein-Weyl. All of them are 3-dimensional and
have a 2-dimensional holonomy algebra.

Keywords: Weyl connection; recurrent curvature; holonomy group; Riccati
equation; Einstein-Weyl structure; conformally flat manifold; homogeneous
space

1Department of Mathematics and Statistics, Masaryk University, Faculty of
Science, Kotlářská 2, 611 37 Brno, Czech Republic

2University of Hradec Králové, Faculty of Science, Rokitanského 62, 500 03
Hradec Králové, Czech Republic
E-mail: anton.galaev(at)uhk.cz

3Department of Mathematics and Statistics, UiT The Arctic University of
Norway, 9037 Tromsø, Norway.

###### Contents

  1. 1 Introduction
  2. 2 The holonomy algebra of recurrent Lorentzian Weyl spaces
  3. 3 The local form of recurrent Weyl structures
  4. 4 The remaining coordinate freedom and symmetries of generic structures
  5. 5 The global form of recurrent Weyl structures
  6. 6 Differential invariants and the local equivalence problem
  7. 7 Recurrent Weyl structures with additional symmetries
  8. 8 Homogeneous spaces
  9. 9 Einstein-Weyl structures with recurrent curvature tensor

##  1\. Introduction

A Lorentzian Weyl manifold is a triple $(M,c,\nabla)$ where $M$ is a smooth
manifold, $c$ is a conformal class of Lorentzian metrics on $M$, and $\nabla$
is a torsion-free affine connection on $M$ such that, for each $g\in c$, there
exists a 1-form $\omega_{g}$ satisfying

(1) $\nabla g=-2\omega_{g}\otimes g.$

In what follows all manifolds will be assumed to be connected. Lorentzian Weyl
manifolds have received a lot of attention in several recent works, e.g., [1,
6, 8, 9, 16]. The connection $\nabla$ is uniquely determined by a metric $g\in
c$ and the corresponding 1-form $\omega_{g}$ by the formula

(2) $\nabla=\nabla^{g}+K^{g},\quad
g(K^{g}_{X}(Y),Z)=g(Y,Z)\omega_{g}(X)+g(X,Z)\omega_{g}(Y)-g(X,Y)\omega_{g}(Z),$

where $\nabla^{g}$ is the Levi-Civita connection of the metric $g$, and
$X,Y,Z$ are vector fields on $M$. If $h=e^{2f}g$ is a different metric in the
conformal class, then the corresponding 1-form $\omega_{h}$ is given by

$\omega_{h}=\omega_{g}-df.$

The Weyl structure $(c,\nabla)$ is called closed if $d\omega_{g}=0$ for some
(and thus every) $g\in c$. If the structure is closed, then $\nabla$ is
locally the Levi-Civita connection for some metric in the conformal class. Let
$R$ denote the curvature tensor of the affine connection $\nabla$. Recall that
an affine connection $\nabla$ is called locally symmetric if $\nabla R=0$. It
turns out that

######  Proposition 1.

If the connection $\nabla$ of a Lorentzian Weyl structure is locally
symmetric, then the Weyl structure is closed.

Since a locally symmetric affine connection is (locally) the Levi-Civita
connection for some metric in the conformal class, the study of locally
symmetric Lorentzian Weyl manifolds reduces to the study of locally symmetric
Lorentzian manifolds. In this paper we therefore relax this condition and
consider Lorentzian Weyl structures with recurrent curvature tensors, i.e.,
curvature tensors satisfying

(3) $\nabla R=\theta\otimes R$

for some 1-form $\theta$. The aim of the paper is to give a description of
such structures. Since the Lorentzian recurrent spaces are classified in [19],
we will focus on the non-closed Weyl structures.

In Section 2 we find the holonomy algebra of a general non-closed recurrent
Lorentzian Weyl manifold. In particular, for $\dim M\geqslant 4$ we have the
following statement.

######  Proposition 2.

Let $(M,c,\nabla)$ be a non-closed recurrent Lorentzian Weyl manifold. Suppose
that $M$ is simply connected and let $\dim M=n+2\geqslant 4$. Then $M$ admits
a $\nabla$-parallel distribution of space-like lines $\mathcal{L}$. The
orthogonal distribution $\mathcal{L}^{\bot}$ is parallel as well and there
exists a parallel isotropic vector field $p$ tangent to $\mathcal{L}^{\bot}$.

We show that there is a preferred metric in the conformal class defined up to
homothety.

######  Proposition 3.

Let $(M,c,\nabla)$ be a non-closed recurrent Lorentzian Weyl manifold and
$\dim M=n+2\geqslant 4$. Suppose that the manifold $M$ is simply connected.
Then there exists a metric $h\in c$ satisfying

(4) $\nabla h=-2\omega_{h}\otimes h,\quad\nabla R=-3\omega_{h}\otimes R.$

The metric $h$ is defined uniquely up to homothety.

Consider the density bundle with weight $w$,

${\mathcal{L}}^{w}=P_{CO}\times_{|\det|^{\frac{w}{n+2}}}\mathbb{R},$

where $P_{CO}$ is the conformal frame bundle. Each metric $g\in c$ defines the
section

$l_{g}=|\operatorname{vol}_{g}|^{-\frac{1}{n+2}}\in{\mathcal{L}}^{1}$

satisfying

$\nabla l_{g}=\omega_{g}\otimes l_{g}.$

We conclude that

$\nabla(R\otimes l^{3}_{h})=0,$

i.e., the tensor field $R$ is weighted parallel with the weight $3$.

######  Corollary 1.

If the curvature tensor of a non-closed Weyl Lorentzian structure is
recurrent, then it is weighted parallel.

The non-closed Weyl manifolds with weighted parallel curvature tensors provide
generalization of pseudo-Riemannian symmetric spaces. Another approach to
symmetric spaces in conformal geometry may be found in [10].

In Section 3 we use the holonomy algebras found in Section 2 and results from
[5] to obtain a local form of the general non-closed recurrent Weyl manifold
$(M,c,\nabla)$.

######  Theorem 1.

Let $(M,c,\nabla)$ be a non-closed recurrent Lorentzian Weyl manifold and
$\dim M=n+2\geqslant 4$. Then in a neighborhood of each point of $M$ the local
conformal class contains the metric

$h=(dt)^{2}+e^{2G}\left(2dvdu+\sum_{i=1}^{n-1}(dx^{i})^{2}\right)$

where

(5) $G=-\ln|u+\psi(t)|+\frac{1}{2}\ln|\psi^{\prime}(t)|,$

$\psi(t)$ is a function with non-vanishing $\psi^{\prime}(t)$, and
$t,v,x^{1},\dots,x^{n-1},u$ are local coordinates. The 1-form corresponding to
$h$ is given by

$\omega_{h}=-\partial_{t}Gdt.$

The pair $(h,\omega_{h})$ satisfies (4).

It is easily checked that the local metric $g=e^{-2G}h$ is flat, i.e., the
conformal class $c$ is flat, and we obtain the following proposition.

######  Proposition 4.

All non-closed recurrent Lorentzian Weyl structures on manifolds of dimension
greater than 3 are conformally flat.

By Theorem 1, a recurrent Lorentzian Weyl structure is locally defined by a
function $\psi(t)$. In (5) we require $|\psi^{\prime}(t)|>0$, which implies
that either $\psi^{\prime}(t)>0$ or $\psi^{\prime}(t)<0$. The transformation
$(\tilde{v},\tilde{u},\tilde{x}^{i},\tilde{t})=(-v,-u,x^{i},t)$ preserves the
coordinate form of the metric, but with $\tilde{\psi}(\tilde{t})=-\psi(t)$.
Since $\frac{d}{d\tilde{t}}\tilde{\psi}(\tilde{t})=-\frac{d}{dt}\psi(t)$, a
Weyl structure given by $\psi(t)$ with $\psi^{\prime}(t)<0$ is equivalent to
the one given by $\tilde{\psi}(\tilde{t})=-\psi(t)$, which satisfies
$\tilde{\psi}^{\prime}(\tilde{t})>0$. By this reason we may assume that
$\psi^{\prime}(t)>0$, which lets us replace $|\psi^{\prime}(t)|$ in (5) with
$\psi^{\prime}(t)$ without loosing any Weyl structures. Notice that the domain
of definition is transformed under this coordinate transformation. If the Weyl
structure satisfies $u+\psi(t)>0$, then it follows that
$\tilde{u}+\tilde{\psi}(\tilde{t})<0$. Similarly, $u+\psi(t)<0$ implies
$\tilde{u}+\tilde{\psi}(\tilde{t})>0$.

Section 4 is devoted to the investigation of the remaining coordinate freedom.
In Proposition 7 we find the vector fields preserving the coordinate form of
Theorem 1. By integrating the vector fields we see that the local equivalence
problem of non-closed recurrent Lorentzian Weyl manifolds of dimension
$\geqslant 4$ can be reduced to the local equivalence problem of the space
$\\{\psi\in C^{\infty}(\mathbb{R})\mid\psi^{\prime}(t)>0\\}$ with respect to
the action (26) of the Lie group
$\mathrm{SAff}_{1}(\mathbb{R})\times\mathrm{PSL}_{2}(\mathbb{R})\times\mathbb{Z}_{2}$.

######  Theorem 2.

Two local recurrent Lorentzian Weyl structures defined by functions $\psi_{1}$
and $\psi_{2}$ with $\psi_{i}^{\prime}(t)>0$ for $i\in\\{1,2\\}$ are
equivalent if and only if

$\psi_{2}(t)=\mathrm{sgn}(B)\frac{a\psi_{1}(A+Bt)+b}{c\psi_{1}(A+Bt)+d},$

for some constants $a,b,c,d,A,B$ satisfying $ad-bc=1,B\neq 0$.

The local equivalence problem is solved for generic non-closed recurrent
Lorentzian Weyl structures in Section 6 by showing (in Proposition 10) that
the differential invariants are generated by the two differential invariants

$\frac{(\psi_{1}^{2}\psi_{4}-4\psi_{1}\psi_{2}\psi_{3}+3\psi_{2}^{3})^{2}}{(2\psi_{1}\psi_{3}-3\psi_{2}^{2})^{3}},\qquad\frac{\psi_{1}(\psi_{1}^{2}\psi_{5}-5\psi_{1}\psi_{2}\psi_{4}+5\psi_{2}^{2}\psi_{3})}{(2\psi_{1}\psi_{3}-3\psi_{2}^{2})^{2}},$

where $\psi_{1},\cdots,\psi_{5}$ are coordinates on the space of $5$-jets of
$\psi(t)$.

Let us fix a metric $h$ satisfying (4). If the manifold $M$ is simply
connected, then there exists a non-vanishing section $X$ of the distribution
$\mathcal{L}$ from Proposition 2. Let us fix a section $X$ satisfying

$h(X,X)=1.$

Note that if $h$ is not fixed, then the section $X$ is defined uniquely up to
multiplication by a non-zero constant. In Section 5 we prove the following
global result.

######  Theorem 3.

Let $(M,c,\nabla)$ be a non-closed recurrent Lorentzian Weyl structure and
$\dim M=n+2\geqslant 4$. Suppose that the manifold $M$ is simply connected.
Fix a metric $h$. Suppose that the vector field $X$ is complete. Then the
manifold $M$ is diffeomorphic to the product

$M\cong\mathbb{R}\times N,$

where $N$ is the integral manifold of the $\nabla$-parallel distribution
orthogonal to $X$. The distribution $p^{\bot}$ on $N$ orthogonal to the
$\nabla$-parallel isotropic vector field $p$ is involutive. The corresponding
codimension-one foliation ${\mathcal{F}}$ on $N$ admits a transversal
projective structure. There is a foliated atlas on $N$ such that to each chart
$U$ with coordinates $v,x^{1},\dots,x^{n-1},u$, where $u$ is the transversal
coordinate, corresponds a regular function $\psi\in C^{\infty}(\mathbb{R})$
such that on $\mathbb{R}\times U$, $h$ and $\omega_{h}$ are as in Theorem 1.
Given two such charts $U_{1}$ and $U_{2}$ with non-trivial intersection, the
transverse coordinates are transposed as

$u_{2}=\frac{au_{1}+b}{cu_{1}+d},\quad ad-bc=1$

and the corresponding functions satisfy

$\psi_{2}=\frac{a\psi_{1}-b}{-c\psi_{1}+d}.$

A diffeomorphism $\varphi$ on $U\subset M$ is a local symmetry of the
recurrent Weyl structure $(c,\nabla)$ on $M$ if $\varphi$ is a conformal
transformation that preserves the affine connection $\nabla$, and a global
symmetry if the same holds for $U=M$. In the case that the Weyl structure is
defined by a metric $h\in c$ and a 1-form $\omega_{h}$, the condition on
$\varphi$ may be expressed as $\varphi^{*}h=e^{2f}h$ and
$\varphi^{*}\omega_{h}=\omega_{h}-df$ for some function $f\in C^{\infty}(U)$.
A vector field $X$ on $U\subset M$ is called a local infinitesimal symmetry of
the recurrent Weyl structure on $M$ defined by $(h,\omega_{h})$ if it
satisfies $L_{X}h=2\lambda h$ and $L_{X}\omega_{h}=-d\lambda$ for some
function $\lambda\in C^{\infty}(U)$, and a global infinitesimal symmetry if
the same holds for $U=M$. The flow of an infinitesimal symmetry is a symmetry.
We will often drop the adjective “infinitesimal” when it is clear that the
symmetry referred to is a vector field.

The generic non-closed recurrent Lorentzian Weyl manifold of dimension
$n+2\geqslant 4$ has $(2n-1)+\binom{n-1}{2}$ independent infinitesimal
symmetries, and they span a Lie algebra abstractly isomorphic to
$\mathfrak{heis}(n-1)\rtimes\mathfrak{so}(n-1)$, but there are some non-closed
recurrent Lorentzian Weyl manifolds with additional infinitesimal symmetries.
These are studied in Section 7. We show that there is exactly one class of
locally homogeneous spaces and 5 classes of cohomogeneity-one spaces. All
other spaces have cohomogeneity two. More precisely, we obtain the following
result.

######  Theorem 4.

Let $(M,c,\nabla)$ be a non-closed recurrent Lorentzian Weyl manifold and
$\dim M=n+2\geqslant 4$.

  * 1.

The infinitesimal symmetries span the tangent space at each point of $M$ if
and only if each local function $\psi(t)$ from Theorem 1 is equivalent to the
function $t$.

  * 2.

The infinitesimal symmetries span a hyperspace of the tangent space at each
point of $M$ if and only if each local function $\psi(t)$ from Theorem 1 is
equivalent to one function of the following functions:

$\displaystyle\psi(t)$ $\displaystyle=e^{t},$ $\displaystyle\psi(t)$
$\displaystyle=\tan(t),$ $\displaystyle\psi(t)$ $\displaystyle=A\ln t,\quad
A>0,$ $\displaystyle\psi(t)$ $\displaystyle=\tan(A\ln t),\quad A>0,$
$\displaystyle\psi(t)$ $\displaystyle=|t|^{A},\quad At>0,\;A\neq 1.$

  * 3.

In all other cases, the infinitesimal symmetries span a subspace of
codimension-two of the tangent space at each point of $M$.

Denote by $\operatorname{Aut}(M,c,\nabla)$ the group of global symmetries and
by $\operatorname{Aut}^{0}(M,c,\nabla)$ the connected component of the
identity. In Section 8 we show that in each dimension $n+2\geqslant 4$ there
is only simply connected homogeneous non-closed recurrent Lorentzian Weyl
manifold. Define the Lie group

$G_{0}=\mathbb{R}^{+}\ltimes(\mathbb{R}\oplus\mathbb{R}^{n-1}\oplus\mathbb{R})$

with the operation

$(r_{1},w_{1},y_{1},s_{1})\cdot(r_{2},w_{2},y_{2},s_{2})=(r_{1}r_{2},w_{1}+r_{1}^{3}w_{2},y_{1}+r_{1}^{2}y_{2},s_{1}+r_{1}s_{2}).$

Let $b$ be the left-invariant Lorentzian metric on $G_{0}$ with the value

$b_{e}=\left(4(dr)^{2}+2dwds+\sum_{i=1}^{n-1}(dy^{i})^{2}+2(ds)^{2}\right)_{e}$

at the identity $e=(1,0,0,0)\in G_{0}$. Consider the left-invariant 1-form
$\omega$ defined by the value

$\omega_{e}=\left(-\frac{\sqrt{2}}{2}ds+2dr\right)_{e}.$

We show that the Weyl connection $\nabla$ defined by $b$ and $\omega$ has
recurrent curvature tensor.

######  Theorem 5.

Let $(M,c,\nabla)$ be a non-closed recurrent Lorentzian Weyl structure and
$\dim M=n+2\geqslant 4$. Suppose that the manifold $M$ is simply connected and
$\operatorname{Aut}(M,c,\nabla)$-homogeneous. Then $M$ is diffeomorphic to
$G_{0}$. Under this diffeomorphism, $c=[b]$, and $\nabla$ is defined by $b$
and $\omega$. Moreover,

(6) $\operatorname{Aut}^{0}(M,c,\nabla)=\operatorname{Isom}^{0}(G_{0},b)\cong
G_{0}\cdot\text{\rm SO}(n-1)\cdot\mathbb{R}^{n-1}.$

Several of the results mentioned so far (concerning local coordinate
expressions, symmetries and differential invariants) have analogues for $\dim
M=3$, and they can be found in respective sections together with the results
for $\dim M\geqslant 4$.

A special type of Lorentzian Weyl manifolds that has received a lot of
attention (see for example [1, 8]) are the Einstein-Weyl manifolds, and in
Section 9 we investigate whether some these manifolds coincide with the non-
closed recurrent Weyl manifolds. For $\dim M\geqslant 4$ it turns out that
there are no Einstein-Weyl manifolds among the non-closed recurrent Lorentzian
Weyl manifolds. However, there does exist non-closed Lorentzian Einstein-Weyl
manifolds of dimension 3 with recurrent curvature tensor. By Proposition 18
they are exactly the nonclosed recurrent Lorentzian Weyl manifolds with
2-dimensional holonomy algebra, and they can be locally defined by the pair
$(g,\omega)$ given by

$g=2dvdu+(dx)^{2}+\left(a(u)vx+\frac{1}{12}a^{2}(u)x^{4}-\frac{1}{3}\dot{a}(u)x^{3}+c(u)x\right)(du)^{2},\qquad\omega=a(u)xdu,$

where the function $a$ is non-vanishing. As shown in Section 7, the most
symmetric of these have a 2-dimensional Lie algebra of infinitesimal
symmetries, and they are locally equivalent to the one defined by

$a(u)\equiv 1,\qquad c(u)\equiv 0,$

which admits the infinitesimal symmetries

$\partial_{u},\qquad u\partial_{u}-3v\partial_{v}-x\partial_{x}.$

Finally note that the non-closed recurrent Lorentzian Weyl structures have
another remarkable property: they admit a rather large number of weighted
parallel spinors [5].

