Locally conformally flat Kähler and para-Kähler manifolds

Abstract

We complete the classification of locally conformally flat Kähler and para-Kähler manifolds, describing all possible non-flat curvature models for Kähler and para-Kähler surfaces.

1 Introduction

Let $(M,g,J_\varepsilon )$ be a (para)-Kähler manifold, where $J_\varepsilon$ ($J_\varepsilon ^2=-\varepsilon {\text {Id}}$) denotes the parallel (para) complex structure and $g(J_\varepsilon X,J_\varepsilon Y)=\varepsilon g(X,Y)$ for $\varepsilon =\pm 1$. Since the (para) complex structure is parallel, one has the well-known curvature identity $R(X,Y)\cdot J_\varepsilon =J_\varepsilon \cdot R(X,Y)$ which strongly restricts the curvature tensor. A consequence of the above relation is that the Ricci tensor of a 2n-dimensional locally conformally flat (para)-Kähler manifold satisfies $(2n-4)\rho (X,Y)=-\frac{\tau }{2n-1}g(X,Y)$, and thus, it is flat if $n\ge 3$. Moreover, the Ricci tensor is parallel, and thus, (M, g) is locally symmetric in the four-dimensional case [17, 22]. In the positive definite case, it was shown by Tanno [22] that locally conformally flat four-dimensional Kähler metrics are either flat or a product $M_1(c)\times M_2(-c)$ of two surfaces of constant opposite curvature. This result, which strongly depends on the commutation ${\text {Ric}}\cdot J_+=J_+\cdot {\text {Ric}}$ and the diagonalizability of the Ricci operator, does not cover all the possibilities in the pseudo-Riemannian case of neutral signature $(--++)$ as already pointed out by Patterson in [20].

Motivated by some recent interest on locally conformally flat (para)-Kähler surfaces [2, 3, 14], the main purpose of this paper is to complete the above local classification showing the existence of two additional possibilities. Recall that a pseudo-Riemannian manifold (M, g) is said to be indecomposable if there is no non-degenerate subspace that is invariant under the action of the holonomy group. Moreover, the holonomy group may act indecomposably without acting irreducibly.

Theorem 1.1

Let $(M,g,J_-)$ be an indecomposable locally conformally flat para-Kähler surface. Then it is locally isometric to the cotangent bundle $T^*\Sigma$ of a flat affine surface $(\Sigma ,D)$ with paracomplex structure determined by $J_-\mid _{{\text {ker}}\pi _*}={\text {Id}}$, where $\pi :T^*\Sigma \rightarrow \Sigma$ is the canonical projection, and the metric g is given by

• (i) $g=\iota T\circ \iota {\text {Id}}+g_D$, where T is a parallel nilpotent (1, 1)-tensor field on $(\Sigma ,D)$, or

• (ii) $g=\iota T\circ \iota {\text {Id}}+g_D$, where T is a parallel (1, 1)-tensor field on $(\Sigma ,D)$ satisfying $T^2=-k^2{\text {Id}}$.

Moreover, in both cases the para-Kähler two-form $\Omega _-(X,Y)=g(J_-X,Y)$ is the canonical symplectic form of the cotangent bundle.

Para-Kähler surfaces corresponding to case (i) above have been already reported by Patterson [20] (see also [17]), while case (ii) above seems to be missing in previous works. Considering the Ricci operator in case (ii) above, $\frac{1}{k}{\text {Ric}}$ defines a self-adjoint complex structure which is parallel (hence a complex Riemannian structure). Since the Ricci operator and the paracomplex structure $J_-$ commute with each other, one has that $J_+=\frac{1}{k}{\text {Ric}}J_-$ is a complex structure so that $(g,J_+)$ becomes a locally conformally flat Kähler structure.

Theorem 1.2

Let $(M,g,J_+)$ be an indecomposable locally conformally flat Kähler surface. Then it is locally isometric to the cotangent bundle $(T^*\Sigma ,g)$ of a Riemannian surface $(\Sigma ,g_\Sigma )$ of constant curvature with metric

• (i) $g=g_{\nabla ^\Sigma }$ if the Gauss curvature is nonzero, or

• (ii) $g=\iota J_\Sigma \circ \iota {\text {Id}}+g_{\nabla ^\Sigma }$ if the Gauss curvature vanishes,

where $\nabla ^\Sigma$ denotes the Levi–Civita connection of $(\Sigma ,g_\Sigma )$ and $J_\Sigma$ is the Kähler structure on $\Sigma$ corresponding to the Riemannian volume form.

Moreover, the complex structure $J_+$ on $T^*\Sigma$ is determined by the symplectic form $\Omega _+=-d\iota J_\Sigma$.

While the metric tensor (and hence the curvature structure) corresponding to case (ii) in Theorems 1.1 and 1.2 is the same, the corresponding symplectic structures are different. Moreover, the metrics corresponding to cases (i) in the two theorems above correspond to different curvature models which are treated in Sect. 2.

Kähler metrics in Theorem 1.2-(i) are locally isometric (up to reversing the metric) to those studied by Guilfoyle and Klingenberg [14] on the space ${\mathbb {L}}$ of oriented affine lines in ${\mathbb {R}}^3$ by means of the minitwistor correspondence. Analogously, metrics in Theorem 1.1-(i) are locally isometric (up to reversing the metric) to the space ${\mathbb {L}}^\pm$ of oriented spacelike (or timelike) lines in ${\mathbb {R}}^3_1$ (see [2, 3]). On the other hand, metrics corresponding to Assertion (ii) in theorems 1.1 and 1.2 are locally isometric (up to reversing the metric) to the non-Einstein para-Kähler and Kähler metrics in the space of spacelike and timelike oriented geodesics of the de Sitter space ${\mathbb {L}}^\pm (d{\mathbb {S}}^3)$ constructed by Anciaux [3]. Furthermore, the non-locally conformally flat Kähler–Einstein metrics in ${\mathbb {L}}^\pm (d{\mathbb {S}}^3)$ constructed in [3] correspond to those in Remark 3.10.

Para-Kähler manifolds are Walker, since the eigenspaces corresponding to the $\pm 1$ eigenvalues of the paracomplex structure determine parallel degenerate distributions. Moreover, since locally symmetric indefinite Kähler surfaces with non-diagonalizable Ricci operator are Walker as well (cf. Lemma 5.1), we devote Sect. 3.2 to analyse the geometry of locally symmetric self-dual Walker surfaces. Examples of these metrics have been recently constructed on tangent bundles [4, 13].

The proofs of Theorems 1.1 and 1.2 are finally given in Sects. 4 and 5, respectively.

2 Algebraic preliminaries: curvature models

We work at the purely algebraic level. Let $(V,\langle \,.,. \rangle , J_\pm )$ be a (para)-Hermitian inner product space. An algebraic curvature tensor is a multilinear map ${\mathcal {A}}:V\times V\times V\times V\rightarrow {\mathbb {R}}$ satisfying the algebraic identities of the curvature tensor. Let $\rho _{\mathcal {A}}(x,y)={\text {tr}}\{ z\mapsto {A}(x,z)y\}$ be the associated Ricci tensor, where A(x,y)z represents the curvature operator $\langle A(x,y)z,w\rangle ={\mathcal {A}}(x,y,z,w)$, and let $W_{\mathcal {A}}$ be the corresponding Weyl curvature tensor. If the Weyl curvature tensor vanishes, then the algebraic curvature tensor ${\mathcal {A}}$ is determined by the Ricci tensor $\rho _{\mathcal {A}}$.