######  Corollary 2.

Let $(c,\nabla)$ be a non-closed recurrent Lorentzian Weyl spin structure on a
simply connected manifold $M$ of dimension $n+2\geqslant 4$, then it admits
the space of weighted parallel spinors of complex dimension
$2^{[\frac{n}{2}]}$.

A similar result is known for Lorentzian symmetric spaces [14]: each simply
connected indecomposable Cahen-Wallach spin manifold of dimension $n+2$ admits
the space of parallel spinors of complex dimension $2^{[\frac{n}{2}]}$ (Cahen-
Wallach spaces are symmetric Lorentzian spaces represented by a special class
of pp-waves).

##  2\. The holonomy algebra of recurrent Lorentzian Weyl spaces

We start by recalling the classification of the holonomy algebras of
Lorentzian Weyl structures [4].

Let $(M,c,\nabla)$ be a Weyl manifold of Lorentzian signature $(1,n+1)$,
$n\geqslant 1$. Then its holonomy algebra is contained in the conformal
Lorentzian algebra

$\mathfrak{co}(1,n+1)=\mathbb{R}\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}\oplus\mathfrak{so}(1,n+1).$

The Weyl structure is closed if and only if the holonomy algebra is contained
in $\mathfrak{so}(1,n+1)$ and this case is well-studied [14]. By that reason
we suppose that the Weyl structure is non-closed, and the holonomy algebra is
not contained in $\mathfrak{so}(1,n+1)$.

Fix a Witt basis $p,e_{1},\dots,e_{n},q$ of the Minkowski space
$\mathbb{R}^{1,n+1}$. With respect to that basis the subalgebra of
$\mathfrak{so}(1,n+1)$ preserving the null line $\mathbb{R}p$ has the
following matrix form:

$\mathfrak{so}(1,n+1)_{\mathbb{R}p}=\left\\{\left.\begin{pmatrix}a&X^{t}&0\\\
0&A&-X\\\ 0&0&-a\end{pmatrix}\right|\begin{matrix}a\in\mathbb{R}\\\
A\in\mathfrak{so}(n)\\\ X\in\mathbb{R}^{n}\end{matrix}\right\\}.$

We identify the Lie algebra $\mathfrak{so}(1,n+1)$ with the space of bivectors
$\wedge^{2}\mathbb{R}^{1,n+1}$ in such a way that

$(X\wedge Y)Z=(X,Z)Y-(Y,Z)X.$

Under this identification the above element of
$\mathfrak{so}(1,n+1)_{\mathbb{R}p}$ corresponds to

$-ap\wedge q+A-p\wedge X.$

We get the decomposition

$\mathfrak{so}(1,n+1)_{\mathbb{R}p}=(\mathbb{R}p\wedge
q\,\oplus\mathfrak{so}(n))\ltimes\,p\wedge\mathbb{R}^{n}.$

The holonomy algebra $\mathfrak{g}\subset\mathfrak{co}(1,n+1)_{\mathbb{R}p}$
of a non-closed Lorentzian Weyl structure of dimension $n+2$, $n\geqslant 1$,
satisfies one of the following 3 conditions:

  * 1.

$\mathfrak{g}\subset\mathfrak{co}(1,n+1)$ is irreducible. In this case
$\mathfrak{g}=\mathfrak{co}(1,n+1)$.

  * 2.

$\mathfrak{g}\subset\mathfrak{co}(1,n+1)$ preserves an orthogonal
decomposition

$\mathbb{R}^{1,n+1}=\mathbb{R}^{1,k+1}\oplus\mathbb{R}^{n-k},\quad-1\leqslant
k\leqslant n-1.$

In this case $\mathfrak{g}$ is one of the following:

    * $\bullet$

$\mathbb{R}\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}\oplus\mathfrak{so}(1,k+1)\oplus\mathfrak{so}(n-k)$,
$\quad-1\leqslant k\leqslant n-1$;

    * $\bullet$

$\mathbb{R}(\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}+p\wedge
q)\oplus\mathfrak{k}\oplus\mathfrak{so}(n-k)\ltimes
p\wedge\mathbb{R}^{k}\subset\mathfrak{co}(1,n+1)_{\mathbb{R}p}$, $0\leqslant
k\leqslant n-1$;

    * $\bullet$

$\mathbb{R}\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}\oplus\mathbb{R}p\wedge
q\oplus\mathfrak{k}\oplus\mathfrak{so}(n-k)\ltimes
p\wedge\mathbb{R}^{k}\subset\mathfrak{co}(1,n+1)_{\mathbb{R}p}$, $1\leqslant
k\leqslant n-1$.

Here $\mathfrak{k}\subset\mathfrak{so}(k)$ is the holonomy algebra of a
Riemannian manifold.

  * 3.

$\mathfrak{g}$ is contained in $\mathfrak{co}(1,n+1)_{\mathbb{R}p}$, and it
does not preserve any proper non-degenerate subspace of $\mathbb{R}^{1,n+1}$.
Such algebras may be divided into 6 types. For the results of the current
paper it is enough to know that each such $\mathfrak{g}$ contains the ideal
$p\wedge\mathbb{R}^{n}$.

######  Theorem 6.

Let $(M,c,\nabla)$ be a recurrent non-closed Lorentzian Weyl manifold of
dimension $n+2\geqslant 3$, then one of the following conditions holds:

  * •

$n=1$ and the holonomy algebra of $\nabla$ is one of the following

$\mathfrak{g}=\mathbb{R}(\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}+p\wedge
q),\quad\mathbb{R}(2\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}+p\wedge q)\ltimes\mathbb{R}p\wedge e_{1};$

  * •

$n\geqslant 2$ and the holonomy algebra of $\nabla$ is

$\mathbb{R}(\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}+p\wedge q)\ltimes
p\wedge\mathbb{R}^{n-1}\subset\mathfrak{co}(1,n+1)_{\mathbb{R}p}.$

Proof of Theorem 6. Let $(M,c,\nabla)$ be a recurrent non-closed Lorentzian
Weyl manifold and let $\mathfrak{g}\subset\mathfrak{co}(1,n+1)$ be its
holonomy algebra at a point $x\in M$. Let us denote the tangent space $T_{x}M$
by $\mathbb{R}^{1,n+1}$. Suppose that $R_{x}\neq 0$. Since $R$ is recurrent,
from the Ambrose-Singer Theorem it follows that

(7) $\mathfrak{g}={\rm span}\\{R_{x}(X,Y)|X,Y\in\mathbb{R}^{1,n+1}\\}.$

Consider the space of algebraic tensors of type $\mathfrak{g}$:

$\displaystyle{\mathcal{R}}(\mathfrak{g})=\bigl{\\{}R\in\operatorname{Hom}(\wedge^{2}\mathbb{R}^{1,n+1},\mathfrak{g})\mid{}$
$\displaystyle R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0\quad\forall
X,Y,Z\in\mathbb{R}^{1,n+1}\bigr{\\}}.$

From the Bianchi identity it follows that
$R_{x}\in{\mathcal{R}}(\mathfrak{g})$. The holonomy algebra $\mathfrak{g}$
acts on the space ${\mathcal{R}}(\mathfrak{g})$ in the following natural way

(8) $\xi:R\mapsto\xi\cdot R,\quad(\xi\cdot R)(X,Y)=[\xi,R(X,Y)]-R(\xi
X,Y)-R(X,\xi Y).$

Since the curvature tensor $R$ is recurrent, by the holonomy principle,
$\mathfrak{g}$ preserves the line $\mathbb{R}R_{x}$, i.e., there exists a
1-form $\rho$ on $\mathfrak{g}$ such that

(9) $\xi\cdot R_{x}=\rho(\xi)R_{x}\quad\forall\,\xi\in\mathfrak{g}.$

Now we are going to describe the holonomy algebras $\mathfrak{g}$ of non-
closed Weyl connections and elements $R\in{\mathcal{R}}(\mathfrak{g})$
satisfying the properties (7) and (9). Consider the 3 possibilities for
$\mathfrak{g}$ from above.

Case 1. Suppose that $\mathfrak{g}=\mathfrak{co}(1,n+1)_{\mathbb{R}p}$. There
exists an isomorphism of $\mathfrak{so}(1,n+1)$-modules

${\mathcal{R}}(\mathfrak{co}(1,n+1))\cong{\mathcal{R}}(\mathfrak{so}(1,n+1))\oplus\,\wedge^{2}\mathbb{R}^{1,n+1},$

where an element $A\in\wedge^{2}\mathbb{R}^{1,n+1}$ defines an algebraic
curvature tensor $R_{A}$ by the equality

(10) $R_{A}(X,Y)=AX\wedge Y+X\wedge AY+2(AX,Y)\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}.$

Let $R\in{\mathcal{R}}(\mathfrak{co}(1,n+1)).$ Property (9) implies that
$R\in{\mathcal{R}}(\mathfrak{so}(1,n+1))$. Such $R$ does not satisfy property
(7).

Case 3. Suppose that $\mathfrak{g}\subset\mathfrak{co}(1,n+1)_{\mathbb{R}p}$.
Then $\mathfrak{g}$ contains the ideal $p\wedge\mathbb{R}^{n}$. First suppose
that $n\geqslant 2$, Let $\xi=p\wedge Z$, $Z\in\mathbb{R}^{n}$. Let
$R=R_{0}+R_{A}$, where $R_{0}\in{\mathcal{R}}(\mathfrak{so}(1,n+1))$ and an
element $A\in\wedge^{2}\mathbb{R}^{1,n+1}$ defines $R_{A}$ by (10). The
element $A$ may be rewritten in the form

$A=ap\wedge q+p\wedge X+q\wedge Y+B,\quad
a\in\mathbb{R},\,\,X,Y\in\mathbb{R}^{n},\,\,B\in\mathfrak{so}(n).$

The equality (9) implies

$[\xi,A]=\rho(\xi)A,$

or, equivalently,

$aZ-BZ=\rho(\xi)X,\quad Z\wedge
Y=\rho(\xi)B,\quad\rho(\xi)a=0,\quad\rho(\xi)b=0.$

If $\rho(\xi)\neq 0$ for some $\xi\in p\wedge\mathbb{R}^{n}$, then $A=0$, and
$R$ does not satisfy property (7). This implies that $\rho(\xi)=0$ for all
$\xi\in p\wedge\mathbb{R}^{n}$. Consequently, $a=0$, $B=0$, $Y=0$, i.e.,
$A=p\wedge X$. Combining this with the description of the element
$R\in\mathfrak{co}(1,n+1)_{\mathbb{R}p}$ given in [4], we conclude that $R$ is
determined by the equalities

$\displaystyle R(p,q)=-\lambda p\wedge q-p\wedge X_{0},\quad R(p,V)=0,$
$\displaystyle R(U,V)=-p\wedge\big{(}P(V)U-P(U)V-2g(V,X)U+2g(U,X)V\big{)},$
$\displaystyle R(U,q)=\gamma(U)\mathop{\text{missing}}{\rm
id}\nolimits-g(U,2X+X_{0})p\wedge q+P(U)-p\wedge K(U),$

where

$\lambda\in\mathbb{R},\quad X_{0},\in\mathbb{R}^{n},\quad
P\in\mathop{\text{missing}}{\rm
Hom}\nolimits(\mathbb{R}^{n},\mathfrak{so}(n)),\quad
K\in\odot^{2}\mathbb{R}^{n},\quad S\in{\mathcal{R}}(\mathfrak{so}(n))$

are fixed, and $U,V\in\mathbb{R}^{n}$ are arbitrary. The condition
$p\wedge\mathbb{R}^{n}\,\cdot R=0$ implies $\lambda=0$, $S=0$, and

$2P(V)U-P(U)V+2g(U,X)V+g(V,X_{0})U+(U,V)X_{0}=0\quad\forall\,U,V\in\mathbb{R}^{n}.$

Using this equality and the similar equality, where $U$ is interchanged with
$V$, we get

$3P(V)U+g(U,4X+X_{0})V+2g(V,X+X_{0})V+3g(U,V)X_{0}=0\quad\forall\,U,V\in\mathbb{R}^{n}.$

The condition $P(V)\in\mathfrak{so}(n)$ implies $g(P(V)V,W)=-g(P(V)W,V)$ for
all $V,W\in\mathbb{R}^{n}$. Using this, we get $X=-X_{0}$, and $P(V)=-V\wedge
X_{0}$. Thus it holds

(11) $\displaystyle\begin{split}&R(p,q)=-p\wedge X_{0},\quad R(p,V)=0,\\\
&R(U,V)=-p\wedge\big{(}3(U\wedge V)X_{0}\big{)},\\\
&R(U,q)=-2g(U,X_{0})\mathop{\text{missing}}{\rm
id}\nolimits-3g(U,X_{0})p\wedge q-U\wedge X_{0}-p\wedge K(U).\end{split}$

The condition (7) implies $X_{0}\neq 0$. We may suppose that $X_{0}=e_{1}$.
Since

$R(p,q)=-p\wedge e_{1},\quad R(e_{i},e_{1})=3p\wedge e_{i},\quad
R(e_{i},q)=-e_{i}\wedge e_{1}-p\wedge K(e_{i}),\quad i>1,$

it holds $-e_{i}\wedge e_{1}\in\mathfrak{g}$. From (8) and (9) for
$\xi=-e_{i}\wedge e_{1}$, $X=p$, $Y=q$ it follows that

$(\xi\cdot R)(p,q)=-p\wedge e_{i}=-\rho(\xi)p\wedge e_{1},$

which gives a contradiction, i.e., it is impossible that $\mathfrak{g}$ is
contained in $\mathfrak{co}(1,n+1)_{\mathbb{R}p}$ and $n\geqslant 2$.

Now suppose that $n=1$. Results of [4] show that $\mathfrak{g}$ is one of the
following:

  * a.

$\mathbb{R}\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}\oplus\mathbb{R}p\wedge
q\ltimes\mathbb{R}p\wedge e_{1}$;

  * b.

$\mathbb{R}\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}\ltimes\mathbb{R}p\wedge e_{1}$;

  * c.

$\mathbb{R}(\alpha\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}+p\wedge q)\ltimes\mathbb{R}p\wedge e_{1}$, $\
\alpha\in\mathbb{R}$.

Let $\xi=p\wedge e_{1}$. If $\rho(p\wedge e_{1})\neq 0$, then it is easy to
show that the Weyl structure is closed. If $\rho(p\wedge e_{1})=0$, then as
above we obtain the following equalities for $R$:

(12) $R(p,q)=-\beta p\wedge e_{1},\quad R(p,e_{1})=0,\quad
R(e_{1},q)=\beta(2\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}+p\wedge q)-kp\wedge
e_{1},\quad\beta,k\in\mathbb{R}.$

This and the property (7) imply that $\mathfrak{g}$ is the algebra from point
c. Moreover, it holds $\alpha=2$, $\rho(2\mathop{\text{missing}}{\rm
id}\nolimits+p\wedge q)=-5$, and $k=0$.

Case 2. First consider the holonomy algebra
$\mathfrak{g}=\mathbb{R}\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}\oplus\mathfrak{so}(1,k+1)\oplus\mathfrak{so}(n-k)$,
$-1\leqslant k\leqslant n-1$. In [4], it is shown that

${\mathcal{R}}(\mathfrak{g})\cong{\mathcal{R}}(\mathfrak{so}(1,k+1))\oplus{\mathcal{R}}(\mathfrak{so}(n-k))\oplus\,\mathbb{R}^{1,k+1}\otimes\mathbb{R}^{n-k}.$

There exists an invariant line in the
$\mathfrak{so}(1,k+1)\oplus\mathfrak{so}(n-k)$-module
$\mathbb{R}^{1,k+1}\otimes\mathbb{R}^{n-k}$ if and only if $1\leqslant
k+2\leqslant 2$ and $n-k=1$. Hence, for $n\geqslant 2$ there is no
$R\in{\mathcal{R}}(\mathfrak{g})$ satisfying (7) and (9). The only possible
situation is $k=0$, $n=1$ and

$\mathfrak{g}=\mathbb{R}\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}\oplus\mathfrak{so}(1,1)\oplus\mathfrak{so}(1)\cong\mathbb{R}\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}\oplus\mathbb{R}p\wedge
q\subset\mathfrak{co}(1,2)_{\mathbb{R}p}.$

According to [4], $R$ is uniquely determined by the following equalities:

$R(p,q)=-\lambda p\wedge q,\quad R(p,e_{1})=0,\quad
R(e_{1},q)=\gamma(\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}+p\wedge q),\quad\lambda,\gamma\in\mathbb{R}.$

From (9) for $\xi=a\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}$, $a\in\mathbb{R}$ it follows that
$\rho(a\mathop{\text{missing}}{\rm id}\nolimits_{\mathbb{R}^{1,2}})=-2a$.
Next, we use (8) and (9) for $\xi=bp\wedge q$, $b\in\mathbb{R}$. Substituting
$X=e_{1}$, $Y=q$, we obtain $\rho(bp\wedge q)=-b$. Also, for $X=p$, $Y=q$ we
have $\rho(bp\wedge q)R(p,q)=0$ for all $b\in\mathbb{R}$; hence $\lambda=0$
and $R$ does not satisfy property (7) ($\mathfrak{g}$ does not contain
$\mathbb{R}p\wedge q$).