A straightforward calculation shows that the Ricci operator ${\text {Ric}}_{\mathcal {A}}$ of a locally conformally flat (para)-Kähler surface is either diagonalizable or its Jordan normal form (following the discussion in [18]) is as follows.

• (i) The Ricci operator has two $2\times 2$ Jordan blocks. At each point there is a basis $\{ u_1,v_1,u_2,v_2\}$ of the tangent space so that the Ricci operator and the nonzero inner products are given by

  $$\begin{aligned} \text {Ric}=\left( \begin{array}{llll} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \end{array}\right) ,\qquad \langle u_i,v_i\rangle =\varepsilon _i,\quad \varepsilon _i^2=1\, \quad (i=1,2)\,. \end{aligned}$$

  Moreover (up to reversing the metric), there are two distinct possibilities as $\varepsilon _1\varepsilon _2=\pm 1$. Let $(V,\langle \,.,. \rangle , {\mathcal {A}})$ be an algebraic curvature tensor with vanishing Weyl tensor whose Ricci operator corresponds to one of the previous situations.

  • (i.1) If $\varepsilon _1\varepsilon _2=1$, there is a unique (up to sign) Ricci-commuting Hermitian structure $(\langle \,.,. \rangle , J_+)$ given by $J_+u_1=-u_2,\, J_+v_1=-v_2$. Moreover, there are no Ricci commuting para-Hermitian structures.

  • (i.2) If $\varepsilon _1\varepsilon _2=-1$, there is a unique (up to sign) Ricci-commuting para-Hermitian structure $(\langle \,.,. \rangle , J_-)$ given by $J_-u_1 = v_2,\, J_-v_1 = v_2$. Moreover, there are no Ricci commuting Hermitian structures.

• (ii) The Ricci operator is complex diagonalizable with eigenvalues $\pm ik$. At each point there is a basis $\{ u_1,v_1,u_2,v_2\}$ of the tangent space so that the Ricci operator and the nonzero inner products are given by

  $$\begin{aligned} \text {Ric}=\left( \begin{array}{cccc} 0 &{}\quad k &{}\quad 0 &{}\quad 0\\ -k &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad k\\ 0 &{}\quad 0 &{}\quad -k &{}\quad 0 \end{array}\right) , \;\;\langle u_i,u_i\rangle =1=-\langle v_i,v_i\rangle \, \quad (i=1,2)\,. \end{aligned}$$

  Moreover (up to reversing the metric), one may assume $k>0$ and one has that there is a unique (up to sign) Ricci commuting para-Hermitian structure $(\langle \,.,. \rangle , J_-)$ given by $J_-u_1 = v_2,\,J_-v_1=-u_2$ and a unique (up to sign) Ricci commuting Hermitian structure $(\langle \,.,. \rangle , J_+)$ given by ${J_+u_1} = {u_2},\, {J_+v_1} = -{v_2}$.

Hence, we introduce the following locally conformally flat algebraic curvature models $(V,\langle \,.,. \rangle , {\mathcal {A}})$ given by ${\mathcal {A}}=\frac{1}{2}\langle \,.,.\rangle \odot \rho _{{\mathcal {A}}}$, where $\rho _{{\mathcal {A}}}$ is the Ricci tensor corresponding to the Ricci operators above, and $\odot$ denotes the Kulkarni–Nomizu’s product:

(${\mathfrak {M}}^+$)::

$(V,\langle \,.,. \rangle , {\mathcal {A}})$ given by

$$\begin{aligned} {\mathcal {A}}_{1413}={\mathcal {A}}_{3231}=\frac{1}{2} \end{aligned}$$

with respect to a basis $\{ u_1,u_2,u_3,u_4\}$ where the nonzero inner products are $\langle u_1,u_2\rangle =1=\langle u_3,u_4\rangle$.

$({\mathfrak {M}}^-)$::

$(V,\langle \,.,. \rangle , {\mathcal {A}})$ given by

$$\begin{aligned} {\mathcal {A}}_{1413}={\mathcal {A}}_{3231}=-\frac{1}{2} \end{aligned}$$

with respect to a basis $\{ u_1,u_2,u_3,u_4\}$ where the nonzero inner products are $\langle u_1,u_2\rangle =1=-\langle u_3,u_4\rangle$.

(${\mathfrak {N}}_k$)::

$(V,\langle \,.,. \rangle , {\mathcal {A}})$ given by

$$\begin{aligned} {{\mathcal {A}}_{1413}={\mathcal {A}}_{1442}={\mathcal {A}}_{3224}={\mathcal {A}}_{3231} = \frac{k}{2}} \end{aligned}$$

with respect to an orthonormal basis $\{ u_1,u_2,u_3,u_4\}$ where $u_1,u_3$ are spacelike vectors and $u_2,u_4$ are timelike vectors. Further, observe that while the curvature models $({\mathfrak {N}}_k)$ are not isometric, they all are homothetic to the curvature model $({\mathfrak {N}}_1)$.

The curvature tensor of the locally conformally flat para-Kähler manifolds corresponding to Assertions (i) and (ii) in Theorem 1.1 is modelled in $({\mathfrak {M}}^-)$ and $({\mathfrak {N}}_k)$, respectively. The curvature tensor of the locally conformally flat Kähler manifolds corresponding to Assertions (i) and (ii) in Theorem 1.2 is modelled in $({\mathfrak {M}}^{+})$ and $({\mathfrak {N}}_k)$, respectively.

3 Walker structures: Riemannian extensions

Let $(M,g,{\mathfrak {D}})$ be a four-dimensional Walker manifold, i.e. a pseudo-Riemannian manifold (M, g) of neutral signature admitting a parallel degenerate plane field ${\mathfrak {D}}$ of maximal dimension. Walker showed in [23] the existence of local coordinates $(x^1,x^2,x_{1'},x_{2'})$ so that the parallel degenerate distribution ${\mathfrak {D}}={\text {span}}\{\partial _{x_{1'}},\partial _{x_{2'}}\}$ and the metric expresses as

$$\begin{aligned} g=dx^i\otimes dx_{i'}+dx_{i'}\otimes dx^i+g_{ij}(x^1,x^2,x_{1'},x_{2'}) dx^i\otimes dx^j\,. \end{aligned}$$

(1)

Riemannian extensions not only provide simple examples of Walker structures but also are the underlying structure in many important situations. We briefly schedule their construction as follows. Let $\Sigma$ be a surface and let $\pi : T^*\Sigma \rightarrow \Sigma$ be the projection from the cotangent bundle. Let ${\tilde{p}}=(p,\omega )\in T^*\Sigma$, where $p\in \Sigma$ and $\omega \in T_p^*\Sigma$, denote a point of $T^*\Sigma$. For each vector field X on $\Sigma$, the evaluation map is the function $\iota X\in {\mathcal {C}}^\infty (T^*\Sigma )$ defined by $\iota X(p,\omega )=\omega (X_p)$. For any vector field Z on $\Sigma$, the complete lift $Z^C$ is the vector field determined by the identity $Z^C(\iota X)=\iota [Z,X]$. In the same way as vector fields on $T^*\Sigma$ are characterized by their action on evaluation maps, tensor fields of type (0, s) on $T^*\Sigma$ are characterized by their action on complete lifts of vector fields. In particular, any tensor field T of type (1, 1) on $\Sigma$ induces a one-form $\iota T$ on $T^*\Sigma$ characterized by $\iota T(X^C)=\iota (TX)$. We refer to Yano-Ishihara [24] for details concerning this material.

Riemannian extensions of torsion-free connections were introduced by Patterson and Walker in [21] as metrics on $T^*\Sigma$ characterized by $g_D(X^C,Y^C)=-\iota (D_XY+D_YX)$, where D is a torsion-free affine connection on the base manifold $\Sigma$. These metrics have been further generalized in [8] as follows. Let $\Phi$ be a symmetric (0, 2)-tensor field and let T, S be tensor fields of type (1, 1) on an affine surface $(\Sigma ,D)$. The modified Riemannian extension is the neutral signature metric on $T^*\Sigma$ defined by

$$\begin{aligned} g_{D,\Phi ,T,S}=\iota T\circ \iota S+g_D+\pi ^*\Phi , \end{aligned}$$

(2)

where ‘$\circ$’ denotes the symmetric product.

Let $(x^1,x^2)$ be local coordinates in a neighbourhood ${\mathcal {U}}$ of $\Sigma$ and let $(x^1,x^2,x_{1'},x_{2'})$ be the induced coordinates in $\pi ^{-1}({\mathcal {U}})$. Then one has (see [8])

$$\begin{aligned} \begin{array}{lll} g_{D,\Phi ,T,S}&{}=&{}dx^i\otimes dx_{i'}+dx_{i'}\otimes dx^{i} \\ &{}&{}+\left\{ \frac{1}{2} x_{r'} x_{s'} (T_i{}^r S_j{}^s + T_j{}^r S_i{}^s)-2x_{k'}{}^D\Gamma _{ij}{}^k+\Phi _{ij} \right\} dx^i\otimes dx^j, \end{array} \end{aligned}$$

where $T=T_i{}^j dx^i\otimes \partial _{x^j}$, $S=S_i{}^j dx^i\otimes \partial _{x^j}$, $\Phi =\Phi _{ij}dx^i\otimes dx^j$ and ${}^D\Gamma _{ij}{}^k$ are the Christoffel symbols of the affine connection $D$. Moreover, the Walker distribution ${\mathfrak {D}}={\text {ker}}\pi _*$.

3.1 Self-dual Walker manifolds

The existence of a two-dimensional degenerate distribution ${\mathfrak {D}}$ on a neutral signature four-dimensional manifold (M, g) naturally induces an orientation as follows. Let $p\in M$ and let $\{ u,v\}$ be an arbitrary basis of ${\mathfrak {D}}_p$. Then the Hodge-star operator satisfies $\star (u^*\wedge v^*)=\pm (u^*\wedge v^*)$, where $u^*, v^*\in T_p^*M$ are the dual forms. Hence, any four-dimensional Walker manifold is naturally oriented by the self-duality of $u^*\wedge v^*$ (see, for example, [11]). Let $(M,g,{\mathfrak {D}})$ be a four-dimensional Walker manifold and let $(x^1,x^2,x_{1'},x_{2'})$ be local coordinates as in (1). Then the Walker orientation determined by $\star (dx_{1'}\wedge dx_{2'})=dx_{1'}\wedge dx_{2'}$ corresponds to the volume element ${\text {vol}}_g=dx^1\wedge dx^2\wedge dx_{1'}\wedge dx_{2'}$. Self-dual Walker metrics have been described in [8] as follows.

Theorem 3.1

([8], Theorem 7.1) A four-dimensional Walker manifold is self-dual if and only if it is locally isometric to the cotangent bundle $T^*\Sigma$ of an affine surface $(\Sigma ,D)$ with metric

$$\begin{aligned} g=\iota X (\iota {\text {Id}} \circ \iota {\text {Id}}) + \iota T \circ \iota {\text {Id}} + g_D + \pi ^*\Phi \end{aligned}$$

(3)

where X is a vector field on $\Sigma$ and T and $\Phi$ are a (1, 1)-tensor field and a symmetric (0, 2)-tensor field on $\Sigma$, respectively.

A special case of the above theorem describes the local structure of paracomplex space forms as follows.

Theorem 3.2

([8], Theorem 2.2) A para-Kähler surface of nonzero constant paraholomorphic sectional curvature c is locally isometric to the cotangent bundle of a flat affine surface, equipped with the modified Riemannian extension $g=c\, \iota {\text {Id}} \circ \iota {\text {Id}} + g_D$.

Let $\theta _{(p,\omega )}=\pi ^*\omega _p(=x_{\ell '}dx^\ell )$ be the tautological one-form of $T^*\Sigma$, and let $\Omega =d\theta (=dx_{\ell '}\wedge dx^{\ell })$ be the canonical symplectic form of $T^*\Sigma$. Using the modified Riemannian extension $g=c\, \iota {\text {Id}} \circ \iota {\text {Id}} + g_D$, one naturally constructs the paracomplex structure $J_-$ determined by $\Omega (X,Y)=g(J_-X,Y)$ so that

$$\begin{aligned} J_-\partial _{x_{i'}}=\partial _{x_{i'}}, \qquad J_-\partial _{x^i}= -\partial _{x^i}+ c\, x_{i'}x_{j'}\partial _{x_{j'}}\,. \end{aligned}$$

Note that the Walker distribution ${\mathfrak {D}}={\text {ker}}\pi _*$ corresponds to the eigenspace ${\mathcal {D}}_+={\text {ker}}(J_--{\text {Id}})$ of the paracomplex structure $J_-$.

3.2 Locally symmetric self-dual Walker surfaces

Since any locally conformally flat para-Kähler surface is locally symmetric, as well as any locally conformally flat Kähler surface with non-diagonalizable Ricci operator, we firstly analyse the self-dual Walker surfaces in Theorem 3.1 which are locally symmetric.

Let $\rho ^D$ denote the Ricci tensor of $(\Sigma ,D)$ and decompose it as $\rho ^D=\rho ^D_s+\rho ^D_{sk}$, where $\rho ^D_s$ (resp., $\rho ^D_{sk}$) denotes the symmetric (resp., skew-symmetric) part of $\rho ^D$.

Lemma 3.3

Let (M, g) be a locally symmetric self-dual Walker manifold. Then the Riemannian extension g in Theorem 3.1satisfies $g=\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi$ for some parallel (1, 1)-tensor field T on $(\Sigma ,D)$.

Furthermore, if the scalar curvature is nonzero, then (M, g) is locally isometric to a paracomplex space form as in Theorem 3.2.