Suppose that

$\mathfrak{g}=\mathbb{R}(\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}+p\wedge
q)\oplus\mathfrak{k}\oplus\mathfrak{so}(n-k)\ltimes
p\wedge\mathbb{R}^{k}\subset\mathfrak{co}(1,n+1)_{\mathbb{R}p},\quad
0\leqslant k\leqslant n-1.$

According to [4], each $R\in{\mathcal{R}}(\mathfrak{g})$ may be written in the
form $R=R_{1}+R_{2}+R_{3}$, where $R_{1}\in{\mathcal{R}}(\mathfrak{k}\ltimes
p\wedge\mathbb{R}^{k}),$ $R_{2}\in{\mathcal{R}}(\mathfrak{so}(n-k))$, and
after a proper choice of the basis it holds

(13) $\displaystyle\begin{split}&R_{3}(e_{i},e_{k+1})=ap\wedge e_{i},\quad
1\leqslant i\leqslant k,\\\ &R_{3}(e_{k+1},q)=-a(\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}+p\wedge q),\\\
&R_{3}(e_{j},q)=-ae_{k+1}\wedge e_{j},\quad k+2\leqslant j\leqslant
n\end{split}$

for some $a\in\mathbb{R}$. If $n-k\geqslant 2$ and $j>k+1$ then $e_{k+1}\wedge
e_{j}\in\mathfrak{g}$ and from (9) we get

$\displaystyle((e_{k+1}\wedge e_{j})\cdot R_{3})(e_{j},q)=-a[e_{k+1}\wedge
e_{j},e_{k+1}\wedge e_{j}]+$ $\displaystyle R_{3}(e_{k+1},q)=$
$\displaystyle=$ $\displaystyle R_{3}(e_{k+1},q)=\rho(e_{k+1}\wedge
e_{j})R_{3}(e_{j},q).$

From the last equation it follows that $a=0$, $R_{3}=0$ and $R$ does not
satisfy property (7). We conclude that $n-k=1$ and

$\mathfrak{g}=\mathbb{R}(\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}+p\wedge q)\oplus\mathfrak{k}\ltimes
p\wedge\mathbb{R}^{n-1}.$

Using arguments as in Case 1, it is not hard to show that $\mathfrak{k}=0$,
and it holds $R=R_{3}$, where the non-zero values of $R_{3}$ are the
following:

(14) $R_{3}(U,e_{n})=ap\wedge U,\quad
R_{3}(e_{n},q)=-a(\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}+p\wedge q)$

for all $U\in\mathbb{R}^{n-1}$.

Finally applying the arguments we used just above, it is easy to check that
the holonomy algebra

$\mathfrak{g}=\mathbb{R}\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}\oplus\mathbb{R}p\wedge
q\oplus\mathfrak{k}\oplus\mathfrak{so}(n-k)\ltimes
p\wedge\mathbb{R}^{k}\subset\mathfrak{co}(1,n+1)_{\mathbb{R}p},\quad
1\leqslant k\leqslant n-1$

does not satisfy the required properties. This proves the theorem. ∎

##  3\. The local form of recurrent Weyl structures

In what follows the dot over the function denotes the partial derivative in
the direction of the coordinate $u$.

######  Theorem 7.

Let $(M,c,\nabla)$ be a non-closed Lorentzian Weyl manifold of dimension
$n+2\geqslant 4$ with recurrent curvature tensor. Then around each point of
$M$ there exist coordinates $v,x^{1},\dots,x^{n},u$ such that the class $c$ is
locally represented by the metric

(15)
$g=2dvdu+\sum_{i=1}^{n-1}(dx^{i})^{2}+e^{-2F}(dx^{n})^{2}+a(u)\sum_{i=1}^{n-1}(x^{i})^{2}(du)^{2},$

the corresponding 1-form $\omega$ is given by

$\omega=\dot{F}du,$

where $F=F(x^{n},u)$ is a function such that $\dot{F}$ satisfies the Riccati

(16) $\ddot{F}-\dot{F}^{2}=-a(u)$

and such that $\partial_{n}\dot{F}$ is non-vanishing.

It is remarkable that the Riccati equation is also related to the Einstein-
Weyl equation in dimension 2 [2, 3]. Similarly, the Einstein-Weyl equation in
Lorentzian signature and dimension 3 is equivalent to the dKP equation, and
that has led to many interesting results [1, 6, 7, 8].

Proof of Theorem 7. Let $(M,c,\nabla)$ be a non-closed Lorentzian Weyl
manifold of dimension $n+2\geqslant 4$ with recurrent curvature tensor. By
Theorem 6 its holonomy algebra coincides with

$\mathbb{R}(\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,n+1}}+p\wedge q)\ltimes
p\wedge\mathbb{R}^{n-1}\subset\mathfrak{co}(1,n+1)_{\mathbb{R}p}.$

From [5, Theorem 8] it follows that around each point of $M$ there exists a
coordinate neighborhood $U$ with coordinates $v,x^{1},\dots,x^{n},u$ and a
metric $g\in c$ such that

$g=2dvdu+h+H(du)^{2},$
$h=\sum_{i,j=1}^{n-1}\delta_{ij}dx^{i}dx^{j}+e^{-2F}(dx^{n})^{2},$
$\omega=fdu,\quad f=\dot{F},$

where $H=H(x^{1},\dots,x^{n-1},u)$ and $F=F(x^{n},u)$ are functions. The
Christoffel symbols of the connection $\nabla$ are the following:

$\displaystyle\Gamma_{v}$ $\displaystyle=0,$ $\displaystyle\Gamma_{i}$
$\displaystyle=\begin{pmatrix}0&-(\delta_{ik}\dot{F})^{n}_{k=1}&\frac{1}{2}\partial_{i}H\\\
0&0&((\delta_{ik}\dot{F})^{n}_{k=1})^{t}\\\ 0&0&0\end{pmatrix},\quad
i=1,\ldots,n-1$ $\displaystyle\Gamma_{n}$
$\displaystyle=(-\delta_{bn}\delta_{cn}\partial_{n}F)_{b,c=v,1,\ldots,n,u},$
$\displaystyle\Gamma_{u}$
$\displaystyle=\begin{pmatrix}0&(\frac{1}{2}\partial_{k}H)^{n-1}_{k=1}&0&0\\\
0&\dot{F}E_{n-1}&0&-((\frac{1}{2}\partial_{k}H)^{n-1}_{k=1})^{t}\\\ 0&0&0&0\\\
0&0&0&2\dot{F}\end{pmatrix}.$

The components of the curvature tensor are as follows:

$\displaystyle R(\partial_{v},\partial_{u})=0,\quad
R(\partial_{v},\partial_{i})=0,\quad R(\partial_{v},\partial_{n})=0,\quad
R(\partial_{i},\partial_{j})=0,\quad i,j=1,\ldots,n-1,$ $\displaystyle
R(\partial_{i},\partial_{n})=\partial_{n}\dot{F}\begin{pmatrix}0&(\delta_{ik})^{n}_{k=1}&0\\\
0&0&-((\delta_{ik})^{n}_{k=1})^{t}\\\ 0&0&0\end{pmatrix},$ $\displaystyle
R(\partial_{i},\partial_{u})=\begin{pmatrix}0&Z_{iu}&0\\\ 0&0&-Z_{iu}^{t}\\\
0&0&0\end{pmatrix},\quad\text{where}\quad
Z_{iu}=\bigg{(}\bigg{(}\frac{1}{2}\partial_{i}\partial_{k}H+\delta_{ik}(\ddot{F}-\dot{F}^{2})\bigg{)}^{n-1}_{k=1},0\bigg{)}$
$\displaystyle
R(\partial_{n},\partial_{u})=\partial_{n}\dot{F}\begin{pmatrix}0&0&-H\\\
0&1&0\\\ 0&0&2\end{pmatrix}.$

Consider the field of frames

$p=\partial_{v},\quad e_{i}=\partial_{i},\quad
q=\partial_{u}-\frac{1}{2}H\partial_{v}.$

In notation of the proof of Theorem 6 it holds $R=R_{3}$, consequently
$R(\partial_{i},\partial_{u})$ must be equal to zero, in other words,

$\frac{1}{2}\partial_{i}\partial_{k}H+\delta_{ik}(\ddot{F}-\dot{F}^{2})=0,\quad\text{for
all}\quad i,k=1,\ldots,n-1.$

From this we find out that

$H(x^{1},\dots,x^{n-1},u)=a(u)\sum_{i=1}^{n-1}(x^{i})^{2}+\sum_{i=1}^{n-1}b_{i}(u)x^{i}+c(u),$

where $a(u)$, $b_{i}(u)$, $c(u)$ are functions and

(17) $\ddot{F}-\dot{F}^{2}=-a(u).$

It is easy to see that the coordinates $v,x^{1},\dots,x^{n-1}$ may be changed
in such a way that $c(u)=0$ and all $b_{i}(u)=0$. Now it is easy to check that
the only non-zero component of $\nabla R$ is
$\nabla_{n}R(\partial_{i},\partial_{n})$, and it holds

$\partial_{n}\dot{F}\cdot\nabla_{n}R(\partial_{i},\partial_{n})=\left(\partial^{2}_{n}\dot{F}+\partial_{n}F\cdot\partial_{n}\dot{F}\right)R(\partial_{i},\partial_{n}).$

Let $\theta=\theta_{v}dv+\sum_{i=1}^{n}\theta_{i}dx^{i}+\theta_{u}du$. We get
the equality

$\theta_{n}\partial_{n}\dot{F}=\partial^{2}_{n}\dot{F}+\partial_{n}F\cdot\partial_{n}\dot{F}.$

Let $U_{0}\subset U$ be the open subset, where $\partial_{n}\dot{F}$ is non-
vanishing. Suppose that $U_{0}\neq\emptyset$. Then on $U_{0}$ it holds

$\theta=\theta_{n}dx^{n},\quad\theta_{n}=\partial_{n}F+\partial_{n}\ln|\partial_{n}\dot{F}|.$

Since $\theta_{n}$ is a smooth function on $U$, $U_{0}=U$. Note that
$d\omega=\partial_{n}\dot{F}dx^{n}\wedge du$. We conclude that for each
coordinate neighborhood $U$, either $d\omega|_{U}=0$ or $d\omega|_{U}$ is non-
vanishing on $U$. Since $M$ is connected, we conclude that $d\omega$ and
$\partial_{n}\dot{F}$ are non-vanishing on $M$. ∎

######  Proposition 5.

In the settings of the statement of Theorem 7, the conformal class of the
local metric $g$ contains a unique up to homothety metric $h$ that satisfies
(4).

###### Proof.

The proof of Theorem 7 shows that the curvature tensor $R$ satisfies

$\nabla R=\theta\otimes
R,\quad\theta=\partial_{n}(F+\ln|\partial_{n}\dot{F}|)dx^{n}.$

It holds $d\theta=-3d\omega$, and

$\theta=-3\omega+d\varphi,\quad\varphi=F+\ln|\partial_{n}\dot{F}|.$

Consider the metric

$h=e^{\frac{2}{3}\varphi}g.$

The corresponding 1-from is given by

$\omega_{h}=\omega-\frac{1}{3}d\varphi=-\frac{1}{3}\theta.$

Thus, $h$ satisfies (4). It is obvious that any other metric from the local
conformal class satisfying (4) is homothetic to $h$. ∎

######  Proposition 6.

In the settings of Theorem 7, the coordinates and the local metric may be
chosen in such a way that $a(u)=0$. Then

(18) $F=-\ln|u+\psi(x^{n})|+c(x^{n})$

for some functions $\psi(x^{n})$ and $c(x^{n})$ such that
$\psi^{\prime}(x^{n})$ is non-vanishing.

###### Proof.

Following [11] we consider the coordinate transformation:

$\displaystyle v$
$\displaystyle=\bar{v}-\frac{1}{2}\varphi^{\prime}(\bar{u})\sum_{i=1}^{n-1}(\bar{x}^{i})^{2},$
$\displaystyle x^{i}$ $\displaystyle=e^{\varphi(\bar{u})}\bar{x}^{i},\quad
i=1,\dots,{i-1},$ $\displaystyle x^{n}$ $\displaystyle=\bar{x}^{n},\quad
u=u(\bar{u}),$

where $\varphi(\bar{u})$ is a function and it holds

$u^{\prime}(\bar{u})=e^{2\varphi(\bar{u})}.$

With respect to the new coordinate system we obtain

$g=e^{2\varphi(\bar{u})}\bar{g},$

where

$\bar{g}=e^{-2G}(d\bar{x}^{n})^{2}+2d\bar{v}d\bar{u}+\sum_{i=1}^{n-1}(d\bar{x}^{i})^{2}+b(\bar{u})\sum_{i=1}^{n-1}(\bar{x}^{i})^{2}(d\bar{u})^{2},$
$G(\bar{x}^{n},\bar{u})=F(\bar{x}^{n},u(\bar{u}))+\varphi(\bar{u}),$

and

$b(\bar{u})=e^{4\varphi(\bar{u})}a(u(\bar{u}))+(\varphi^{\prime}(\bar{u}))^{2}-\varphi^{\prime\prime}(\bar{u}).$

Requiring the condition

$b(\bar{u})=0$

we obtain the Riccati equation

$\varphi^{\prime\prime}-(\varphi^{\prime})^{2}=ae^{4\varphi}.$

The function $G(\bar{x}^{n},\bar{u})$ satisfies

$\ddot{G}=\dot{G}^{2},$

where dot denotes the partial derivative with respect to $\bar{u}$. We obtain
the general solution

$G(\bar{x}^{n},\bar{u})=-\ln|\bar{u}+\psi(\bar{x}^{n})|+c(\bar{x}^{n}),$

where $\psi(x^{n})$ and $c(x^{n})$ are some functions.

Changing the notation in the obvious way we obtain the following metric from
the local conformal class:

$g=e^{-2F}(dx^{n})^{2}+2dvdu+\sum_{i=1}^{n-1}(dx^{i})^{2},\quad
F=-\ln|u+\psi(x^{n})|+c(x^{n}).$

Note that

$\partial_{x^{n}}\dot{F}=\frac{\psi^{\prime}(x^{n})}{(u+\psi(x^{n}))^{2}}.$

This implies that $\psi^{\prime}$ is non-vanishing. This proves the
proposition. ∎

The Weyl structure obtained in Proposition 6 depends on two functions
$\psi(x^{n})$ and $c(x^{n})$. We will see now that after a proper change of
the coordinates the Weyl structure depends on a single function.

Proof of Theorem 1. By Theorem 7 and Proposition 6 there exists a local metric
$g$ from the conformal class $c$ given by (15) with $F$ given by (18). Note
that (16) implies

$\partial_{x^{n}}\ddot{F}-2\dot{F}\partial_{x^{n}}\dot{F}=0.$

Consequently,

$\partial_{u}(2F-\ln|\partial_{x^{n}}\dot{F}|)=0,$

i.e., the function $2F-\ln|\partial_{x^{n}}\dot{F}|$ depends only on the
variable $x^{n}$. Consider the new coordinate $t$ related to the coordinate
$x^{n}$ by a function $x^{n}=x^{n}(t)$ such that

(19) $\frac{\partial x^{n}}{\partial
t}=e^{\frac{1}{3}(2F-\ln|\partial_{x^{n}}\dot{F}|)}.$

Then it holds

$\partial_{t}=e^{\frac{1}{3}(2F-\ln|\partial_{x^{n}}\dot{F}|)}\partial_{x^{n}},\quad
dx^{n}=e^{\frac{1}{3}(2F-\ln|\partial_{x^{n}}\dot{F}|)}dt.$

Consequently, with respect to the coordinates $t,v,x^{1},\dots,x^{n-1},u$ the
metric $g$ may be written as

$g=e^{-2G}(dt)^{2}+2dvdu+\sum_{i=1}^{n-1}(dx^{i})^{2},$

where

$G(t,u)=\frac{1}{3}(F(x^{n}(t),u)+\ln|\partial_{x^{n}}\dot{F}(x^{n}(t),u)|).$

From (18) and (19) it follows that the function $c$ satisfies

$c=\frac{3}{2}\ln\frac{\partial x^{n}}{\partial
t}+\frac{1}{2}\ln|\partial_{x^{n}}\psi|.$

Using this and the definition of $G$ we obtain

$G=\frac{1}{3}\left(-\ln|u+\psi|+c+\ln\frac{|\partial_{x^{n}}\psi|}{(u+\psi^{2})^{2}}\right)=-\ln|u+\psi|+\frac{1}{3}(c+\ln|\partial_{x^{n}}\psi|)\\\
=-\ln|u+\psi|+\frac{1}{2}\ln\left|\frac{\partial x^{n}}{\partial
t}\partial_{x^{n}}\psi\right|=-\ln|u+\psi|+\frac{1}{2}\ln|\partial_{t}\psi|.$

Thus $G$ is given by (5). This implies that

$|\partial_{t}\dot{G}|=e^{2G}.$

According to the proof of Proposition 5,

$h=e^{\frac{2}{3}(G+\ln|\partial_{t}\dot{G}|)}g=e^{2G}g=(dt)^{2}+e^{2G}\left(2dvdu+\sum_{i=1}^{n-1}(dx^{i})^{2}\right),$

and

$\omega_{h}=-\frac{1}{3}\theta=-\frac{1}{3}\partial_{x^{n}}(F+\ln|\dot{F}^{\prime}|)dx^{n}=-\partial_{x^{n}}Gdx^{n}=-\partial_{t}Gdt.$

The theorem is proved. ∎

Let us consider the recurrent Lorentzian Weyl structures in dimension 3.

######  Theorem 8.

Let $(M,c,\nabla)$ be a connected non-closed Lorentzian Weyl manifold of
dimension $3$ with recurrent curvature tensor. Then one of the following
holds:

  * $\bullet$

around each point of $M$ there exist coordinates $v,x,u$ such that $c=[g]$,

$g=2dvdu+e^{-2F}(dx)^{2},\quad\omega=\dot{F}du,$

where $F=F(x,u)$ is an arbitrary function with non-vanishing
$\partial_{x}\dot{F}$;

  * $\bullet$

around each point of $M$ there exist coordinates $v,x,u$ such that $c=[g]$,

$g=2dvdu+(dx)^{2}+H(du)^{2},\quad\omega=a(u)xdu,$
$H=H(v,x,u)=a(u)vx+\frac{1}{12}a^{2}(u)x^{4}-\frac{1}{3}\dot{a}(u)x^{3}+c(u)x,$

where the function $a(u)$ is non-vanishing.

Proof of Theorem 8. Let $(M,c,\nabla)$ be a non-closed Lorentzian Weyl
manifold of dimension $n+2=3$ with recurrent curvature tensor. Then by Theorem
6 its holonomy algebra is one of the following:

$\mathfrak{g}=\mathbb{R}(\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}+p\wedge
q),\quad\mathfrak{g}=\mathbb{R}(2\mathop{\text{missing}}{\rm
id}\nolimits_{\mathbb{R}^{1,2}}+p\wedge q)\ltimes\mathbb{R}p\wedge e_{1}.$

Consider the first case. Using the results from [5] we obtain that around each
point of $M$ there exist coordinates $v,x,u$ and a metric $g\in c$ such that

$g=2dvdu+e^{-2F}(dx)^{2}+H(du)^{2},\quad\omega=\dot{F}du,$

where $H=H(u)$ and $F=F(x,u)$ are functions. Applying a simple coordinate
transformation, we may assume that $H=0$. The only non-zero Christoffel
symbols of the connection $\nabla$ are the following:

$\displaystyle\Gamma^{x}_{xx}=-\partial_{x}F,\quad\Gamma^{u}_{uu}=2\dot{F}.$

The components of the curvature tensor are as follows

$R(\partial_{v},\partial_{x})=0,\quad R(\partial_{v},\partial_{u})=0,\quad
R(\partial_{x},\partial_{u})=\partial_{x}\dot{F}A,\quad
A=\begin{pmatrix}0&0&0\\\ 0&1&0\\\ 0&0&2\end{pmatrix}.$

For $\nabla R$ we obtain

$\displaystyle\nabla R(\partial_{v},\partial_{x})$ $\displaystyle=\nabla
R(\partial_{v},\partial_{u})=0,\quad\nabla_{v}R(\partial_{x},\partial_{u})=0$
$\displaystyle\nabla_{x}R(\partial_{x},\partial_{u})$
$\displaystyle=\left((\partial_{x}F)(\partial_{x}\dot{F})+\partial^{2}_{x}\dot{F}\right)A,$
$\displaystyle\nabla_{u}R(\partial_{x},\partial_{u})$
$\displaystyle=\left(\partial_{x}\ddot{F}-2(\partial_{x}\dot{F})\dot{F}\right)A.$

The condition $\nabla R=\theta\otimes\mathbb{R}$ is equivalent to the
following system of equations:

(20)
$\theta_{v}\partial_{x}\dot{F}=0,\quad\partial^{2}_{x}\dot{F}+(\partial_{x}F)(\partial_{x}\dot{F})=\theta_{x}\partial_{x}\dot{F},\quad\partial_{x}\ddot{F}-2(\partial_{x}\dot{F})\dot{F}=\theta_{u}\partial_{x}\dot{F},$

where $\theta_{\alpha}=\theta(\partial_{\alpha})$, $\alpha=v,x,u$. Let $U_{0}$
be the set of non-zero points of the function $\partial_{x}\dot{F}$. Then on
$U_{0}$ it holds

$\theta_{v}=0,\quad\theta_{x}=\partial_{x}(F+\ln|\partial_{x}\dot{F}|),\quad\theta_{u}=\partial_{u}(-2\dot{F}+\ln|\partial_{x}\dot{F}|).$

Hence, if $U_{0}\neq\emptyset$, then $U_{0}$ is the entire coordinate
neighbourhood.

Consider the second case. From the results of [5] we obtain that around each
point of $M$ there exist coordinates $v,x,u$ and a metric $g\in c$ such that

$g=2dvdu+(dx)^{2}+H(du)^{2},$

where $H$ is a function and the corresponding 1-form $\omega$ satisfies

$\omega=fdu,\quad\partial_{v}H=f.$

The Christoffel symbols for the connection $\nabla$ are as follows

$\displaystyle\Gamma_{v}=\begin{pmatrix}0&0&\frac{1}{2}f\\\ 0&0&0\\\
0&0&0\end{pmatrix},\quad\Gamma_{x}=\begin{pmatrix}0&-f&\frac{1}{2}\partial_{x}H\\\
0&0&f\\\
0&0&0\end{pmatrix},\quad\Gamma_{u}=\begin{pmatrix}\frac{1}{2}f&\frac{1}{2}\partial_{x}H&-\frac{1}{2}fH+\frac{1}{2}\dot{H}\\\
0&f&-\frac{1}{2}\partial_{x}H\\\ 0&0&\frac{3}{2}f\end{pmatrix}.$

Consider the field of frames

$p=\partial_{v},\quad e_{1}=\partial_{x},\quad
q=\partial_{u}-\frac{1}{2}H\partial_{v}.$

It holds $R^{x}_{uvx}=\partial_{v}f$. On the other hand, from (12) it follows
that $R(\partial_{v},\partial_{x})=R(p,e_{1})=0$. Hence, $\partial_{v}f=0$.
Using this we find the components of the curvature tensor

(21) $\displaystyle\begin{split}&R(\partial_{v},\partial_{x})=0,\quad
R(\partial_{v},\partial_{u})=\frac{1}{2}\partial_{x}f\begin{pmatrix}0&1&0\\\
0&0&-1\\\ 0&0&0\end{pmatrix},\\\
&R(\partial_{x},\partial_{u})=\begin{pmatrix}\frac{1}{2}\partial_{x}f&-\frac{1}{2}f^{2}+\dot{f}+\frac{1}{2}\partial_{x}^{2}H&-\frac{1}{2}(\partial_{x}f)H\\\
0&\partial_{x}f&\frac{1}{2}f^{2}-\dot{f}-\frac{1}{2}\partial_{x}^{2}H\\\
0&0&\frac{3}{2}\partial_{x}f\end{pmatrix}.\end{split}$

From (12) it follows that

$\frac{1}{2}f^{2}-\dot{f}-\frac{1}{2}\partial_{x}^{2}H=0\quad\text{and}\quad
R(\partial_{x},\partial_{u})=\frac{1}{2}\partial_{x}f\begin{pmatrix}1&0&-H\\\
0&2&0\\\ 0&0&3\end{pmatrix}.$

The covariant derivatives of $R$ are as follows

$\displaystyle\nabla_{a}R(\partial_{v},\partial_{x})=0,\quad a=v,x,u,$
$\displaystyle\nabla_{v}R(\partial_{v},\partial_{u})=0,\quad\nabla_{x}R(\partial_{v},\partial_{u})=\frac{1}{2}\partial_{x}^{2}f\begin{pmatrix}0&1&0\\\
0&0&-1\\\ 0&0&0\end{pmatrix},$
$\displaystyle\nabla_{u}R(\partial_{v},\partial_{u})=\left(\frac{1}{2}\partial_{x}\dot{f}-\frac{5}{4}f\partial_{x}f\right)\begin{pmatrix}0&1&0\\\
0&0&-1\\\ 0&0&0\end{pmatrix},$
$\displaystyle\nabla_{v}R(\partial_{x},\partial_{u})=\frac{1}{2}\partial_{x}^{2}f\begin{pmatrix}0&1&0\\\
0&0&-1\\\
0&0&0\end{pmatrix},\quad\nabla_{x}R(\partial_{x},\partial_{u})=\frac{1}{2}\partial_{x}^{2}f\begin{pmatrix}1&0&-H\\\
0&2&0\\\ 0&0&3\end{pmatrix},$
$\displaystyle\nabla_{u}R(\partial_{x},\partial_{u})=\left(\frac{1}{2}\partial_{x}\dot{f}-\frac{5}{4}f\partial_{x}f\right)\begin{pmatrix}1&0&-H\\\
0&2&0\\\ 0&0&3\end{pmatrix}.$

The condition $\nabla R=\theta\otimes R$ is equivalent to the following system
of equations:

$\partial_{x}^{2}f=0,\quad\theta_{v}\partial_{x}f=0,\quad\partial_{x}^{2}f=\theta_{x}\partial_{x}f,\quad\partial_{x}\dot{f}-\frac{5}{2}f\partial_{x}f=\theta_{u}\partial_{x}f,$

where $\theta_{\alpha}=\theta(\partial_{\alpha})$, $\alpha=v,x,u$. We see that
$f=f(x,u)=a(u)x+b(u)$. Applying a conformal rescaling of the metric and a
simple coordinate transformation we may assume that $b(u)=0$. Let $U_{0}$ be
the set of points where $a(u)$ is non-vanishing. Then on $U_{0}$,

$\theta_{u}=\partial_{u}\ln|a(u)|-\frac{5}{2}f.$

Since $\theta_{u}$ is smooth, $U_{0}$ is the entire coordinate neighborhood.
This proves the theorem. ∎

The curvature tensor of the first Weyl structure from Theorem 8 satisfies

$\nabla R=\theta\otimes
R,\quad\theta=\partial_{x}(F+\ln|\partial_{x}\dot{F}|)dx+\partial_{u}(-2F+\ln|\partial_{x}\dot{F}|)du.$

Again we obtain $\nabla(R\otimes l^{3}_{h})=0$ for
$h=e^{\frac{2}{3}\varphi}g$, $\varphi=F+\ln|\partial_{x}\dot{F}|$.

For the second structure from Theorem 8 it holds

$\nabla R=\theta\otimes
R,\quad\theta=\left(\partial_{u}\ln|a(u)|-\frac{5}{2}a(u)x\right)du.$

In this case, $\nabla(R\otimes l^{\frac{5}{2}}_{h})=0$ for
$h=e^{\frac{4}{5}\varphi}g$, $\varphi=\ln|a(u)|$.

##  4\. The remaining coordinate freedom and symmetries of generic structures

In Section 3, we used coordinate transformations to simplify the expressions
for the recurrent Weyl structures, resulting in Theorem 1. Here we investigate
the remaining coordinate freedom that preserves these expressions. In doing
so, we will also find the infinitesimal symmetries of generic recurrent Weyl
structures.

We start with $\dim M\geqslant 4$.

######  Proposition 7.

Let $(M,h,\omega_{h})$ be a non-closed recurrent Lorentzian Weyl manifold of
dimension $n+2\geqslant 4$. In a neighborhood of each point in $M$, there
exists local coordinates so that the pair $(h,\omega_{h})$ takes the form

(22) $\displaystyle h$
$\displaystyle=(dt)^{2}+\frac{\psi^{\prime}(t)}{(u+\psi(t))^{2}}\left(2dudv+\sum_{i=1}^{n-1}(dx^{i})^{2}\right),$
$\displaystyle\omega_{h}$
$\displaystyle=\left(\frac{\psi^{\prime}(t)}{u+\psi(t)}-\frac{\psi^{\prime\prime}(t)}{2\psi^{\prime}(t)}\right)dt,$

with $\psi^{\prime}(t)>0$. Furthermore, the Lie algebra of vector fields
preserving this form is spanned by

(23) $\partial_{v},\qquad\partial_{x^{i}},\qquad
x^{i}\partial_{v}-u\partial_{x^{i}},\qquad
x^{i}\partial_{x^{j}}-x^{j}\partial_{x^{i}},$

all of which are Killing fields of the metric $h$ that act trivially on
$\omega_{h}$, in addition to the five vector fields

(24) $\begin{gathered}\partial_{t},\qquad
2t\partial_{t}+3\sum_{i=1}^{n-1}x^{i}\partial_{x^{i}}+6v\partial_{v},\\\
\partial_{u},\qquad
2u\partial_{u}+\sum_{i=1}^{n-1}x^{i}\partial_{x^{i}},\qquad
u^{2}\partial_{u}+u\sum_{i=1}^{n-1}x^{i}\partial_{x^{i}}-\frac{1}{2}\sum_{i=1}^{n-1}(x^{i})^{2}\partial_{v}.\end{gathered}$

###### Proof.

According to Theorem 1 and the paragraph following the theorem, if
$(M,c,\nabla)$ is a non-closed recurrent Lorentzian Weyl manifold with $\dim
M\geqslant 4$, then there exists coordinates so that a representative $h\in c$
takes the form

$h=(dt)^{2}+\frac{\psi^{\prime}(t)}{(u+\psi(t))^{2}}\left(2dudv+\sum_{i=1}^{n-1}(dx^{i})^{2}\right),$

while the corresponding 1-form is given by

$\omega_{h}=\left(\frac{\psi^{\prime}(t)}{u+\psi(t)}-\frac{\psi^{\prime\prime}(t)}{2\psi^{\prime}(t)}\right)dt.$

Furthermore, the representative $h\in c$ is unique up to homothety, and
$\omega_{h}$ defines a unique 1-dimensional distribution
$\langle\partial_{t}\rangle$. This distribution is parallel:
$\nabla\partial_{t}\in\langle\partial_{t}\rangle$.

If $\varphi_{s}$ is a 1-parameter group of transformations preserving the
coordinate form of the recurrent Weyl structure given by $(h,\omega_{h})$,
then we have

$\displaystyle\varphi_{s}^{*}h$
$\displaystyle=e^{2\lambda_{s}}\left((dt)^{2}+\frac{\mu_{s}^{\prime}(t)}{(u+\mu_{s}(t))^{2}}\left(2dudv+\sum_{i=1}^{n-1}(dx^{i})^{2}\right)\right),$
$\displaystyle\varphi_{s}^{*}\omega_{h}$
$\displaystyle=\left(\frac{\mu_{s}^{\prime}(t)}{u+\mu_{s}(t)}-\frac{\mu_{s}^{\prime\prime}(t)}{2\mu_{s}^{\prime}(t)}\right)dt-d\lambda_{s}.$

Since the representative $h$ of the conformal class is unique up to homothety,
it is clear that $\lambda_{s}$ is constant, while $\mu_{s}$ is a function of
$t$ only. The equations for the vector field $Y$ whose flow is $\varphi_{s}$
are then given by

(25)
$\begin{split}L_{Y}h&=2\tilde{\lambda}h+\left(\frac{\tilde{\mu}^{\prime}(t)(u+\psi(t))-2\psi^{\prime}(t)\tilde{\mu}(t)}{(u+\psi(t))^{3}}\right)\left(2dudv+\sum_{i=1}^{n-1}(dx^{i})^{2}\right),\\\
L_{Y}\omega_{h}&=\left(\frac{\tilde{\mu}^{\prime}(t)(u+\psi(t))-\psi^{\prime}(t)\tilde{\mu}(t)}{(u+\psi(t))^{2}}-\frac{\tilde{\mu}^{\prime\prime}(t)\psi^{\prime}(t)-\psi^{\prime\prime}(t)\tilde{\mu}^{\prime}(t)}{2\psi^{\prime}(t)^{2}}\right)dt-d\tilde{\lambda},\end{split}$

where $\tilde{\lambda}=\frac{d}{ds}\big{|}_{s=0}\lambda_{s}$ is a constant and
$\tilde{\mu}=\frac{d}{ds}\big{|}_{s=0}\mu_{s}$ is a function of $t$. The
right-hand-side of these equations follows from the definition of the Lie
derivative: $L_{Y}h=\frac{d}{ds}\big{|}_{s=0}(\varphi_{s}^{*}h)$ and
$L_{Y}\omega_{h}=\frac{d}{ds}\big{|}_{s=0}(\varphi_{s}^{*}\omega_{h})$. The
vector fields (23) and (24) can be found by a direct analysis of (25), but the
computations are easier to follow if we focus on the parallel vector fields
and distributions admitted by (22).

The general recurrent Weyl structure (22) admits a unique parallel (null)
vector field $\partial_{v}$. It also admits the parallel distributions
$\langle\partial_{t}\rangle$,
$\langle\partial_{t}\rangle^{\perp}=\langle\partial_{u},\partial_{v},\partial_{x^{1}},\dots,\partial_{x^{n-1}}\rangle$,
$\langle\partial_{v}\rangle^{\perp}\cap\langle\partial_{t}\rangle^{\perp}=\langle\partial_{v},\partial_{x^{1}},\dots,\partial_{x^{n-1}}\rangle$,
and it is clear that $Y$ must preserve these distributions.

Let us write $Y$ in coordinates:

$Y=A\partial_{v}+\sum_{i=1}^{n-1}B^{i}\partial_{x^{i}}+B^{n}\partial_{t}+C\partial_{u},$

where $A,B^{1},\dots,B^{n},C$ are smooth functions. They must satisfy the
following conditions:

$\displaystyle L_{Y}\partial_{v}=-A_{0}\partial_{v}\qquad$
$\displaystyle\Leftrightarrow\qquad\partial_{v}A=A_{0},\partial_{v}B^{1}=\cdots=\partial_{v}B^{n}=\partial_{v}C=0,$
$\displaystyle L_{Y}\partial_{t}\in\langle\partial_{t}\rangle\qquad$
$\displaystyle\Leftrightarrow\qquad\partial_{t}A=\partial_{t}B^{1}=\cdots=\partial_{t}B^{n-1}=\partial_{t}C=0,$
$\displaystyle
L_{Y}\partial_{x^{i}}\in\langle\partial_{v},\partial_{x^{i}},\dots,\partial_{x^{n-1}}\rangle\qquad$
$\displaystyle\Leftrightarrow\qquad\partial_{x^{i}}C=\partial_{x^{i}}B^{n}=0,$
$\displaystyle L_{Y}\partial_{u}\in\langle\partial_{t}\rangle^{\perp}\qquad$
$\displaystyle\Leftrightarrow\qquad\partial_{u}B^{n}=0,$
$\displaystyle(L_{Y}h)(\partial_{u},\partial_{u})=0\qquad$
$\displaystyle\Leftrightarrow\qquad\partial_{u}A=0,$
$\displaystyle(L_{Y}h)(\partial_{x^{i}},\partial_{x^{j}})=0\qquad$
$\displaystyle\Leftrightarrow\qquad\partial_{x^{i}}B^{j}+\partial_{x^{j}}B^{i}=0,$
$\displaystyle(L_{Y}h)(\partial_{u},\partial_{x^{i}})=0\qquad$
$\displaystyle\Leftrightarrow\qquad\partial_{x^{i}}A+\partial_{u}B^{i}=0,$
$\displaystyle(L_{Y}h)(\partial_{x^{i}},\partial_{x^{i}})=(L_{Y}h)(\partial_{x^{j}},\partial_{x^{j}})\qquad$
$\displaystyle\Leftrightarrow\qquad\partial_{x^{i}}B^{i}=\partial_{x^{j}}B^{j},$
$\displaystyle(L_{Y}h)(\partial_{u},\partial_{v})=(L_{Y}h)(\partial_{x^{i}},\partial_{x^{i}})\qquad$
$\displaystyle\Leftrightarrow\qquad\partial_{v}A+\partial_{u}C=2\partial_{x^{i}}B^{i},$
$\displaystyle(L_{Y}h)(\partial_{t},\partial_{t})=2\tilde{\lambda}\qquad$
$\displaystyle\Leftrightarrow\qquad\partial_{t}B^{n}=\tilde{\lambda}.$

Here $A_{0}$ and $\tilde{\lambda}$ are constants. The general solution of this
system of differential equations is parametrized by $(2n-1)+\binom{n-1}{2}+6$
constants, including $\tilde{\lambda}$ and $A_{0}$. Assuming that $Y$
satisfies these conditions, the requirement that a function $\tilde{\mu}(t)$
satisfying the first line of (25) exists gives one additional relation between
the constants, namely $A_{0}=3\tilde{\lambda}$ (this is independent of
$\psi$). It reduces the number of parameters to $(2n-1)+\binom{n-1}{2}+5$, and
the vector fields satisfying all of these conditions are exactly those listed
in the proposition. They automatically satisfy the second line of (25).