Proof

The scalar curvature of any Riemannian extension (3) is given by $\tau =3 {\text {tr}}T+12\iota X$. It now follows from the constancy of the scalar curvature that $X=0$ and ${\text {tr}}T=\kappa$ for some $\kappa \in {\mathbb {R}}$. Since any self-dual Walker manifold is locally isometric to a Riemannian extension given by (3), the covariant derivatives of the curvature operator are polynomials on the fibre coordinates $(x_{1'},x_{2'})$. One has

$$\begin{aligned} \begin{array}{rcl} (\nabla _{\partial _{x^1}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x^1}) &{} = &{} -\frac{1}{8}\kappa \left( T_1{}^2\right) ^2 x_{2'}^3 + \text { other terms}, \\ (\nabla _{\partial _{x^2}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x^1}) &{} = &{} \frac{1}{4}\kappa \left( T_2{}^1\right) ^2 x_{1'}^3 + \text { other terms}, \\ (\nabla _{\partial _{x^2}}R)(\partial _{x_{2'}},\partial _{x^1},\partial _{x^1},\partial _{x^1}) &{} = &{} -\frac{\kappa }{8}\left( \kappa -2T_2{}^2\right) x_{1'} + \text { other terms}. \end{array} \end{aligned}$$

Assume first that the scalar curvature $\tau =3\kappa \ne 0$. If $\nabla R = 0$, then the previous expansions show that the tensor field T is a multiple of the identity, $T=c{\text {Id}}$. Further calculations now show that

$$\begin{aligned} \begin{array}{l} (\nabla _{\partial _{x^1}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x^1}) = c \{2\rho _{s}^D(\partial _{x^1},\partial _{x^1}) -\frac{1}{2}c\Phi _{11}\}x_{2'}+ \text { other terms}, \\ (\nabla _{\partial _{x^2}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x^1}) = \frac{3}{2}c \{2\rho _{s}^D(\partial _{x^1},\partial _{x^2}) -\frac{1}{2}c\Phi _{12}\}x_{2'}+ \text { other terms}, \\ (\nabla _{\partial _{x^2}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x_{1'}}) = \frac{1}{2}c^2 \{2\rho _{s}^D(\partial _{x^2},\partial _{x^2}) -\frac{1}{2}c\Phi _{22}\}x_{1'}^3+ \text { other terms}, \end{array} \end{aligned}$$

from where it follows that the symmetric (0, 2)-tensor field $\Phi$ is determined by $\Phi =\frac{4}{c}\rho _{s}^D$. In this situation, the Ricci operator ${\text {Ric}} = \frac{3c}{2}{\text {Id}}$, and the metric is Einstein by ([8], Thm.2.1). Furthermore, one has that for any unit vector field X, the Jacobi operators $R_X=R(\,\cdot , X)X$ have eigenvalues $\{ 0, c,\frac{1}{4}c,\frac{1}{4}c\}$ with ${\text {ker}}(R_X-c{\text {Id}})$ timelike. Hence, (M, g) is locally a paracomplex space form (see [12]) and thus locally isometric to a modified Riemannian extension given by Theorem 3.2.

Next assume the scalar curvature $\tau =0$. Let DT denote the covariant derivative of the (1,1)-tensor field T with respect to the affine connection D. We set $DT=DT_{j;i}{}^k dx^i\otimes dx^j\otimes \partial _{x^k}$, where $DT_{j;i}{}^k=\partial _{x^i}T_j{}^k+T_j{}^\ell {}^D\Gamma _{i\ell }{}^k$. Since the scalar curvature vanishes, ${\text {tr}}T=0$. Hence, $T_1{}^1 = - T_2{}^2$ and thus $DT_{1;1}{}^1=-DT_{2;1}{}^2$ and $DT_{1;2}{}^1=-DT_{2;2}{}^2$. A straightforward calculation now shows

$$\begin{aligned} \begin{array}{l} (\nabla _{\partial _{x^1}}R)(\partial _{x_{2'}},\partial _{x^1},\partial _{x^1},\partial _{x^1}) = { \frac{1}{2} DT_{1;1}{}^2}, \\ (\nabla _{\partial _{x^1}}R)(\partial _{x_{2'}},\partial _{x^2},\partial _{x^2},\partial _{x^1}) = { \frac{1}{2} DT_{2;1}{}^1}, \\ (\nabla _{\partial _{x^2}}R)(\partial _{x_{2'}},\partial _{x^2},\partial _{x^1},\partial _{x^2}) = \frac{1}{2} DT_{1;2}{}^2, \\ (\nabla _{\partial _{x_{1'}}}R)(\partial _{x^1},\partial _{x^2},\partial _{x^2},\partial _{x^1}) = - \frac{1}{2} DT_{2;2}{}^1, \\ (\nabla _{\partial _{x_{1'}}}R)(\partial _{x^1},\partial _{x^2},\partial _{x^1},\partial _{x^1}) = DT_{2;2}{}^2+\frac{1}{2}DT_{2;1}{}^1, \\ (\nabla _{\partial _{x_{2'}}}R)(\partial _{x^1},\partial _{x^2},\partial _{x^2},\partial _{x^2}) =DT_{2;1}{}^2-\frac{1}{2}DT_{1;2}{}^2, \end{array} \end{aligned}$$

from where it follows that the trace-free (1, 1)-tensor field T is $D$-parallel. $\square$

The existence of parallel (1, 1)-tensor fields on affine surfaces $(\Sigma ,D)$ was considered in [9] showing that (besides the trivial case $T=0$) a parallel trace-free (1, 1)-tensor field corresponds to one of the following:

• (a) An affine para-Kähler structure (${\text {det}}T=-k^2<0$), which in suitable adapted coordinates becomes $T=k(\partial _{x^1}\otimes dx^1-\partial _{x^2}\otimes dx^2)$.

• (b) An affine nilpotent Kähler structure ($T^2=0$), which in suitable adapted coordinates becomes $T=k\partial _{x^1}\otimes dx^2$.

• (c) An affine Kähler structure (${\text {det}}T=k^2>0$), which in suitable adapted coordinates becomes $T=k(\partial _{x^2}\otimes dx^1-\partial _{x^1}\otimes dx^2)$.

Each of the three possibilities above gives rise to different geometric structures which are locally conformally flat as follows.

3.2.1 Locally symmetric self-dual Walker surfaces determined by an affine para-Kähler structure

This case leads to product manifolds as the following shows.

Lemma 3.4

Let $(T^*\Sigma ,g=\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi )$ be a locally symmetric self-dual Walker manifold determined by an affine para-Kähler structure T on $(\Sigma ,D)$. Then $(T^*\Sigma ,g)$ is locally conformally flat and locally isometric to a product of two Lorentzian surfaces of constant opposite Gauss curvature.

Proof

Let $(\Sigma ,D)$ be an affine surface and choose local coordinates $(x^1,x^2)$ so that the parallel tensor field T is locally given by $T = k(\partial _{x^1}\otimes dx^1-\partial _{x^2}\otimes dx^2)$. Then T is parallel if and only if the Christoffel symbols satisfy ${}^D\Gamma _{11}{}^2$ = ${}^D\Gamma _{12}{}^1 = {}^D\Gamma _{12}{}^2 = {}^D\Gamma _{22}{}^1=0$ (see [9]). Let $(x^1,x^2,x_{1'},x_{2'})$ be the induced coordinates on $T^*\Sigma$. Then the symmetric and skew-symmetric Ricci tensors of $(\Sigma ,D)$ are given by

$$\begin{aligned} \rho ^D_s=-(\partial _{x^1}{}^D\Gamma _{22}{}^2+\partial _{x^2}{}^D\Gamma _{11}{}^1)dx^1\circ dx^2, \quad \rho ^D_{sk}=\frac{1}{2}(\partial _{x^1}{}^D\Gamma _{22}{}^2-\partial _{x^2}{}^D\Gamma _{11}{}^1) dx^2\wedge dx^1\,. \end{aligned}$$