The result was checked with the help of Maple’s DifferentialGeometry package.
It is straight-forward to verify that (23) are Killing fields of the metric
$h$ for any $\psi$. ∎

Proposition 7 says that all non-closed recurrent Lorentzian Weyl structures on
manifolds of dimension greater than 3 admit the infinitesimal symmetries (23),
in some local coordinate system. These $(2n-1)+\binom{n-1}{2}$ vector fields
span a Lie algebra abstractly isomorphic to
$\mathfrak{heis}(n-1)\rtimes\mathfrak{so}(n-1)$.

Additional infinitesimal symmetries are linear combinations of the vector
fields (24), but for a generic non-closed recurrent Lorentzian Weyl structure
the flow of any such linear combination acts nontrivially. Thus a generic non-
closed recurrent Lorentzian Weyl structure admits exactly
$(2n-1)+\binom{n-1}{2}$ infinitesimal symmetries.

Integrating the first two vector fields of (24) results in the transformations

$u\mapsto u,\quad x^{i}\mapsto e^{3s_{2}}x^{i},\quad t\mapsto
e^{2s_{2}}t+s_{1},\quad v\mapsto e^{6s_{2}}v,\qquad s_{1},s_{2}\in\mathbb{R},$

while integrating the last three vector fields of (24) results in the
transformations

$u\mapsto\frac{au+b}{cu+d},\;\;x^{i}\mapsto\frac{x^{i}}{cu+d},\;\;t\mapsto
t,\;\;v\mapsto v+\frac{c/2}{cu+d}\sum_{i=1}^{n-1}(x^{i})^{2},\quad
a,b,c,d\in\mathbb{R},\;ad-bc=1.$

Under these transformations, the function $\psi$ is transformed according to

$\psi(t)\mapsto\psi(e^{-2s_{2}}(t-s_{1}))),\qquad\psi(t)\mapsto\frac{a\psi(t)-b}{-c\psi(t)+d},$

respectively. Notice that the transformations parametrized by $a,b,c,d$
commute with the ones that are parametrized by $s_{1},s_{2}$. The structure of
this transformation group is
$\mathrm{SAff}_{1}(\mathbb{R})\times\mathrm{PSL}_{2}(\mathbb{R})$. This is the
connected Lie group whose action gives rise to the Lie algebra of vector
fields spanned by (24). This group can be extended with the transformation

$u\mapsto-u,\quad x^{i}\mapsto x^{i},\quad t\mapsto-t,\quad v\mapsto-v$

which also preserves the form (22) and the condition $\psi^{\prime}(t)>0$.
Under this transformation $\psi$ is transformed as
$\tilde{\psi}(\tilde{t})=-\psi(-t)$.

Notice that the two transformations $(u,x^{i},t,v)\mapsto(u,x^{i},-t,v)$ and
$(u,x^{i},t,v)\mapsto(-u,x^{i},t,-v)$ of the extended groups
$\mathrm{Aff}_{1}(\mathbb{R})$ and $\mathrm{PGL}_{2}(\mathbb{R})$,
respectively, also preserve the coordinate form of Theorem 1, but not the
condition $\psi^{\prime}(t)>0$.

To sum up, we have the following transformations, along with combinations of
such:

(26) $\displaystyle(t,\psi(t))$
$\displaystyle\mapsto\left(e^{2s_{2}}t+s_{1},\psi(e^{-2s_{2}}(t-s_{1}))\right),$
$\displaystyle(t,\psi(t))$
$\displaystyle\mapsto\left(t,\frac{a\psi(t)+b}{c\psi(t)+d}\right),\quad ad-
bc=1,$ $\displaystyle(t,\psi(t))$ $\displaystyle\mapsto(-t,-\psi(-t)).$

This can be viewed as an action of the group
$\mathrm{SAff}_{1}(\mathbb{R})\times\mathrm{PSL}_{2}(\mathbb{R})\times\mathbb{Z}_{2}\subset\mathrm{Aff}_{1}(\mathbb{R})\times\mathrm{PGL}_{2}(\mathbb{R})$
on the space $\\{\psi\in C^{\infty}(\mathbb{R})\mid\psi^{\prime}(t)>0\\}$.

######  Corollary 3.

The local equivalence problem of non-closed recurrent Lorentzian Weyl
manifolds of dimension $\geqslant 4$ can be reduced to the local equivalence
problem of functions $\\{\psi\in
C^{\infty}(\mathbb{R})\mid\psi^{\prime}(t)>0\\}$ under the Lie group action
(26).

Next, we set $\dim M=3$ and start by considering the first case from Theorem 8
where the recurrent Weyl structure $(g,\omega_{g})$ is given by

(27) $g=2dvdu+e^{-2F(x,u)}(dx)^{2},\qquad\omega_{g}=\dot{F}du$

with $\partial_{x}\dot{F}$ non-vanishing.

######  Proposition 8.

A vector field preserves the coordinate form of the general recurrent Weyl
structure (27) if and only if it takes the form

$A(x)\partial_{x}+B(u)\partial_{u}+(C_{0}+C_{1}v)\partial_{v}$

where $A,B$ are smooth functions and $C_{0},C_{1}$ are constants. Moreover,
the vector field $\partial_{v}$ is an infinitesimal symmetry for every
recurrent Weyl structure of the form (27), and a generic recurrent Weyl
structure of this form admits no additional infinitesimal symmetries.

###### Proof.

Each structure of the form (27) admits the unique $\nabla$-parallel vector
field $\partial_{v}$ and the unique $\nabla$-parallel distribution
$\langle\partial_{x}\rangle$. Any infinitesimal symmetry $Y$ is required to
preserve these structures which implies $L_{Y}\partial_{v}=0$ and
$L_{Y}\partial_{x}=\alpha\partial_{x}$ for some function $\alpha$. This gives

$Y=A(x,u)\partial_{x}+B(u)\partial_{u}+(C_{0}(u)+C_{1}v)\partial_{v}.$

The condition $(L_{Y}g)(\partial_{u},\partial_{u})=0$ implies that $C_{0}(u)$
is constant while $(L_{Y}g)(\partial_{x},\partial_{u})=0$ implies that $A$ is
a function of $x$ only. The general vector field $Y$ of this form satisfies
the conditions

$\displaystyle L_{Y}g$ $\displaystyle=2\lambda g-2\mu e^{-2F(x,u)}(dx)^{2},$
$\displaystyle L_{Y}\omega_{g}$ $\displaystyle=\dot{\mu}du-d\lambda,$

with

$\lambda=\frac{C_{1}+\dot{B}(u)}{2},\qquad\mu=B(u)\dot{F}(x,u)+A(x)\partial_{x}F(x,u)+\lambda-a^{\prime}(x),$

which means that it preserves the coordinate form (27). It is straight-forward
to show that $\partial_{v}$ is a Killing vector for all Weyl structures of the
form (27), and to verify that generic recurrent Weyl structures of the form
(27) admit no additional infinitesimal symmetries. The latter will also be
clear from the computations in Section 7. ∎

The Lie pseudogroup preserving the coordinate form of (27) is given by

$(v,x,u)\mapsto(C_{1}v+C_{0},\alpha(x),\beta(u)),\qquad\alpha^{\prime}(x)\neq
0,\quad\beta^{\prime}(u)\neq 0,\quad C_{1}\beta^{\prime}(u)>0.$

Here $\alpha,\beta$ are local diffeomorphisms on the line, and $C_{0},C_{1}$
are constants. The connected component of the Lie pseudogroup is constrained
by $\alpha^{\prime}(x)>0,\beta^{\prime}(u)>0,C>0$ (for each $(x,u)$ where the
transformations are defined). However, the extension given by
$\alpha^{\prime}(x)\neq 0,\beta^{\prime}(u)\neq 0,C\beta^{\prime}(u)>0$ also
preserves the coordinate form of the Weyl structure, giving us a Lie
pseudogroup consisting of 4 connected components.

Thus we get the following Lie pseudogroup action on the the space
$C^{\infty}(\mathbb{R}^{2})$:

(28)
$\begin{split}(x,u,F(x,u))&\mapsto\left(\alpha(x),\beta(u),F\left(\alpha^{-1}(x),\beta^{-1}(u)\right)-\frac{1}{2}\ln\left(\frac{\beta^{\prime}(u)C_{1}}{\alpha^{\prime}(x)^{2}}\right)\right),\\\
\alpha^{\prime}(x)&\neq 0,\qquad\beta^{\prime}(u)\neq 0,\qquad
C_{1}\beta^{\prime}(u)>0.\end{split}$

Next we consider the recurrent Weyl structures from Theorem 8 that locally
take the form

(29) $\displaystyle g=2dvdu+(dx)^{2}+H(du)^{2},\qquad\omega_{g}=a(u)xdu,$
$\displaystyle
H=a(u)vx+\frac{1}{12}a^{2}(u)x^{4}-\frac{1}{3}\dot{a}(u)x^{3}+c(u)x,$

where the function $a(u)$ is non-vanishing.

######  Proposition 9.

The Lie algebra of vector fields preserving the coordinate form of the general
recurrent Weyl structure (29) is spanned by the four vector fields

$\partial_{u},\qquad\partial_{v},\qquad u\partial_{u}-v\partial_{v},\qquad
u\partial_{u}+v\partial_{v}+x\partial_{x}.$

A generic Weyl structure of this form admits no infinitesmial symmetries.

###### Proof.

Taking into account that an infinitesimal symmetry must preserve the unique
parallel distribution spanned by $\partial_{v}$, we see that an infinitesimal
symmetry must be of the form

$Y=A(v,x,u)\partial_{v}+B(x,u)\partial_{x}+C(x,u)\partial_{u}.$

The functions

$(L_{Y}g)(\partial_{v},\partial_{x}),\quad(L_{Y}g)(\partial_{x},\partial_{u})$

vanish if and only if $\partial_{x}C=0$ and $\partial_{x}A+\partial_{u}B=0$.
Furthermore, the conditions

$(L_{Y}g)(\partial_{x},\partial_{x})=2B_{0}=(L_{Y}g)(\partial_{u},\partial_{v})$

are equivalent to $\partial_{x}B=B_{0}$ and
$\partial_{v}A+\partial_{u}C=2B_{0}$. Solving this overdetermined system of 4
PDEs gives

$A=F_{3}(u)+(2B_{0}-F_{2}^{\prime}(u))v-F_{1}^{\prime}(u)x,\quad
B=B_{0}x+F_{1}(u),\quad C=F_{2}(u).$

The condition $L_{Y}\omega_{g}=\tilde{a}(u)xdu$ implies $F_{1}(u)\equiv 0$. In
order for the equations

$\displaystyle L_{Y}g$
$\displaystyle=2\tilde{\lambda}g+\left(\tilde{a}(u)vx+\frac{2}{12}a(u)\tilde{a}(u)x^{4}-\frac{1}{3}\tilde{a}^{\prime}(u)x^{3}+\tilde{c}(u)x\right)(du)^{2},$
$\displaystyle L_{Y}\omega$ $\displaystyle=\tilde{a}(u)xdu-d\tilde{\lambda}$

to hold, for some pair of functions $\tilde{a}(u),\tilde{c}(u)$, we get the
additional requirements that $F_{3}$ is constant, $F_{2}$ is affine in $u$ and
$F_{1}$ vanishes identically. This means that $Y$ is a linear combination of
the four vector fields from the proposition. It is easy to verify that generic
recurrent Weyl structures of the form (29) admit no infinitesimal symmetries.
This will also be clear from the computations in Section 7. ∎

The Lie group preserving the coordinate form of (29) is given by

$(v,x,u)\mapsto\left(\frac{A_{4}}{A_{3}}v+A_{2},A_{4}x,A_{3}A_{4}u+A_{1}\right),\qquad
A_{1},A_{2}\in\mathbb{R},\quad A_{3},A_{4}\in\mathbb{R}\setminus\\{0\\}.$

Thus we are left with a 4-dimensional Lie subgroup of $\mathrm{Aff}(3)$
generated by

(30)
$\begin{split}(u,a(u),c(u))\mapsto&\left(A_{3}A_{4}u+A_{1},\frac{a\left(\frac{u-A_{1}}{A_{3}A_{4}}\right)}{A_{3}A_{4}^{2}},\frac{c\left(\frac{u-A_{1}}{A_{3}A_{4}}\right)}{A_{3}^{2}A_{4}}-\frac{A_{2}a\left(\frac{u-A_{1}}{A_{3}A_{4}}\right)}{A_{3}A_{4}^{2}}\right),\\\
&\;\;A_{1},A_{2}\in\mathbb{R},\quad
A_{3},A_{4}\in\mathbb{R}\setminus\\{0\\},\end{split}$

acting on the pair $(a(u),c(u))$ of functions. This Lie-group consists of 4
connected components.

######  Corollary 4.

(1) The local equivalence problem of non-closed recurrent Lorentzian Weyl
manifolds of dimension $3$ with 1-dimensional holonomy algebra can be reduced
to the local equivalence problem of functions in $C^{\infty}(\mathbb{R}^{2})$
under the Lie pseudogroup action (28). (2) The local equivalence problem of
non-closed recurrent Lorentzian Weyl manifolds of dimension $3$ with
2-dimensional holonomy algebra can be reduced to the local equivalence problem
of a pair of functions in $C^{\infty}(\mathbb{R})^{2}$ under the Lie group
action (30).

##  5\. The global form of recurrent Weyl structures

Let $(M,c,\nabla)$ be a non-closed recurrent Lorentzian Weyl manifold and
$\dim M=n+2\geqslant 4$. In this section we assume that $M$ is simply
connected.

Proof of Proposition 3. Let us choose an open cover $\\{U_{\alpha}\\}$ of $M$
such that for each $U_{\alpha}$ there exists a metric $h_{\alpha}$ as in
Proposition 5. On the intersection $U_{\alpha}\cap U_{\beta}$ it holds that

$h_{\alpha}=e^{2c_{\alpha\beta}}h_{\beta}$

for some $c_{\alpha\beta}\in\mathbb{R}$. Clearly this defines a Čech
1-cocycle. Since the manifold is simply connected, there exist numbers
$b_{\alpha}$ such that $b_{\alpha}-b_{\beta}=c_{\alpha\beta}$. The local
metrics $e^{-2b_{\alpha}}h_{\alpha}$ define a global metric $h$ on $M$.

Let us show that $h\in c$. We consider again the cover $\\{U_{\alpha}\\}$. Let
now $h_{\alpha}$ be the restriction of $h$ to $U_{\alpha}$. In the proof of
Theorem 7 we used the special local coordinates constructed in [5]. This
construction implies that for each $h_{\alpha}$ there exists $g_{\alpha}\in c$
and a function $f_{\alpha}$ on $U_{\alpha}$ such that

$h_{\alpha}=e^{2f_{\alpha}}g_{\alpha}|_{U\alpha}.$

Let us fix an index $\gamma$. Since for any $\alpha$ we have
$g_{\alpha},g_{\gamma}\in c$, there exists a function $\varphi_{\alpha\gamma}$
on $M$ such that

$g_{\alpha}=e^{2\varphi_{\alpha\gamma}}g_{\gamma}.$

Since on the intersection $U_{\alpha}\cap U_{\beta}$ it holds
$h_{\alpha}=h_{\beta}$, we conclude that

$f_{\alpha}+\varphi_{\alpha\gamma}=f_{\beta}+\varphi_{\beta\gamma}$

on $U_{\alpha}\cap U_{\beta}$, i.e., the functions
$f_{\alpha}+\varphi_{\alpha\gamma}$ define a global function $f$ on $M$ and

$h=e^{2f}g_{\gamma}\in c.$

∎

Proof of Theorem 3. Fix a metric $h\in c$ as in Proposition 3. It is clear
that $h$ is defined uniquely up to homothety. Fix a vector field $X$ as in the
Introduction. Consider a local Weyl structure as in Theorem 1 defined on an
open subset of $M$. The vector field $\partial_{t}$ satisfies
$h(\partial_{t},\partial_{t})=1$. This shows that locally $X=\pm\partial_{t}$.
The coordinate $t$ may be chosen in such a way that $X=\partial_{t}$. Consider
the dual 1-from $\eta$ on $M$ defined by $\eta(Y)=h(X,Y)$. Locally $\eta=dt$,
consequently,

$d\eta=0$

on $M$. Next,

$\eta(X)=h(X,X)=1.$

Recall that by the assumption, the vector field $X$ is complete, and the
manifold $M$ is simply connected. By [15, Proposition 8], $M$ is globally
diffeomorphic to the product

$M\cong\mathbb{R}\times N,$

where $N$ is a leaf of the foliation defined by the kernel of the 1-from
$\eta$. Denote by $t$ the coordinate on $\mathbb{R}$. Under the
diffeomorphism, the vector field $X$ is identified with $\partial_{t}$. It is
obvious that the just constructed coordinate $t$ is the globalization of the
local coordinate $t$ considered in Theorem 1. Consequently, in a neighborhood
$U$ of each point from $N$ there exist coordinates $v,x^{1},\dots,x^{n-1},u$
such that on $\mathbb{R}\times U$ the metric $h$ is as in Theorem 1 with a
global function $\psi$ on $\mathbb{R}$. Recall that the holonomy algebra of
the connection $\nabla$ annihilates an isotropic vector. Since $M$ is simply
connected, there exists a $\nabla$-parallel isotropic vector field $p$ on $M$.
Locally $p$ is proportional to $\partial_{v}$, and we may assume that all
coordinate systems are choosen in such a way that locally it holds
$\partial_{v}=p$. The orthogonal distribution on $N$ to $p$ defines a
foliation ${\mathcal{F}}$ on $N$. It is clear that the coordinates
$v,x^{1},\dots,x^{n-1},u$ are foliated with $u$ being the transversal
coordinate. The rest of the proof follows immediately from the results of
Section 4. ∎

##  6\. Differential invariants and the local equivalence problem

By Theorems 1 and 8, any non-closed recurrent Lorentzian Weyl structure on a
manifold $M$ is locally defined by a function $\psi(t)$ if $\dim M\geqslant
4$, or by either one function $F(u,x)$ or a pair $(a(u),c(u))$ of functions
when $\dim M=3$. In Section 4 we found the Lie pseudogroups representing the
remaining coordinate transformations acting on these structures. In this
section we compute the differential invariants of these Lie pseudogroups.