A straightforward calculation now shows that the Ricci operator of $(T^*\Sigma ,g)$, when expressed on the coordinate basis, satisfies

$$\begin{aligned} {\text {Ric}} = \left( \begin{array}{cccc} k &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -k &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad k\Phi _{12}+2\rho ^D_{s}(\partial _{x^1},\partial _{x^2}) &{}\quad k &{}\quad 0 \\ -k\Phi _{12}+2\rho ^D_{s}(\partial _{x^1},\partial _{x^2}) &{}\quad 0 &{}\quad 0 &{}\quad -k \end{array}\right) , \end{aligned}$$

from where it follows that the Ricci curvatures are $\pm k$. Moreover, a straightforward calculation shows that $({\text {Ric}}-k{\text {Id}})({\text {Ric}}+k{\text {Id}})=0$, and hence, the Ricci operator is diagonalizable in an orthonormal basis. If it is parallel, then the manifold is locally isometric to a product of two Lorentzian surfaces of constant opposite Gauss curvature and hence locally conformally flat. $\square$

Remark 3.5

Let $(\Sigma ,D,T)$ be an affine para-Kähler surface. Let $(x^1,x^2)$ be local coordinates on $\Sigma$ so that the tensor field T expresses locally as $T = k\,(\partial _{x^1}\otimes dx^1-\partial _{x^2}\otimes dx^2)$. Then there exists an appropriate locally defined deformation tensor $\Phi$ so that $(T^*\Sigma ,g=\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi )$ is locally symmetric. A straightforward calculation shows that

$$\begin{aligned} (\nabla _{\partial _{x^2}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x_{1'}}) =-k^2\{k\Phi _{12}+2\rho ^D_{sk}(\partial _{x^1},\partial _{x^2})\}x_{1'}^2x_{2'}+\text {other terms}. \end{aligned}$$

Setting $\Phi _{12}=-\frac{2}{k}\rho ^D_{sk}(\partial _{x^1},\partial _{x^2})$, one has

$$\begin{aligned} (\nabla _{\partial _{x^1}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x_{1'}})= & {} \frac{k}{2}\left\{ {k\partial _{x^2}\Phi _{11}} -2(D_{\partial _{x^1}}\rho ^D)(\partial _{x^2},\partial _{x^1})\right\} x_{1'}^2 \\&+\text {other terms}, \\ (\nabla _{\partial _{x^2}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x_{1'}})= & {} -\frac{k}{2} \left\{ {k\partial _{x^1}\Phi _{22}} +2(D_{\partial _{x^2}}\rho ^D)(\partial _{x^1},\partial _{x^2}) \right\} x_{1'}^2 \\&+\text {other terms}, \end{aligned}$$

which determines $\Phi _{11}$ and $\Phi _{22}$. A straightforward calculation now shows that the covariant derivative $\nabla R$ vanishes when choosing the deformation tensor $\Phi$ as above.

3.2.2 Locally symmetric self-dual Walker surfaces determined by an affine nilpotent Kähler structure

The situation corresponding to an affine nilpotent Kähler structure $(\Sigma ,D,T)$ (where ${\text {det}}T=0$, $T\ne 0$) gives rise to a single curvature model, as the following shows.

Lemma 3.6

Let $(T^*\Sigma ,g=\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi )$ be a locally symmetric self-dual Walker manifold determined by an affine nilpotent Kähler structure T on $(\Sigma ,D)$. Then $(T^*\Sigma ,g)$ is locally conformally flat modelled on $({\mathfrak {M}}^-)$, and locally isometric to a modified Riemannian extension $g=\iota T\circ \iota {\text {Id}}+g_D$ where $(\Sigma ,D)$ is a flat affine surface.

Proof

Let $(x^1,x^2)$ be local coordinates on $\Sigma$ in which the tensor field T is locally expressed as $T = k\,\partial _{x^1}\otimes dx^2$. Then it follows from the work in [9] that the Christoffel symbols satisfy ${}^D\Gamma _{11}{}^1={}^D\Gamma _{11}{}^2 = {}^D\Gamma _{12}{}^2 = 0$ and ${}^D\Gamma _{12}{}^1={}^D\Gamma _{22}{}^2$. The symmetric and skew-symmetric components of the Ricci tensor of $(\Sigma ,D)$ are given by

$$\begin{aligned} \rho ^D_{s}=(\partial _{x^1}{}^D\Gamma _{22}{}^1-\partial _{x^2}{}^D\Gamma _{22}{}^2)dx^2\otimes dx^2, \qquad \rho ^D_{sk}=\partial _{x^1}{}^D\Gamma _{22}{}^2 dx^2\wedge dx^1\,. \end{aligned}$$

Let $(x^1,x^2,x_{1'},x_{2'})$ be the induced local coordinates on $T^*\Sigma$. A straightforward calculation shows that the component of the Weyl tensor

$$\begin{aligned} \begin{array}{rcl} W(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x_{1'}}) &{}=&{} \frac{1}{k}\partial _{x_{1'}}(\nabla _{\partial _{x^1}}R)(\partial _{x^1},\partial _{x^2},\partial _{x^2},\partial _{x^1}) \\ &{}=&{} \frac{k}{4}\Phi _{11}-\rho ^D_{sk}(\partial _{x^1},\partial _{x^2})\,. \end{array} \end{aligned}$$

Assuming that $(T^*\Sigma ,g)$ is locally symmetric, one has $\Phi _{11}=\frac{4}{k}\rho ^D_{sk}(\partial _{x^1},\partial _{x^2})$. Furthermore, using the previous expression of $\Phi _{11}$ one gets that the only nonzero component of the Weyl tensor (up to the usual symmetries) satisfies

$$\begin{aligned} \begin{array}{rcl} W(\partial _{x^1},\partial _{x^2},\partial _{x^1},\partial _{x^2}) &{}=&{}\frac{1}{4k}x_{1'}\partial _{x_{1'}}\partial _{x_{1'}} (\nabla _{\partial _{x^2}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^2},\partial _{x_{1'}}) \\ &{}&{}-\frac{1}{k^2}\partial _{x_{1'}}\partial _{x_{1'}}\partial _{x^1} (\nabla _{\partial _{x^2}}R)(\partial _{x^1},\partial _{x^2},\partial _{x^2},\partial _{x_{2'}}) \\ &{}&{}+\frac{1}{4k^2}\partial _{x_{1'}}\partial _{x_{1'}}\partial _{x^2} (\nabla _{\partial _{x^2}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^2},\partial _{x_{1'}}) \\ &{}&{}-\frac{1}{2k^2}{}^D\Gamma _{22}{}^2\partial _{x_{1'}}\partial _{x_{1'}} (\nabla _{\partial _{x^2}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^2},\partial _{x_{1'}})\,. \end{array} \end{aligned}$$

This shows that if $(T^*\Sigma ,g)$ is locally symmetric, then it is locally conformally flat. Moreover, using that $\Phi _{11}=\frac{4}{k}\rho ^D_{sk}(\partial _{x^1},\partial _{x^2})$ one gets

$$\begin{aligned} (\nabla _{\partial _{x^1}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^2},\partial _{x^1})= & {} \frac{k}{2}\partial _{x^1}\Phi _{12} -(D_{\partial _{x^2}}\rho ^D)(\partial _{x^1},\partial _{x^2}) \\&+\partial _{x^1}\rho ^D_{s}(\partial _{x^2},\partial _{x^2})- \partial _{x^2}\rho ^D_{sk}(\partial _{x^1},\partial _{x^2}), \end{aligned}$$

from where it follows that $\Phi _{12}=\frac{2}{k}\left( ({}^D\Gamma _{22}{}^2)^2-\partial _{x^2}{}^D\Gamma _{22}{}^2-\partial _{x^1}{}^D\Gamma _{22}{}^1\right) +\phi _{12}(x^2)$, for some function $\phi _{12}(x^2)$. Using the previous expressions for $\Phi _{11}$ and $\Phi _{12}$ one has

$$\begin{aligned} (\nabla _{\partial _{x^2}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^2},\partial _{x^1})= \partial _{x^1}\left( \frac{k}{2}\Phi _{22}+2\partial _{x^2}{}^D\Gamma _{22}{}^1-2{}^D\Gamma _{22}{}^2{}^D\Gamma _{22}{}^1 \right) -\frac{k}{2}\phi _{12}'(x^2), \end{aligned}$$

which gives $\Phi _{22}=\frac{4}{k}({}^D\Gamma _{22}{}^2{}^D\Gamma _{22}{}^1-\partial _{x^2}{}^D\Gamma _{22}{}^1)+x^1 \phi _{12}'(x^2)+\phi _{22}(x^2)$, for some function $\phi _{22}(x^2)$.