Let us start with briefly outlining the approach. For more details on
differential invariants and the theory of jet bundles, we refer to [18, 12,
13] and references therein. Consider a bundle $\pi\colon E\to B$, and let $G$
be a Lie pseudogroup of local diffeomorphisms on $E$ preserving the fibers of
$\pi$. For each $p\in B$ and $k\geqslant 0$, let $J_{p}^{k}\pi$ denote the
space of $k$-jets of sections of $\pi$ at the point $p$, and define the $k$-th
order jet space $J^{k}\pi=\sqcup_{p\in B}J_{p}^{k}\pi$. The space $J^{k}\pi$
is a bundle over $J^{l}\pi$ for $0\leqslant l<k$, and over $B$. We have an
identification $J^{0}\pi\simeq E$. A section $s$ of $\pi$ is naturally
prolonged to a section $j^{k}s$ of $J^{k}\pi$.

The Lie pseudogroup $G$ prolongs uniquely to $J^{k}\pi$ for any $k\geqslant
1$, determined by how it transforms sections of $\pi$. There exists formulas
for these prolongations (see for example (1) of [12] to see how a vector field
on $E$ is prolonged to $J^{k}\pi$), but this type of computations is very
tedious and is best to leave to a computer algebra system. For the
computations done here, Maple with its DifferentialGeometry and JetCalculus
packages was used extensively.

A differential invariant of order $k$ is a function on $J^{k}\pi$ which is
constant on $G$-orbits. If $I$ is a differential invariant of order $k$ and
$s$ is a section of $\pi$, we get a function $I\circ j^{k}s$ on $B$. The
examples we work with here satisfy the conditions of [13] and it is therefore
sufficient to consider differential invariants that are rational functions on
the fibers of $J^{k}\pi\to J^{0}\pi$, as they will separate generic orbits in
$J^{k}\pi$. After finding generators for the differential invariants in each
of our cases, we will show how to use them to solve the equivalence problem.

Note that it is possible to compute differential invariants directly from the
local form of the recurrent Weyl structures, but then one works on a bundle
with 3-dimensional base, rather than a bundle with 1- or 2-dimensional base
which we are doing here. Furthermore, the sections of this bundle would be
constrained by a nontrivial system of PDEs. The choice we have made here
simplifies the exposition but gives the same end result.

The space of functions on $\mathbb{R}$ can be identified with the space of
sections of the trivial bundle
$\pi\colon\mathbb{R}\times\mathbb{R}\to\mathbb{R}$. Let
$t,\psi_{0},\psi_{1},\dots,\psi_{k}$ be coordinates on the space $J^{k}\pi$,
and let $D_{t}$ denote the total derivative operator:

$D_{t}=\partial_{t}+\sum_{i=0}^{\infty}\psi_{i+1}\partial_{\psi_{i}}.$

######  Proposition 10.

The field of rational differential invariants of functions on $\mathbb{R}$
under the Lie group action (26) are generated by the two differential
invariants

$I=\frac{(\psi_{1}^{2}\psi_{4}-4\psi_{1}\psi_{2}\psi_{3}+3\psi_{2}^{3})^{2}}{(2\psi_{1}\psi_{3}-3\psi_{2}^{2})^{3}},\qquad
J=\frac{\psi_{1}(\psi_{1}^{2}\psi_{5}-5\psi_{1}\psi_{2}\psi_{4}+5\psi_{2}^{2}\psi_{3})}{(2\psi_{1}\psi_{3}-3\psi_{2}^{2})^{2}},$

and the invariant derivation $\hat{\partial}_{I}=D_{t}(I)^{-1}D_{t}$.

###### Proof.

The dimension of a generic orbit on $J^{k}\pi$ is equal to
$5=\dim\mathfrak{g}$ for $k\geqslant 3$. Since $\dim J^{5}\pi=7$ there exist
two algebraically independent invariants on $J^{5}(\mathbb{R})$. It is not
difficult to verify that $I$ and $J$ are invariants on $J^{5}\pi$ and that
they are independent. For each $k\geqslant 5$, we can construct a new
independent invariant on $J^{k}(\mathbb{R})$ by applying the invariant
derivation $\hat{\partial}_{I}$ $k-5$ times to $J$. Thus we have a
transcendence basis of differential invariants of any fixed order $k$. Since
all the invariants, except for $I$, are affine in the highest-order variables,
it is not difficult to see that they generate the whole field, i.e., that
there exists no algebraic extension consisting of rational invariants only. ∎

For the next proposition, we work over the bundle
$\pi\colon\mathbb{R}\times\mathbb{R}^{2}\to\mathbb{R}$, and use coordinates
$u$, $a_{0}$, $c_{0}$, $a_{1}$, $c_{1}$, $\dots$, $a_{k}$, $c_{k}$ on
$J^{k}\pi$. Let $D_{u}$ denote the total derivative operator:

$D_{u}=\partial_{u}+\sum_{i=0}^{\infty}(a_{i+1}\partial_{a_{i}}+b_{i+1}\partial_{b_{i}}).$

######  Proposition 11.

The field of rational differential invariants of pairs of functions on
$\mathbb{R}$ under the group action (30) is generated by the differential
invariants

$\displaystyle I=\frac{(a_{0}c_{1}-c_{0}a_{1})a^{4}}{a_{1}^{4}},\quad
J=\frac{a_{0}a_{2}}{a_{1}^{2}},\quad
K=\frac{(a_{0}c_{2}-c_{0}a_{2})a^{5}}{a_{1}^{5}},$

and the invariant derivation $\hat{\partial}_{I}=D_{u}(I)^{-1}D_{u}$.

###### Proof.

Generic orbits on $J^{k}\pi$ are 4-dimensional for $k\geqslant 1$. It can
easily be verified that $I,J,K$ and $\hat{\partial}_{I}$ are invariant, and
that they generate $2k-1=\dim J^{k}\pi-4$ algebraically independent invariants
on $J^{k}$. These $2k-1$ algebraically independent invariants can be taken to
be affine in the highest-order variables, implying that they generate the
whole field of rational invariants on $J^{k}\pi$ for any $k$. ∎

In the next case we work over the bundle
$\pi\colon\mathbb{R}^{2}\times\mathbb{R}\to\mathbb{R}^{2}$ and use the
coordinates $u$, $x$, $F$, $F_{u}$, $F_{x}$, $\cdots$, $F_{ux^{k-1}}$,
$F_{x^{k}}$ on $J^{k}\pi$. Let $D_{u}$ and $D_{x}$ denote the total derivative
operators (here $F_{u^{0}x^{0}}=F$):

$D_{u}=\partial_{u}+\sum_{k=0}^{\infty}\sum_{i=0}^{k}F_{u^{k-i+1}x^{i}}\partial_{F_{u^{k-i}x^{i}}},\quad
D_{x}=\partial_{x}+\sum_{k=0}^{\infty}\sum_{i=0}^{k}F_{u^{k-i}x^{i+1}}\partial_{F_{u^{k-i}x^{i}}}.$

######  Proposition 12.

The field of rational differential invariants of functions on $\mathbb{R}^{2}$
under the Lie pseudogroup action (28) is generated by the differential
invariants

$\displaystyle I$
$\displaystyle=\frac{(-2F_{u}F_{ux}+F_{uux})(F_{x}F_{ux}+F_{uxx})}{F_{ux}^{3}},$
$\displaystyle J$
$\displaystyle=\frac{(-2F_{u}F_{x}F_{ux}-2F_{u}F_{uxx}+F_{x}F_{uux}+F_{uuxx})}{F_{ux}^{2}},$

and the invariant derivations

$\nabla_{1}=\frac{F_{uxx}+F_{x}F_{ux}}{F_{ux}^{2}}D_{u},\qquad\nabla_{2}=\frac{F_{uux}-2F_{u}F_{ux}}{F_{ux}^{2}}D_{x}.$

###### Proof.

The Lie pseudogroup acts transitively on $J^{2}\pi$, while the codimension of
a generic orbit on $J^{k}$ is $\binom{k}{2}-2$ for $k\geqslant 3$. It is easy
to verify that $I,J,\nabla_{1},\nabla_{2}$ are invariant, and that the 4
invariants $I,J,\nabla_{1}(I),\nabla_{2}(I)$ are algebraically independent on
$J^{4}\pi$. It is possible to make algebraic combinations of the 4th-order
invariants to obtain 3 independent invariants $K_{1},K_{2},K_{3}$ in such a
way that $K_{1}$ depends only on $F_{uuux}$ and lower-order variables, $K_{2}$
depends only on $F_{uuxx}$ and lower-order variables and $K_{3}$ depends only
on $F_{uxxx}$ and lower-order variables. Due to the special form of
$\nabla_{1}$ and $\nabla_{2}$ it is now obvious that we can generate 4 new
independent invariants of order 5, 5 new independent invariants of order 6,
and so on. This gives us in total $\binom{k}{2}-2$ invariants on $J^{k}\pi$.
All invariants above are (or can be chosen to be) affine in the highest-order
variables, implying that they generate the whole field of rational invariants
on $J^{k}\pi$ for any $k$. ∎

We finish this section with explaining how the differential invariants can be
used for solving the local equivalence problem of generic non-closed recurrent
Lorentzian Weyl structures. The idea is essentially that of “classifying
manifolds” in [18]. Let us focus on the case covered by Proposition 12. This
is the least trivial case, since the base of $\pi$ is 2-dimensional.

First, let us note that there are other ways of generating the field of
differential invariants. If we have two independent differential invariants
$I_{1}$ and $I_{2}$ of order $k$, we can construct invariant derivations in
the following way. Take $\hat{\partial}_{I_{i}}=a_{i}D_{u}+b_{i}D_{x}$ for
$i=1,2$ where $a_{i},b_{i}$ are undetermined functions on $J^{k}\pi$ for some
$k$. The condition $\hat{\partial}_{I_{i}}(I_{j})=\delta_{ij}$ determines
$a_{1},b_{1},a_{2},b_{2}$ uniquely. The invariant derivations
$\hat{\partial}_{I_{1}},\hat{\partial}_{I_{2}}$ are called the Tresse
derivatives corresponding to the pair $I_{1},I_{2}$. In general, the Tresse
derivatives can not be constructed from every pair of algebraically
independent invariants. We must require that the linear system above has a
unique solution. If it does, it follows that the 2-form $d(I_{1}\circ
j^{k}s)\wedge d(I_{2}\circ j^{k}s)$ is nonzero for a generic section
$s\in\Gamma(\pi)$, i.e., the functions $I_{1}\circ j^{k}s$ and $I_{2}\circ
j^{k}s$ are functionally independent functions on $B$.

Now, consider the two invariants $I,J$ of Proposition 12, and define
$K=\nabla_{1}(I),L=\nabla_{2}(I)$. Using the Tresse derivatives corresponding
to $I,J$ from Proposition 12, we can generate all invariants by applying
$\hat{\partial}_{I}$ and $\hat{\partial}_{J}$ to $K$ and $L$. For a section
$s\in\Gamma(\pi)$ we can use the four invariants to define four functions on
$B$:

$I_{s}=I\circ j^{3}s,\quad J_{s}=J\circ j^{4}s,\quad K_{s}=K\circ j^{4}s,\quad
L_{s}=L\circ j^{4}s.$

If $s$ is sufficiently generic, the functions $I_{s}$ and $J_{s}$ are
functionally independent, and can be used as local coordinates. The remaining
two functions $K_{s}$ and $L_{s}$ can be expressed in terms of them:
$K_{s}=\kappa_{s}(I_{s},J_{s}),L_{s}=\lambda_{s}(I_{s},J_{s})$. Two generic
sections $s_{1}$ and $s_{2}$ are locally equivalent if and only if
$(\kappa_{s_{1}},\lambda_{s_{1}})$ is equal to
$(\kappa_{s_{2}},\lambda_{s_{2}})$ in some neighborhood.

For Proposition 10 the local equivalence problem is solved for generic
sections $s_{1}$ and $s_{2}$ by comparing the function
$J_{s_{1}}=\kappa_{s_{1}}(I_{s_{1}})$ to $J_{s_{2}}=\kappa_{s_{2}}(I_{s_{2}})$
in a similar way. For Proposition 11 it is solved by comparing the pairs
$(\kappa_{s_{1}},\lambda_{s_{1}})$ and $(\kappa_{s_{2}},\lambda_{s_{2}})$
where $\kappa_{s}$ and $\lambda_{s}$ are defined by $J_{s}=\kappa_{s}(I_{s})$
and $K_{s}=\lambda_{s}(I_{s})$.

##  7\. Recurrent Weyl structures with additional symmetries

In this section we give a local description of the recurrent Weyl manifolds
with additional infinitesimal symmetries. We start with those of dimension
$\geqslant 4$.

As we saw in Proposition 7, the symmetry algebra of any non-closed recurrent
Lorentzian Weyl structure on a manifold of dimension $n+2\geqslant 4$ contains
the Lie algebra spanned by (23), which is of dimension $(2n-1)+\binom{n-1}{2}$
and abstractly isomorphic to $\mathfrak{heis}(n-1)\rtimes\mathfrak{so}(n-1)$.
The remaining vector fields preserving the coordinate form of the Weyl
structures are given in (24), and span a Lie algebra isomorphic to
$\mathfrak{aff}_{1}(\mathbb{R})\oplus\mathfrak{sl}_{2}(\mathbb{R})$. We repeat
them here for convenience:

$\begin{gathered}Z_{1}=\partial_{t},\quad
Z_{2}=2t\partial_{t}+3\sum_{i=1}^{n-1}x^{i}\partial_{x^{i}}+6v\partial_{v},\\\
Z_{3}=\partial_{u},\quad
Z_{4}=2u\partial_{u}+\sum_{i=1}^{n-1}x^{i}\partial_{x^{i}},\quad
Z_{5}=u^{2}\partial_{u}+u\sum_{i=1}^{n-1}x^{i}\partial_{x^{i}}-\frac{1}{2}\sum_{i=1}^{n-1}(x^{i})^{2}\partial_{v}.\end{gathered}$

The general linear combination $\sum_{i=1}^{5}a_{i}Z_{i}$ is a symmetry of the
recurrent Weyl structure defined by the function $\psi$ if and only if $\psi$
satisfies

(31) $(a_{1}+2ta_{2})\psi^{\prime}(t)+a_{5}\psi(t)^{2}-2a_{4}\psi(t)+a_{3}=0.$

If either $a_{1}=a_{2}=0$ or $a_{3}=a_{4}=a_{5}=0$ it follows that $\psi$ is
constant. Since we require $\psi^{\prime}(t)>0$ it is clear that the Lie
algebra
$\mathfrak{g}\subset\mathfrak{aff}_{1}(\mathbb{R})\oplus\mathfrak{sl}_{2}(\mathbb{R})$
of additional symmetries satisfies
$\mathfrak{g}\cap\mathfrak{aff}_{1}(\mathbb{R})=\\{0\\}$ and
$\mathfrak{g}\cap\mathfrak{sl}_{2}(\mathbb{R})=\\{0\\}$. This implies in
particular that $\dim\mathfrak{g}\leqslant 2$ and that both projections

$\mathfrak{g}\to\mathfrak{aff}_{1}(\mathbb{R}),\qquad\mathfrak{g}\to\mathfrak{sl}_{2}(\mathbb{R})$

are injective. We summarize this in a Lemma.

######  Lemma 1.

The Lie algebra of infinitesimal symmetries of a non-closed recurrent
Lorentzian Weyl structure is of the form
$(\mathfrak{heis}(n-1)\rtimes\mathfrak{so}(n-1))\rtimes\mathfrak{g}$ with
$\mathfrak{g}\subset\mathfrak{aff}_{1}(\mathbb{R})\oplus\mathfrak{sl}_{2}(\mathbb{R})$,
$\mathfrak{g}\cap\mathfrak{aff}_{1}(\mathbb{R})=\\{0\\}$,
$\mathfrak{g}\cap\mathfrak{sl}_{2}(\mathbb{R})=\\{0\\}$ and
$\dim\mathfrak{g}\leqslant 2$.

We can use our remaining coordinate freedom of
$\mathrm{SAff}_{1}(\mathbb{R})\times\mathrm{PSL}_{2}(\mathbb{R})\times\mathbb{Z}_{2}$
to simplify the search for recurrent Weyl structures with additional
symmetries. Let us first consider the case when $\dim\mathfrak{g}=2$. The
transformations of $\mathrm{PSL}_{2}(\mathbb{R})$ act transitively on the
2-dimensional Lie subalgebras of $\mathfrak{sl}(2)$, so we may assume that the
projection of $\mathfrak{g}$ to $\mathfrak{sl}(2)$ is spanned by $Z_{3}$ and
$Z_{4}$ (after a coordinate change preserving the form (22) and the condition
$\psi^{\prime}(t)>0$). Since both $\langle Z_{1},Z_{2}\rangle$ and $\langle
Z_{3},Z_{4}\rangle$ contain a unique 1-dimensional ideal (spanned by $Z_{1}$
and $Z_{3}$, respectively), it follows that

$\mathfrak{g}=\langle Z_{1}+C_{1}Z_{3},Z_{2}+Z_{4}+C_{2}Z_{3}\rangle,$

where the coefficient of $Z_{4}$ is determined by the requirement that
$\mathfrak{g}$ is closed under the Lie bracket. By further applying
transformations from
$\mathrm{SAff}_{1}(\mathbb{R})\times\mathrm{PSL}_{2}(\mathbb{R})$, and
changing basis, the Lie algebra can be brought to the form $\langle Z_{1}\pm
Z_{3},Z_{2}+Z_{4}\rangle$. Only one of these preserves a Weyl structure
satisfying $\psi^{\prime}(t)>0$, namely $\mathfrak{g}=\langle
Z_{1}-Z_{3},Z_{2}+Z_{4}\rangle$. In this case the Weyl structure (22) is
$\mathfrak{g}$-invariant if and only if $\psi$ satisfies the ODEs

$\psi^{\prime}(t)-1=0,\qquad 2t\psi^{\prime}(t)-2\psi(t)=0.$

The unique solution of this pair of ODEs is $\psi(t)=t$. It follows that all
non-closed recurrent Lorentzian Weyl structures with $\dim\mathfrak{g}=2$ are
locally equivalent to the one defined by $\psi(t)=t$. It is not difficult to
see that the full Lie algebra of infinitesimal symmetries acts transitively,
implying that the Weyl manifold is locally homogeneous.