A straightforward calculation now shows that the metric $g=\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi$, where the deformation tensor $\Phi$ is given by the expressions above is locally symmetric (and hence locally conformally flat). Moreover, the Ricci operator, when expressed on the coordinate basis, takes the form

$$\begin{aligned} {\text {Ric}} =\left( \begin{array}{cccc} 0&{}k&{}0&{}0\\ 0&{}0&{}0&{}0\\ 0&{}2\rho ^D_{sk}(\partial _{x^2},\partial _{x^1})&{}0&{}0\\ 2\rho ^D_{sk}(\partial _{x^1},\partial _{x^2})&{}2\rho ^D_s(\partial _{x^2},\partial _{x^2})&{}k&{}0 \end{array}\right) \,. \end{aligned}$$

Hence, the Ricci operator is two-step nilpotent and a straightforward calculation shows that its Jordan normal form corresponds to that discussed in Sect. 2-(i.2) with $\varepsilon _1\varepsilon _2=-1$. Hence, the curvature tensor is determined by the 0-model $({\mathfrak {M}}^-)$.

Now let $(\Sigma ,D)$ be a flat affine surface and let T be an affine nilpotent Kähler structure. Setting $\Phi =0$, a direct calculation using the expressions above shows that the Riemannian extension $g=\iota T\circ \iota {\text {Id}}+g_D$ is locally symmetric, thus completing the proof. $\square$

Remark 3.7

Let $(\Sigma ,D,T)$ be an affine nilpotent Kähler surface. Let $(x^1,x^2)$ be local coordinates on $\Sigma$ so that the tensor field T expresses locally as $T = k\,\partial _{x^1}\otimes dx^2$. Then there exists a locally defined tensor field $\Phi$ given as in the proof of Lemma 3.6 so that $(T^*\Sigma ,g=\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi )$ is locally symmetric.

3.2.3 Locally symmetric self-dual Walker surfaces determined by an affine Kähler structure

The situation induced by an affine Kähler structure $(\Sigma ,D,T)$ (where $T^2=-k^2{\text {Id}}$) corresponds to the 0-model $({\mathfrak {N}}_k)$, as the following shows.

Lemma 3.8

Let $(T^*\Sigma ,g=\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi )$ be a locally symmetric self-dual Walker manifold determined by an affine Kähler structure T on $(\Sigma ,D)$. Then $(T^*\Sigma ,g)$ is locally conformally flat modelled on $({\mathfrak {N}}_k)$ and locally isometric to the Riemannian extension $g=\iota T\circ \iota {\text {Id}}+g_D$, where $(\Sigma ,D)$ is a flat affine surface.

Proof

Let $(x^1,x^2)$ be local coordinates on $\Sigma$ so that the tensor field T expresses locally as $T = k\,\left( \partial _{x^2}\otimes dx^1-\partial _{x^1}\otimes dx^2\right)$. Then it follows from the work in [9] that the Christoffel symbols satisfy ${}^D\Gamma _{11}{}^1={}^D\Gamma _{12}{}^2 =- {}^D\Gamma _{22}{}^1$ and ${}^D\Gamma _{12}{}^1=-{}^D\Gamma _{11}{}^2={}^D\Gamma _{22}{}^2$. Furthermore, the symmetric and skew-symmetric parts of the Ricci tensor are given by

$$\begin{aligned} \rho ^D_{s}= & {} (\partial _{x^1}{}^D\Gamma _{22}{}^1-\partial _{x^2}{}^D\Gamma _{22}{}^2)(dx^1\otimes dx^1+dx^2\otimes dx^2), \\ \rho ^D_{sk}= & {} (\partial _{x^2}{}^D\Gamma _{22}{}^1+\partial _{x^1}{}^D\Gamma _{22}{}^2)(dx^2\otimes dx^1-dx^1\otimes dx^2). \end{aligned}$$

Now since

$$\begin{aligned} (\nabla _{\partial _{x^1}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x_{1'}}) = \frac{k^2}{8}\left\{ k(\Phi _{11}+\Phi _{22})-4 \rho ^D_{sk}(\partial _{x^2},\partial _{x^1})\right\} x_{1'}^3+\text {other terms}, \end{aligned}$$

assuming that $(T^*\Sigma ,g)$ is locally symmetric we set $\Phi _{11}=-\Phi _{22}+\frac{4}{k}\rho ^D_{sk}(\partial _{x^2},\partial _{x^1})$. A long but straightforward calculation now shows that the only nonzero component (up to the usual symmetries) of the Weyl tensor is

$$\begin{aligned} W(\partial _{x^1},\partial _{x^2},\partial _{x^1},\partial _{x^2})= & {} \frac{1}{k}x_{1'}\partial _{x_{1'}}\partial _{x_{2'}} (\nabla _{\partial _{x^1}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^2},\partial _{x_{1'}}) \\&+\frac{1}{k}x_{2'}\partial _{x_{2'}}\partial _{x_{2'}} (\nabla _{\partial _{x^2}}R)(\partial _{x^2},\partial _{x^1},\partial _{x^1},\partial _{x_{1'}}) \\&-\frac{1}{3k^2}\partial _{x^2}\partial _{x_{1'}}\partial _{x_{2'}} (\nabla _{\partial _{x^1}}R)(\partial _{x^1},\partial _{x^2},\partial _{x^1},\partial _{x_{2'}}) \\&+\frac{2}{3k^2}{}^D\Gamma _{22}{}^2 \partial _{x_{1'}}\partial _{x_{2'}} (\nabla _{\partial _{x^1}}R)(\partial _{x^1},\partial _{x^2},\partial _{x^1},\partial _{x_{2'}}) \\&-\frac{1}{k^2}\partial _{x^1}\partial _{x_{1'}}\partial _{x_{2'}} (\nabla _{\partial _{x^2}}R)(\partial _{x^1},\partial _{x^2},\partial _{x^1},\partial _{x_{2'}}) \\&-\frac{2}{k^2}{}^D\Gamma _{22}{}^1\partial _{x_{1'}}\partial _{x_{2'}} (\nabla _{\partial _{x^2}}R)(\partial _{x^1},\partial _{x^2},\partial _{x^1},\partial _{x_{2'}})\,. \end{aligned}$$

Therefore, the local symmetry of $(T^*\Sigma ,g)$ implies its local conformal flatness. Moreover, the Ricci operator of $(T^*\Sigma ,g)$, when expressed on the coordinate basis, is given by

$$\begin{aligned} {\text {Ric}}=\left( \begin{array}{cccc} 0&{}-k&{}0&{}0\\ k&{}0&{}0&{}0\\ 2\rho ^D_{s}(\partial _{x^1},\partial _{x^1})&{} 2\rho ^D_{sk}(\partial _{x^2},\partial _{x^1})&{}0&{}k\\ 2\rho ^D_{sk}(\partial _{x^1},\partial _{x^2})&{} 2\rho ^D_{s}(\partial _{x^1},\partial _{x^1})&{}-k&{}0 \end{array} \right) \,. \end{aligned}$$

Hence, the Ricci curvatures are $\pm ik$, and moreover, ${\text {Ric}}^2=-k^2{\text {Id}}$, which shows that the Ricci operator is complex diagonalizable. Since the curvature tensor is completely determined by the Ricci operator, it corresponds to the 0-model $({\mathfrak {N}}_k)$.

To conclude, let $(\Sigma ,D)$ be a flat affine surface and T an affine Kähler structure. A direct calculation shows that the Riemannian extension $g=\iota T\circ \iota {\text {Id}}+g_D$ is locally symmetric, thus completing the proof. $\square$

Remark 3.9

Let $(\Sigma ,D,T)$ be an affine Kähler surface. Then there exist locally defined suitable deformation tensors $\Phi$ so that $(T^*\Sigma ,g=\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi )$ is locally symmetric. A straightforward calculation as in the proof of Lemma 3.8 gives that $\Phi _{11}=-\Phi _{22}+\frac{4}{k}\rho ^D_{sk}(\partial _{x^2},\partial _{x^1})$ and the coefficients $\Phi _{12}$ and $\Phi _{22}$ satisfy

$$\begin{aligned} \begin{array}{l} \partial _{x^1}\Phi _{22}-\partial _{x^2}\Phi _{12}= \frac{2}{k}\left\{ (D_{\partial _{x^1}}\rho ^D_{sk})(\partial _{x^1},\partial _{x^2}) +(D_{\partial _{x^2}}\rho ^D_{s})(\partial _{x^1},\partial _{x^1}) +\partial _{x^1}\rho ^D_{sk}(\partial _{x^2},\partial _{x^1})\right\}, \\ \partial _{x^2}\Phi _{22}+\partial _{x^1}\Phi _{12}=\frac{2}{k}\left\{ (D_{\partial _{x^2}}\rho ^D_{sk})(\partial _{x^2},\partial _{x^1}) +(D_{\partial _{x^1}}\rho ^D_{s})(\partial _{x^1},\partial _{x^1}) +\partial _{x^2}\rho ^D_{sk}(\partial _{x^2},\partial _{x^1})\right\}. \end{array} \end{aligned}$$

Now the system of equations above is just the inhomogeneous Cauchy-Riemann equation $\partial _{{\overline{z}}}\phi =f$ for $\phi =\Phi _{22}+i\Phi _{12}$, where the function $f=f_1+i f_2$ is given by the right hand side of the expressions above, which admit local solutions for any affine Kähler surface $(\Sigma ,D)$ (see, for example [15]).

Remark 3.10

Let (M, g) be a locally symmetric Walker metric as in Lemma 3.8. Then the Ricci operator defines a complex structure $S=\frac{1}{k}{\text {Ric}}$ which is self-adjoint and parallel (hence a complex Riemannian structure). Furthermore, the twin metric ${\hat{g}}(X,Y)=g(\frac{1}{k}{\text {Ric}}X,Y)=\frac{1}{k}\rho (X,Y)$ is locally symmetric and Einstein with scalar curvature ${\hat{\tau }}=4k$, which is a special case of the Main Theorem in [6].

3.2.4 Locally symmetric self-dual Walker surfaces with $T=0$

In contrast with the previous cases the class of locally symmetric Riemannian extensions of the form $g=g_D+\pi ^*\Phi$ is much larger and the underlying structure is not necessarily locally conformally flat.

Example 3.11

Let $(\Sigma ,D)$ be the flat plane and let $\Phi$ be the symmetric (0, 2)-tensor field $\Phi =x^1 x^2 (dx^1\otimes dx^2+dx^2\otimes dx^1)$. Then $(T^*\Sigma ,g_D+\pi ^*\Phi )$ is a Ricci-flat locally symmetric manifold which is not locally conformally flat. The curvature tensor is determined by the only nonzero component $R(\partial _{x^1},\partial _{x^2},\partial _{x^1},\partial _{x^2})=1$ (up to the usual symmetries).

It follows from the work in [1] that if $(T^*\Sigma ,g=g_D+\pi ^*\Phi )$ is locally symmetric, then so is $(\Sigma ,D)$. Locally symmetric affine surfaces were described by Opozda [19] (see also [10]) showing that $(\Sigma ,D)$ corresponds to one of the following:

• (i) The flat affine surface modelled in ${\mathbb {R}}^2$ with ${}^D\Gamma _{ij}{}^k=0$.

• (ii) The Levi–Civita connection of the hyperbolic plane ${\mathbb {H}}^2={\mathbb {R}}^+\times {\mathbb {R}}$ given by ${}^D\Gamma _{11}{}^1= {}^D\Gamma _{12}{}^2=- {}^D\Gamma _{22}{}^1=-\frac{1}{x^1}$.

• (iii) The Levi–Civita connection of the Lorentzian hyperbolic plane ${\mathbb {H}^2_1}={\mathbb{R}^+}\times {\mathbb {R}}$ given by ${}^D\Gamma _{11}{}^1= {}^D\Gamma _{12}{}^2= {}^D\Gamma _{22}{}^1=-\frac{1}{x^1}$.

• (iv) The Levi–Civita connection of the standard sphere.

• (v) One of the two non-metrizable affine connections modelled on ${\mathbb {R}}^2$ with Christoffel symbols ${}^D\Gamma _{11}{}^1=1$ and ${}^D\Gamma _{22}{}^1=\pm 1$.

Moreover, the Ricci operator of $(T^*\Sigma ,g=g_D+\pi ^*\Phi )$ vanishes in Case (i) and it is two-step nilpotent otherwise. Furthermore, ${\text {Ric}}$ is of rank two in cases (ii), (iii) and (iv), but of rank one in Case (v).

In order to describe the locally symmetric Riemannian extensions which are also locally conformally flat, we introduce the following algebraic curvature model:

(${\mathfrak {P}}$)::

$(V,\langle \,.,. \rangle , {\mathcal {A}})$ given by

$$\begin{aligned} {\mathcal {A}}_{1313} = {\mathcal {A}}_{1441} = \frac{1}{2} \end{aligned}$$

with respect to a basis $\{ u_1,u_2,u_3,u_4\}$ where the nonzero inner products are $\langle u_1,u_2\rangle =1$ and $\langle u_3,u_3\rangle = 1 = -\langle u_4,u_4\rangle$.

If $(T^*\Sigma ,g=g_D+\pi ^*\Phi )$ is locally conformally flat, then there exist coordinates on $(\Sigma ,D)$ so that one may assume $\Phi =0$. This situation is summarized as follows:

Lemma 3.12

Let $(T^*\Sigma ,g=g_D+\pi ^*\Phi )$ be a locally symmetric and locally conformally flat Riemannian extension. Then it is flat or it is locally isometric to a Riemannian extension $(T^*\Sigma ,g_D)$ where $(\Sigma ,D)$ corresponds to one of the following:

• (i) The Levi–Civita connection of a Riemannian surface of constant curvature, in which case the curvature tensor of $(T^*\Sigma ,g_D)$ is modelled on $({\mathfrak {M}}^+)$.