######  Proposition 13.

Let $(M,h,\omega_{h})$ be a locally homogeneous non-closed recurrent
Lorentzian Weyl structure of dimension $n+2\geqslant 4$. In a neighborhood of
each point in $M$, there exists local coordinates so that the pair
$(h,\omega_{h})$ takes the form

$h=(dt)^{2}+\frac{1}{(u+t)^{2}}\left(2dvdu+\sum_{i=1}^{n-1}(dx^{i})^{2}\right),\qquad\omega_{h}=\frac{dt}{u+t},$

for $u+t>0$. It is invariant under the Lie algebra spanned by the vector
fields

$\displaystyle\partial_{v},\qquad\partial_{x^{i}},\qquad
x^{i}\partial_{v}-u\partial_{x^{i}},\qquad
x^{i}\partial_{x^{j}}-x^{j}\partial_{x^{i}},$
$\displaystyle\partial_{t}-\partial_{u},\qquad
t\partial_{t}+u\partial_{u}+2\sum_{i=1}^{n-1}x^{i}\partial_{x^{i}}+3v\partial_{v}.$

All other non-closed recurrent Lorentzian Weyl structures have symmetry
algebra of strictly lower dimension.

Most of the vector fields in Proposition 13 are Killing fields of the metric
$h$. The only exception to this is the last vector field, which is a
homothety. However, all of these vector fields are Killing fields of the
conformally related metric $\tilde{h}=\frac{1}{(u+t)^{2}}h$. Furthermore, they
are the only Killing fields of $\tilde{h}$, and they annihilate the 1-form
$\omega_{\tilde{h}}$.

Consider now the case when $\dim\mathfrak{g}=1$. By applying
$\mathrm{SAff}_{1}(\mathbb{R})$ to a general element
$a_{1}Z_{1}+a_{2}Z_{2}\in\mathfrak{aff}_{1}(\mathbb{R})$, it is clear that we
can set either $a_{1}=0$ or $a_{2}=0$. Thus we are left with two non-
equivalent 1-dimensional Lie subalgebras of $\mathfrak{aff}_{1}(\mathbb{R})$:

$\langle Z_{1}\rangle,\qquad\langle Z_{2}\rangle.$

Classification of elements in $\mathfrak{sl}(2)$ up to transformations of
$\mathrm{PSL}_{2}(\mathbb{R})$ is equivalent to the classification of binary
quadratic forms, which is well-known (see for example [17]). From this
classification it follows that the 1-dimensional Lie subalgebras of
$\mathfrak{sl}(2)$ can be brought to one of the following, nonequivalent,
normal forms:

$\langle Z_{3}\rangle,\qquad\langle Z_{4}\rangle,\qquad\langle
Z_{3}+Z_{5}\rangle.$

Since we require that both projections
$\mathfrak{g}\to\mathfrak{aff}_{1}(\mathbb{R})$ and
$\mathfrak{g}\to\mathfrak{sl}_{2}(\mathbb{R})$ are nontrivial, we are left
with 6 cases to investigate:

Case 1, $\mathfrak{g}\subset\langle Z_{1}\rangle\oplus\langle Z_{3}\rangle$:

By using a scaling transformation from $\mathrm{SAff}_{1}(\mathbb{R})$ we can
set $\mathfrak{g}=\langle Z_{1}\pm Z_{3}\rangle$. Only one of these preserves
recurrent Weyl structures of the form (22) with $\psi^{\prime}(t)>0$, namely
$\mathfrak{g}=\langle Z_{1}-Z_{3}\rangle$. The invariant recurrent Weyl
structures are given by

$\psi(t)=t+C.$

These Weyl structures are locally homogeneous, and equivalent to the one in
Proposition 13. The integration constant $C$ can be set to 0 by using a
translation from $\mathrm{SAff}_{1}(\mathbb{R})$ (which preserves the Lie
algebra $\mathfrak{g}$).

Case 2, $\mathfrak{g}\subset\langle Z_{1}\rangle\oplus\langle Z_{4}\rangle$:

After applying a scaling transformation in $\mathrm{SAff}_{1}(\mathbb{R})$, we
have $\mathfrak{g}=\langle 2Z_{1}\pm Z_{4}\rangle$. The sign can be fixed by
an application of the transformation $(u,x^{i},t,v)\mapsto(-u,x^{i},-t,-v)$.
This results in the Lie algebra $\mathfrak{g}=\langle 2Z_{1}+Z_{4}\rangle$
which preserves the recurrent Weyl structures given by

$\psi(t)=Ce^{t}.$

The condition $\psi^{\prime}(t)>0$ is equivalent to $C>0$, and the integration
constant can be set to $C=1$ by applying a translation from
$\mathrm{SAff}_{1}(\mathbb{R})$.

Case 3, $\mathfrak{g}\subset\langle Z_{1}\rangle\oplus\langle
Z_{3}+Z_{5}\rangle$:

The Lie algebra can be brought to the form $\langle
Z_{1}\pm(Z_{3}+Z_{5})\rangle$ by a scaling transformation from
$\mathrm{SAff}_{1}(\mathbb{R})$. Of these two, only $\mathfrak{g}=\langle
Z_{1}-(Z_{3}+Z_{5})\rangle$ admits invariant recurrent Weyl structures of the
form (22) with $\psi^{\prime}(t)>0$. They are given by

$\psi(t)=\tan(t+C).$

By applying a translation from $\mathrm{SAff}_{1}(\mathbb{R})$ we can set
$C=0$.

Case 4, $\mathfrak{g}\subset\langle Z_{2}\rangle\oplus\langle Z_{3}\rangle$:

We have $\mathfrak{g}=\langle Z_{2}-2AZ_{3}\rangle$, where we can assume that
$A>0$, due to the transformation $(u,x^{i},t,v)\mapsto(-u,x^{i},-t,-v)$. The
general solution of (31) for $a_{2}=1,a_{3}=-2A,a_{1}=a_{4}=a_{5}=0$ is

$\psi(t)=A\ln|t|+C.$

The integration constant $C$ can be set to $0$ by applying a scaling
transformation from $\mathrm{SAff}_{1}(\mathbb{R})$. The condition
$\psi^{\prime}(t)>0$ is equivalent to $t>0$ when $A>0$.

Case 5, $\mathfrak{g}\subset\langle Z_{2}\rangle\oplus\langle Z_{4}\rangle$:

We have $\mathfrak{g}=\langle Z_{2}+AZ_{4}\rangle$, where $A$ is a nonzero
constant. The general solution of (31) for
$a_{2}=1,a_{4}=A,a_{1}=a_{3}=a_{5}=0$ is

$\psi(t)=C|t|^{A}.$

The condition $\psi^{\prime}(t)>0$ is equivalent to $ACt>0$. The integration
constant $C$ can be set to $1$ by applying a scaling from
$\mathrm{SAff}_{1}(\mathbb{R})$, together with the transformation
$(u,x^{i},t,v)\mapsto(-u,x^{i},-t,-v)$ if necessary. When $A=1$ the recurrent
Weyl manifold is locally homogeneous, and equivalent to the one in Proposition
13.

Case 6, $\mathfrak{g}\subset\langle Z_{2}\rangle\oplus\langle
Z_{3}+Z_{5}\rangle$:

We have $\mathfrak{g}=\langle Z_{2}-2A(Z_{3}+Z_{5})\rangle$, where we can
assume that $A>0$, due to the transformation
$(u,x^{i},t,v)\mapsto(-u,x^{i},-t,-v)$. The general invariant recurrent Weyl
structure of the form (22) is given by

$\psi(t)=\tan(A(\ln|t|+C)).$

We have

$\psi^{\prime}(t)=\frac{A}{t}(1+\psi(t)^{2})$

which means that $\psi^{\prime}(t)>0$ if and only if $t>0$, when $A>0$. By
applying a scaling transformation from $\mathrm{SAff}_{1}(\mathbb{R})$ we can
set $C=0$.

We have thus proved the following proposition.

######  Proposition 14.

Let $(M,h,\omega)$ be a non-closed recurrent Lorentzian Weyl manifold of
dimension $n+2\geqslant 4$. If its symmetry Lie algebra is exactly
$(2n+\binom{n-1}{2})$-dimensional, then in a neighborhood of each point in
$M$, there exists local coordinates such that the pair $(h,\omega)$ takes the
form (22) where $\psi$ is given by one of the following:

$\displaystyle\psi(t)$ $\displaystyle=e^{t},$ $\displaystyle\psi(t)$
$\displaystyle=\tan(t),$ $\displaystyle\psi(t)$ $\displaystyle=A\ln t,\quad
A>0,$ $\displaystyle\psi(t)$ $\displaystyle=\tan(A\ln t),\quad A>0,$
$\displaystyle\psi(t)$ $\displaystyle=|t|^{A},\quad At>0,\;A\neq 1.$

Together, Proposition 13 and Proposition 14 give a complete local description
of non-closed recurrent Lorentzian Weyl structures having a larger Lie algebra
of symmetries than the generic ones.

Next we describe the recurrent Weyl structures on 3-dimensional manifolds with
additional symmetry. Consider first the Weyl structures of the form (27). They
all share the symmetry $\partial_{v}$. We look for recurrent Weyl structures
with additional symmetries of the form

$Y=A(x)\partial_{x}+B(u)\partial_{u}+C_{1}v\partial_{v}.$

The vector field $Y$ is a symmetry of the recurrent Weyl structure defined by
the function $F$ if and only if

$A(x)\partial_{x}F+B(u)\partial_{u}F+(C_{1}+B^{\prime}(u))/2-A^{\prime}(x)=0.$

If either $A(x)\equiv 0$ or $B(u)\equiv 0$ in some neighborhood, we get
$\partial_{x}\dot{F}\equiv 0$ in the same neighborhood, which violates our
assumption that $\partial_{x}\dot{F}$ is non-vanishing.In a neighborhood of a
point where both $A$ and $B$ are non-zero, we may simultaneously change the
$x$ and $u$ coordinates (using transformations from (28)), so that $A(x)\equiv
1$ and $B(u)\equiv 1$. Furthermore, we may set $C_{1}$ equal to either $-2$ or
$0$ by scaling $u$ and $x$ and multiplying $Y$ by a constant. The general
recurrent Weyl structure of the form (27) admitting the symmetry
$\partial_{x}+\partial_{u}$ is given by

$F(x,u)=\tilde{F}(x-u),$

while the recurrent Weyl structure admitting the symmetry
$\partial_{x}+\partial_{u}-2v\partial_{v}$ is given by

$F(x,u)=x+\tilde{F}(x-u).$

In both cases $\tilde{F}^{\prime\prime}(x-u)$ is required to be non-vanishing.

Now, let’s look for recurrent Weyl structures admitting a 2-dimensional Lie
algebra of additional symmetries:

$\langle
Y_{1}=A_{1}(x)\partial_{x}+B_{1}(u)\partial_{u}+C_{1}v\partial_{v},Y_{2}=A_{2}(x)\partial_{x}+B_{2}(u)\partial_{u}+C_{2}v\partial_{v}\rangle.$

Assuming that either $A_{1}(x)$ or $A_{2}(x)$ is nonvanishing in some
neighborhood, we can change the basis and make a local coordinate change in
$x$ to set $A_{1}(x)\equiv 1$ and $A_{2}(x)=x$. This basis satisfies
$[Y_{1},Y_{2}]=Y_{1}$. Since $B_{1}(u)$ can’t vanish identically in this
coordinate neighborhood (as it would lead to $\partial_{x}\dot{F}=0$), we may
in a possibly smaller neighborhood change coordinates to set $B_{1}(u)\equiv
1$. From the commutation relation it follows that $B_{2}(u)=u+C$, where the
constant $C$ can be set to 0 by a translation in $u$. The constant $C_{1}$ is
now required to vanish, due to the commutation relation. The recurrent Weyl
structures of the form (27) admitting the symmetries
$\partial_{x}+\partial_{u}$ and
$x\partial_{x}+u\partial_{u}+C_{2}v\partial_{v}$ are given by

$F(x,u)=\frac{1-C_{2}}{2}\ln|u-x|+C,$

where $C_{2}\neq 1$, and the integration constant $C$ can be set equal to $0$
by simultaneously scaling $x$ and $u$ ($C_{2}=1$ would imply that $F$ is
constant). This defines locally homogeneous recurrent Weyl manifolds:

$g=2dvdu+|u-x|^{C_{2}-1}(dx)^{2},\qquad\omega_{g}=\frac{1-C_{2}}{2(u-x)}du.$

For a given $C_{2}$, this is defined for $u-x>0$ and $u-x<0$, respectively.
The coordinate transformation $(u,x,v)\mapsto(-u,-x,-v)$ lets us identify
these two recurrent Weyl structures, so we end up with

$g=2dvdu+(u-x)^{C_{2}-1}(dx)^{2},\qquad\omega_{g}=\frac{1-C_{2}}{2(u-x)}du,\qquad
u-x>0.$

Lastly, we consider the Weyl structures of the form (29). Assume that such a
Weyl structure has an additional symmetry of the form

$Y=A_{1}\partial_{u}+A_{2}\partial_{v}+A_{3}(u\partial_{u}-v\partial_{v})+A_{4}(u\partial_{u}+v\partial_{v}+x\partial_{x}).$

The vector field $Y$ is a symmetry of the Weyl structure defined by $a(u)$ and
$c(u)$ if and only if the following two equations hold:

(32)
$\begin{split}(A_{1}+(A_{3}+A_{4})u)a^{\prime}(u)+(A_{3}+2A_{4})a(u)=0,\\\
(A_{1}+(A_{3}+A_{4})u)c^{\prime}(u)+(2A_{3}+A_{4})c(u)+A_{2}a(u)=0.\end{split}$

If $A_{4}\neq 0$, we can multiply $Y$ with a constant factor to set $A_{4}$
equal to $1$. If $A_{4}=0$ and $A_{3}\neq 0$, then we can set $A_{3}=1$ in the
same way. Next, unless both $A_{3}=0$ and $A_{4}=0$, we can independently make
translations in $u$ and $v$ to set $A_{1}=0$ and $A_{2}=0$. We end up with two
distinct (families of) vector fields to consider:

$u\partial_{u}+v\partial_{v}+x\partial_{x}+A_{3}(u\partial_{u}-v\partial_{v}),\quad
u\partial_{u}-v\partial_{v}.$

Alternatively, if both $A_{3}=0$ and $A_{4}=0$, then we can multiply $Y$ by a
constant, setting $A_{1}$ equal to $0$ or $1$. Then, if $A_{1}=0$, we can set
$A_{2}$ equal $1$ in the same way. If $A_{1}=1$ and $A_{2}\neq 0$, we can set
$A_{2}$ equal to $\pm 1$ by applying a transformation in the flow of
$2u\partial_{u}+x\partial_{x}$. We obtain the following four distinct vector
fields:

$\partial_{v},\qquad\partial_{u},\qquad\partial_{u}-\partial_{v},\qquad\partial_{u}+\partial_{v}.$

By solving equation (32) for each of these cases, we find the symmetric
recurrent Weyl structures. If $A_{1}=A_{3}=A_{4}=0,A_{2}=1$, then (32) implies
$a(u)\equiv 0$, which is inconsistent with our assumption that $a$ is non-
vanishing. If $A_{2}=A_{3}=A_{4}=0,A_{1}=1$, the solutions are given by
$a(u)\equiv A,c(u)\equiv C$ with $A,C\in\mathbb{R}$, $A\neq 0$. By applying
coordinate transformations from our Lie group, we can normalize the constants
to obtain

(33) $a(u)\equiv 1,\qquad c(u)\equiv 0.$

It turns out that this recurrent Weyl structure admits two symmetries:

$\partial_{u},\qquad u\partial_{u}-3v\partial_{v}-x\partial_{x}.$

If $A_{1}=1=-A_{2},A_{3}=A_{4}=0$ we obtain the general solution $a(u)\equiv
A,c(u)=Au+C$. We can apply coordinate transformations from our Lie group to
obtain

(34) $a(u)\equiv 1,\qquad c(u)=u.$

If $A_{1}=1=A_{2},A_{3}=A_{4}=0$, similar computations lead to

(35) $a(u)\equiv 1,\qquad c(u)=-u.$

Choosing constants $A_{1}=A_{2}=A_{4}=0,A_{3}=1$ gives the general solution
$a(u)=A/|u|,c(u)=C/|u|^{2}$, defined for either $u<0$ or $u>0$. Let us assume
$u>0$ and remove the absolute value sign. Each solution defined for $u<0$ is
equivalent to one of the solutions defined for $u>0$ through the
transformation $(u,x,v)\mapsto(-u,x,-v)$. We can set $A=1$ using the flow of
$u\partial_{u}+v\partial_{v}+x\partial_{x}$, and the transformation
$(u,x,v)\mapsto(u,-x,v)$ if necessary. After this normalization we get the
following solutions:

(36) $a(u)=\frac{1}{u},\quad c(u)=\frac{C}{u^{2}},\quad u>0.$

Now consider the case $A_{1}=A_{2}=0,A_{4}=1$. The general solution to (32)
for this choice of constants is
$a(u)=A|u|^{-\frac{A_{3}+2}{A_{3}+1}},c(u)=C|u|^{-\frac{2A_{3}+1}{A_{3}+1}}$,
which is defined for either $u<0$ or $u>0$. Choosing $u>0$ lets us remove the
absolute value sign. Each solution defined for $u<0$ is equivalent to one of
the solutions defined for $u>0$ through the transformation
$(u,x,v)\mapsto(-u,x,-v)$. By applying a transformation from the flow of
$u\partial_{u}+v\partial_{v}+x\partial_{x}$, and the transformation
$(u,x,v)\mapsto(u,-x,v)$ if necessary, we can set $A=1$:

(37) $a(u)=u^{-\frac{A_{3}+2}{A_{3}+1}},\quad
c(u)=Cu^{-\frac{2A_{3}+1}{A_{3}+1}},\quad u>0.$

The case $A_{3}=-1$ is not covered by the above formulas, but for this case
the system (32) implies $a(u)\equiv 0$, so we disregard it.

In most cases there is only one infinitesimal symmetry. The only exception is
(33) which is also obtained in the special case of (37) given by $A_{3}=-2$
and $C=0$. This can be verified by inserting the solutions (34), (35), (36)
and (37) into (32) and solving for the constants.

To sum up, we have the following propositions when $\dim M=3$.

######  Proposition 15.

Let $(M,g,\omega_{g})$ be a non-closed recurrent Lorentzian Weyl manifold of
dimension 3 admitting a 3-dimensional Lie algebra of infinitesimal symmetries.
In a neighborhood of each point in $M$, there exists local coordinates so that
the pair $(g,\omega_{g})$ takes the form

$g=2dvdu+(u-x)^{C-1}(dx)^{2},\qquad\omega_{g}=\frac{1-C}{2(u-x)}du,\qquad
C\neq 1,$

with $u-x>0$. The Lie algebra of infinitesimal symmetries is transitive, and
spanned by

$\partial_{v},\qquad\partial_{x}+\partial_{u},\qquad
x\partial_{x}+u\partial_{u}+Cv\partial_{v}.$

######  Proposition 16.

Let $(M,g,\omega_{g})$ be a non-closed recurrent Lorentzian Weyl manifold of
dimension 3 admitting a 2-dimensional Lie algebra of infinitesimal symmetries.
Then one of the following statements hold.

  * •

In a neighborhood of a generic point in $M$ there exists local coordinates so
that the pair $(g,\omega_{g})$ takes the form

$g=2dvdu+e^{Cx+F(x-u)}(dx)^{2},\qquad\omega_{g}=F^{\prime}(x-u)du,$

for some $C\in\\{0,1\\}$ and some function $F$ with $F^{\prime\prime}(x-u)$
non-vanishing. In these coordinates, the Lie algebra of symmetries contains
the vector fields spanned by

$\partial_{v},\qquad\partial_{x}+\partial_{u}-2v\partial_{v}.$

  * •

In a neighborhood of any point in $M$, there exists local coordinates so that
the pair $(g,\omega_{g})$ takes the form

$g=2dvdu+(dx)^{2}+x\left(v+\frac{1}{12}x^{3}\right)(du)^{2},\qquad\omega_{g}=xdu.$

In these coordinates, the Lie algebra of symmetries is spanned by

$\partial_{u},\qquad u\partial_{u}-3v\partial_{v}-x\partial_{x}.$

Notice that in the second case the Lie algebra of symmetries is solvable. In
the first case, $C=0$ gives an abelian Lie algebra of symmetries while $C=1$
gives a solvable one.

######  Proposition 17.

Let $(M,g,\omega_{g})$ be a non-closed recurrent Lorentzian Weyl manifold of
dimension 3 admitting a 1-dimensional Lie algebra of infinitesimal symmetries.
Then one of the following statements hold.

  * •

In a neighborhood of a generic point in $M$ there exists local coordinates so
that the pair $(g,\omega_{g})$ takes the form

$g=2dvdu+e^{-2F(x,u)}(dx)^{2},\qquad\omega_{g}=\dot{F}du$

with $\partial_{x}\dot{F}$ non-vanishing. All metrics of this form share the
infinitesimal symmetry $\partial_{v}$.

  * •

In a neighborhood of any point in $M$ there exists local coordinates so that
the pair $(g,\omega_{g})$ takes the form

$\displaystyle g=2dvdu+(dx)^{2}+H(du)^{2},\qquad\omega_{g}=a(u)xdu,$
$\displaystyle
H=a(u)vx+\frac{1}{12}a^{2}(u)x^{4}-\frac{1}{3}\dot{a}(u)x^{3}+c(u)x,$

where $a(u)$ and $c(u)$ are given by either (34), (35) (36) or (37). These
recurrent Weyl structures have the infinitesimal symmetries given,
respectively, by

$\partial_{u}-\partial_{v},\quad\partial_{u}+\partial_{v},\quad
u\partial_{u}-v\partial_{v},\quad
u\partial_{u}+x\partial_{x}+v\partial_{v}+A_{3}(u\partial_{u}-v\partial_{v}).$

##  8\. Homogeneous spaces

Consider the locally homogeneous Weyl structure given by $h$ and $\omega_{h}$
as in Proposition 13, and define $g=e^{2\ln(u+t)}h$. Consider the new
coordinates

$t^{\prime}=\frac{(t+2u)t}{2},\quad
v^{\prime}=v-\frac{t^{3}}{3}-\frac{ut^{2}}{2},$

and denote them again by $t$ and $v$, respectively. With respect to the new
coordinates the metric $g$ and the 1-form $\omega_{g}$ are given by

$g=(dt)^{2}+2dvdu+\sum_{i=1}^{n-1}(dx^{i})^{2},\qquad\omega_{g}=-\frac{du}{\sqrt{2t+u^{2}}}.$

The symmetry algebra of the Weyl structure is spanned by the vector fields

$\displaystyle\partial_{v},\qquad\partial_{x^{i}},\qquad
V_{i}=x^{i}\partial_{v}-u\partial_{x^{i}},\qquad
x^{i}\partial_{x^{j}}-x^{j}\partial_{x^{i}},\quad 1\leqslant i,j\leqslant
n-1,$ $\displaystyle X=\partial_{u}+t\partial_{v}-u\partial_{t},\qquad
Y=2t\partial_{t}+u\partial_{u}+2\sum_{i=1}^{n-1}x^{i}\partial_{x^{i}}+3v\partial_{v}.$

We will denote this Lie algebra by $\mathfrak{g}$. Let $\mathfrak{g}_{0}$ be
the subalgebra of $\mathfrak{g}$ spanned by the vector fields $\partial_{v}$,
$\partial_{x^{1}},\dots,\partial_{x^{n-1}}$, $X$, $Y$. The only non-zero Lie
brackets of $\mathfrak{g}_{0}$ are

(38)
$[Y,\partial_{v}]=-3\partial_{v},\quad[Y,\partial_{x^{i}}]=-2\partial_{x^{i}},\quad
1\leqslant i\leqslant n-1,\quad[Y,X]=-X.$

Consider the new metric

$b=\frac{4}{(2t+u^{2})^{2}}g.$

The corresponding 1-form is given by

$\omega_{b}=\left(\frac{2u}{2t+u^{2}}-\frac{1}{\sqrt{2t+u^{2}}}\right)du+\frac{2}{2t+u^{2}}dt.$

It is easy to check that the symmetry algebra $\mathfrak{g}$ annihilates both
$b$ and $\omega_{b}$. This will imply the equality (6). Let $U$ be the domain
in $\mathbb{R}^{n+2}$ defined by the condition

$2t+u^{2}>0.$

The flows of the vector fields $X$ and $Y$ are given respectively by

$\varphi_{s}(t,v,x,u)=\left(-\frac{s^{2}}{2}-us+t,-\frac{s^{3}}{6}-\frac{us^{2}}{2}+ts+v,x,u+s\right),$
$\psi_{r}(t,v,x,u)=\left(e^{2r}t,e^{3r}v,e^{2r}x,e^{r}u\right).$

It is clear that the flows are defined for each $s,r\in\mathbb{R}$ and
preserve the space $U$. Consequently the symmetry algebra $\mathfrak{g}$ may
be integrated to the action of the corresponding connected Lie subgroup $G$ on
$U$. Consider the connected Lie subgroup $G_{0}$ of $G$ generated by the
translations in the directions of $v,x^{1},\dots,x^{n-1}$ and by the flows
$\varphi_{s}$, $\psi_{r}$. Let us fix $t$ and $u$ satisfying $2t+u^{2}>0$ and
set

$s=u\sqrt{\frac{2}{2t+u^{2}}},\qquad r=-\ln\sqrt{\frac{2}{2t+u^{2}}}.$

Then

$\psi_{r}\varphi_{s}(1,0,0,0)=(t,v_{0},0,u)$

for certain $v_{0}\in\mathbb{R}$. Obviously this implies that $G_{0}$ acts
transitively on $U$. Moreover, the action is simply transitive. By the
construction, $G_{0}$ is simply connected, and its Lie algebra is
$\mathfrak{g}_{0}$ with the Lie brackets multiplied by $-1$. This shows that
the Lie group $G_{0}$ is isomorphic to the Lie group defined in the
Introduction. The explicit isomorphism may be described in the following way.
For $r,v,u\in\mathbb{R}$, $r>0$, and $y\in\mathbb{R}^{n-1}$, denote by
$(r,w,y,s)$ the transformation

$(t,v,x,u)\in U\mapsto\psi_{\ln r}\varphi_{s}\left(t,v+w,x+y,u\right)\in U.$

Fixing the point $(1,0,0,0)$, we may identify the Lie group $G_{0}$ with the
space $U$:

$(r,w,y,s)\in G_{0}\mapsto\psi_{\ln
r}\varphi_{s}\left(1,w,y,0\right)=\left(r^{2}\left(-\frac{s^{2}}{2}+1\right),r^{3}\left(-\frac{s^{3}}{6}+s+w\right),r^{2}y,rs\right)\in
U.$

The pull-back under this identification of the metric $b$ and the 1-form
$\omega_{b}$ defines on $G_{0}$ the left-invariant metric and left-invariant
1-form that we denote by the same symbols. Consequently, and the Weyl
connection $\nabla$ becomes a left-invariant connection on $G_{0}$.

Proof of Theorem 5. Let $(M,c,\nabla)$ be a non-closed recurrent Lorentzian
Weyl manifold. Suppose that the manifold $M$ is simply connected and
$\operatorname{Aut}(M,c,\nabla)$-homogeneous. Denote by $H$ the Lie group
$\operatorname{Aut}^{0}(M,c,\nabla)$ and let $\mathfrak{h}$ denote the
corresponding Lie algebra of vector fields on $M$. It is clear that $M$ is
$H$-homogeneous.

Consider an open subspace $U\subset M$ on which the Weyl structure may be
described as in Proposition 13. Due to rigidity of conformal Killing vector
fields, the restrictions of the vector fields from $\mathfrak{h}$ to $U$ is an
isomorphism of $\mathfrak{h}$ onto a Lie algebra of vector fields on $U$.
These vector fields are local symmetries of the induced Weyl structure on $U$,
hence we get an injection $\mathfrak{h}\hookrightarrow\mathfrak{g}$. The Lie
group $G$ acts on $G_{0}$ by transformations of the Weyl structure on $G_{0}$.
Denote by $G_{e}$ the stabilizer of the identity element $e\in G_{0}$ and by
$\mathfrak{g}_{e}\subset\mathfrak{g}$ the corresponding subalgebra. Since $G$
is connected and $G_{0}$ is simply connected, $G_{e}$ is connected. Fix a
point $x\in M$. Let $H_{x}\subset H$ be the stabilizer of $x$. Again, $H_{x}$
is connected. The injection $\mathfrak{h}\hookrightarrow\mathfrak{g}$ defines
the Lie group homomorphism

$\varphi:H\to G.$

It is clear that the image of $\mathfrak{h}$ in $\mathfrak{g}$ evaluated at a
point of $G_{0}$ coincides with the tangent space at that point, and this
imples that the action of $\varphi(H)$ is transitive on $G_{0}$. It is clear
that under the injection $\mathfrak{h}\hookrightarrow\mathfrak{g}$, the Lie
algebra $\mathfrak{h}_{x}$ is mapped to a subalgebra of $\mathfrak{g}_{e}$,
and, consequently,

$\varphi(H_{x})\subset G_{e}.$

Define the map

$f:M\to G_{0}$

in the following way: take an $y\in M$, then there exists an $h\in H$ such
that $y=hx$, and we set

$f(y)=\varphi(h)e.$

The map $\varphi$ is well-defined. Indeed, if $hx=h_{1}x$, then
$h^{-1}_{1}h\in H_{x}$, and $\varphi(h^{-1}_{1}h)\in G_{e}$, i.e.,
$\varphi(h_{1})e=\varphi(h)e$. Consider the maps $f_{1}:H\to M$ and
$f_{2}:H\to G_{0}$,

$f_{1}(h)=hx,\qquad f_{2}(h)=\varphi(h)e.$

Then,

$f_{2}=f\circ f_{1}.$

Since $f_{1}$ and $f_{2}$ are surjective submersions, $f$ is smooth and it is
an surjective submersion as well. Since both $M$ and $G_{0}$ are simply
connected, $f$ is a diffeomorphism. The diffeomorphism $f$ and the Weyl
structure on $M$ define a $\varphi(H)$-invariant Weyl structure
$(\bar{c},\bar{\nabla})$ on $G_{0}$. Moreover, by Proposition 13, the local
Weyl structure on each open subset $U\subset M$ as above is
$\mathfrak{g}$-invariant. Consequently, the induced Weyl structure
$(\bar{c},\bar{\nabla})$ on $G_{0}$ is $G$-invariant.

######  Lemma 2.

If $(\bar{c},\bar{\nabla})$ is a $G$-invariant recurrent Weyl structure on
$G_{0}$, then it coincides with $(c,\nabla)$.

Proof. Consider on $G_{0}$ the coordinates $t,v,x^{1},\dots,x^{n-1},u$ that we
considered above on $U$. It is easy to see that the vector fields

$\partial_{v},\partial_{x^{i}},V_{j},X,Y,\quad 1\leqslant i,j\leqslant n-1$

span an ideal, which is the radical of $\mathfrak{g}$. We will denote this
ideal by $\mathfrak{a}$. Next,

$\mathfrak{b}=[\mathfrak{a},\mathfrak{a}]=\textrm{span}\\{\partial_{v},\partial_{x^{1}},\dots,\partial_{x^{n-1}},V_{1},\dots,V_{n-1},X\\},$
$[\mathfrak{b},\mathfrak{b}]=\textrm{span}\\{\partial_{v},\partial_{x^{1}},\dots,\partial_{x^{n-1}}\\},$
$[[\mathfrak{b},\mathfrak{b}],\mathfrak{b}]=\mathbb{R}\partial_{v}.$

Consider the structure $(\bar{c},\bar{\nabla})$. In a neighborhood of each
point form $U$ there exist an open subset $V\subset U$ with coordinates
$\bar{t},\bar{v},\bar{x}^{1},\dots,\bar{x}^{n-1},\bar{u}$ and a local metric
$\bar{b}$ from $\bar{c}$ with the 1-form $\omega_{\bar{b}}$ having the same
structure as $b$ and $\omega_{b}$ above. The symmetry algebra
$\bar{\mathfrak{g}}$ coincides with $\mathfrak{g}$. Consequently, the
canonical ideals of $\mathfrak{g}$ coincide with similar ideals of
$\bar{\mathfrak{g}}$. We conclude that $\partial_{\bar{v}}$ is proportional to
$\partial_{v}$, each $\partial_{\bar{x}^{i}}$ is a linear combination of
$\partial_{v},\partial_{x^{1}},\dots,\partial_{x^{n-1}}$, etc. This easily
implies that $\bar{b}$ and $\omega_{\bar{b}}$ written with respect to the
coordinates $t,v,x^{1},\dots,x^{n-1},u$ coincide respectively with $c_{1}b$
and $\omega_{b}$ for some constant $c_{1}\in\mathbb{R}$. This proves the
lemma. ∎

The lemma implies the proof of the theorem. ∎

##  9\. Einstein-Weyl structures with recurrent curvature tensor

An Einstein-Weyl manifold is a Weyl manifold $(M,c,\nabla)$ satisfying

$\mathrm{Ric}^{\mathrm{sym}}=\Lambda g$

for some $g\in c$, where $\mathrm{Ric}^{\mathrm{sym}}$ is the symmetrized part
of the Ricci tensor of $\nabla$ and $\Lambda$ is a smooth function. These
structures have received a lot of attention, in particular in three dimensions
(see for example [1, 8]). In this section we apply our previous results to
give a complete local description of the connected non-closed Einstein-Weyl
structures with recurrent curvature tensor.

It turns out that there are no non-closed Einstein-Weyl manifolds with
recurrent curvature tensor when $\dim M\geqslant 4$. This can be seen either
by computing $\mathrm{Ric}^{\mathrm{sym}}$ for $(M,h,\omega_{h})$ from
Proposition 7 and verifying that the equation
$\mathrm{Ric}^{\mathrm{sym}}=\Lambda h$ is never satisfied, or by looking at
the weighted parallel spinors and applying results from [5]. To find the
three-dimensional Einstein-Weyl manifolds with recurrent curvature tensor, we
use the local coordinate descriptions of Theorem 8.

The Weyl structure defined by

$g=2dvdu+e^{-2F}(dx)^{2},\qquad\omega=\dot{F}du$

with $F=F(x,u)$ has the following symmetrized Ricci tensor:

$\mathrm{Ric}^{\mathrm{sym}}=-\partial_{x}\dot{F}dudx.$

Clearly, the Weyl structure is Einstein-Weyl if and only if $\Lambda\equiv 0$
and $\partial_{x}\dot{F}\equiv 0$. Thus, there are no nonclosed Einstein-Weyl
structures of this form. On the other hand, the Weyl structure given by

$g=2dvdu+(dx)^{2}+H(du)^{2},\qquad\omega=H_{v}du$

with

(39)
$H(v,x,u)=a(u)vx+\frac{1}{12}a^{2}(u)x^{4}-\frac{1}{3}\dot{a}(u)x^{3}+c(u)x$

is Einstein-Weyl for any choices of $a$ and $c$. Indeed, the symmetrized Ricci
tensor is given by

$\mathrm{Ric}^{\mathrm{sym}}=-H_{vv}\left(2dvdu+(dx)^{2}\right)+\frac{1}{2}\left(H_{v}^{2}-2H_{vu}-H_{xx}-HH_{vv}\right)du^{2},$

and it follows that the structure is Einstein-Weyl if and only if

$2H_{vu}+H_{xx}-(HH_{v})_{v}=0.$

In this case $\Lambda=-H_{vv}$. This is essentially a known result: The above
differential equation is equivalent to the dKP equation in [8] under the point
transformation $(v,x,u,H)\mapsto(v,x,-2u,-H)$. It is easy to verify that the
function (39) is a solution of the above PDE for any pair $a,c$. Moreover, if
$H$ is of the form (39), then $\Lambda\equiv 0$ and thus the Ricci tensor is
skew-symmetric. We arrive at the following proposition.

######  Proposition 18.

Let $(M,g,\omega)$ be a non-closed Lorentzian Einstein-Weyl manifold with
recurrent curvature tensor. Then $\dim M=3$ and around each point of $M$ there
exist local coordinates in which $g$ and $\omega$ take the form

$g=2dvdu+(dx)^{2}+H(du),\qquad\omega=H_{v}du$

where $H$ is given by (39) with non-vanishing $a(u)$.

Since the non-closed Lorentzian Einstein-Weyl manifolds with recurrent
curvature tensor are exactly those of the form (29), the symmetry analysis
from the previous section can be directly applied to the recurrent Einstein-
Weyl manifolds. In particular, there is a unique one with maximal
(2-dimensional) Lie algebra of symmetries:

$g=2dvdu+(dx)^{2}+x\left(v+\frac{1}{12}x^{3}\right)(du)^{2},\qquad\omega_{g}=xdu.$

Acknowledgements. The authors thank Boris Kruglikov for helpful discussions.
A.D. was supported by grant MUNI/A/1092/2021 of Masaryk University. A.G.
acknowledges institutional support of University of Hradec Králové. E.S. was
supported by the Norwegian Financial Mechanism 2014-2021 (project registration
number 2019/34/H/ST1/00636) and the Tromsø Research Foundation (project “Pure
Mathematics in Norway”).

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