• (ii) The Levi–Civita connection of a Lorentzian surface of constant curvature, in which case the curvature tensor of $(T^*\Sigma ,g_D)$ is modelled on $({\mathfrak {M}}^-)$.

• (iii) One of the two non-metrizable affine connections modelled on ${\mathbb {R}}^2$ with Christoffel symbols ${}^D\Gamma _{11}{}^1=1$ and ${}^D\Gamma _{22}{}^1=\pm 1$. In this case the curvature tensor of $(T^*\Sigma ,g_D)$ is modelled on $({\mathfrak {P}})$.

4 Locally conformally flat para-Kähler surfaces

Let $(M,g,J_-)$ be a para-Kähler surface. Since $J_-^2={\text {Id}}$ and $J_-^*g=-g$, the eigenspaces ${\mathcal {D}}_\pm ={\text {ker}}(J_-\mp {\text {Id}})$ are totally degenerate. Moreover, $\nabla J_-=0$ shows that ${\mathcal {D}}_\pm$ are parallel, and thus, (M, g, J_) is a Walker manifold with respect to both distributions ${\mathcal {D}}_\pm$. We set the parallel distribution ${\mathfrak {D}}={\mathcal {D}}_+$ so that $J_-\!\mid _{{\mathfrak {D}}}={\text {Id}}$ and

$$\begin{aligned} g=dx^i\otimes dx_{i'}+dx_{i'}\otimes dx^i+g_{ij}(x^1,x^2,x_{1'},x_{2'}) dx^i\otimes dx^j, \end{aligned}$$

(4)

with respect to the Walker coordinates $(x^1,x^2,x_{1'},x_{2'})$. The (locally defined) almost para-Hermitian structures satisfying $J_-\mid _{{\mathfrak {D}}}={\text {Id}}$ are parametrized by a real-valued function $f(x^1,x^2,x_{1'},x_{2'})$ and are given by

$$\begin{aligned} \begin{array}{ll} J_-^f\partial _{x^1} = -\partial _{x^1} + g_{11}\partial _{x_{1'}} + f\partial _{x_{2'}}, &{} J_-^f\partial _{x_{1'}}=\partial _{x_{1'}},\\ J_-^f\partial _{x^2} = -\partial _{x^2} + (2g_{12}-f)\partial _{x_{1'}} + g_{22}\partial _{x_{2'}}, &{} J_-^f\partial _{x_{2'}}=\partial _{x_{2'}}\,. \end{array} \end{aligned}$$

(5)

Let $(g, J_-^f)$ be an almost para-Hermitian structure determined by equations (4)–(5). Then the para-Kähler form $\Omega _f(X,Y)=g(J_-^fX,Y)$ is given by $\Omega _f=(f-g_{12})dx^1\wedge dx^2+dx_{1'}\wedge dx^1+dx_{2'}\wedge dx^2$ and thus

$$\begin{aligned} d\Omega _f=\partial _{x_{1'}}\left( f-g_{12}\right) dx_{1'}\wedge dx^1\wedge dx^2 +\partial _{x_{2'}}\left( f-g_{12} \right) dx_{2'}\wedge dx^1 \wedge dx^2. \end{aligned}$$

Hence, $d\Omega _f=0$ if and only if $f(x^1,x^2,x_{1'},x_{2'})=g_{12}(x^1,x^2,x_{1'},x_{2'})+h(x^1,x^2)$ for some function $h(x^1,x^2)$ and the almost paracomplex structure becomes

$$\begin{aligned} \begin{array}{ll} J_-^h\partial _{x^1} = -\partial _{x^1} + g_{11}\partial _{x_{1'}} + (g_{12}+h)\partial _{x_{2'}}, &{} J_-^h\partial _{x_{1'}}=\partial _{x_{1'}},\\ J_-^h\partial _{x^2} = -\partial _{x^2} + (g_{12}-h)\partial _{x_{1'}} + g_{22}\partial _{x_{2'}}, &{} J_-^h\partial _{x_{2'}}=\partial _{x_{2'}}. \end{array} \end{aligned}$$

(6)

Let $(g,J_-^h)$ be an almost para-Hermitian structure determined by (4) and (6). The para-Kähler two-form is given by $\Omega _h=h\,dx^1\wedge dx^2+dx_{1'}\wedge dx^1+dx_{2'}\wedge dx^2$. We emphasize that the para-Kähler orientation and the Walker orientation are opposite. Indeed, the para-Kähler 2-form $\Omega _h$ is anti-self-dual for the para-Kähler orientation determined by the paracomplex structure $J_-^h$, but it is self-dual for the Walker orientation.

Aimed to describe anti-self-dual para-Kähler surfaces of constant scalar curvature we consider the cotangent bundle $T^*\Sigma$ of an affine surface $(\Sigma ,D)$ with metric $g=\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi$ as discussed in Sect. 3.2 and set the paracomplex structure satisfying the condition $J_-\mid _{{\text {ker}}\pi _*}={\text {Id}}$. The almost para-Hermitian structures $(g,J_-^h)$ defined by (4) and (6) are not para-Kähler in general. In order to express the components of $\nabla J_-^h$ on $T^*\Sigma$ we use the notation $(\nabla _{\partial _{x^\alpha }}J_-^h)\partial _{x^\beta }=(\nabla J_-^h)_{\beta ;\alpha }{}^\gamma \partial _{x^\gamma }$ and $(D_{\partial _{x^i}}\Phi )(\partial _{x^j},\partial _{x^k})=D\Phi _{jk;i}$ to represent the covariant derivative of the symmetric (0, 2)-tensor field $\Phi$ on $\Sigma$. Then a direct calculation shows that:

Lemma 4.1

Let $(T^*\Sigma ,g)$ be a locally symmetric self-dual Walker structure on the cotangent bundle of an affine surface $(\Sigma ,D)$. Let $J_-^h$ be a (locally defined) almost paracomplex structure determined by $J_-^h\mid _{{\text {ker}}\pi _*}={\text {Id}}$ so that $(g,J_-^h)$ is an almost para-Hermitian structure locally given by (6). Then the nonzero components of $\nabla J_-^h$ are determined by

$$\begin{aligned} 2\left( \nabla J_-^h \right) _{1;1}{}^{2'}= & {} \frac{1}{4} x_{1'}^2x_{2'}\left\{ (T_2{}^2)^2-(T_1{}^1)^2\right\} -\frac{1}{2} x_{2'}\left\{ 8\rho ^D(\partial _{x^1},\partial _{x^1}) - 4hT_1{}^2\right\} \\&+\frac{1}{2} x_{1'}\left\{ 8\rho ^D(\partial _{x^2},\partial _{x^1})+ 5hT_1{}^1 + hT_2{}^2\right. \\&\quad \left. +2(T_2{}^1\Phi _{11}-T_1{}^2\Phi _{22}+(T_2{}^2-T_1{}^1)\Phi _{12}) \right\} \\&\quad +2 \left\{ \partial _1 h-h({}^D\Gamma _{11}{}^1+{}^D\Gamma _{12}{}^2)+ D\Phi _{11;2}-D\Phi _{12;1} \right\} ,\end{aligned} $$

$$\begin{aligned} \\&\quad 2(\nabla J_-^h)_{1;2}{}^{2'} = \frac{1}{4} x_{1'}x_{2'}^2 \left\{ (T_2{}^2)^2 -(T_1{}^1)^2 \right\} +\frac{1}{2} x_{1'}\left\{ 8 \rho ^D(\partial _{x^2},\partial _{x^2}) + 4hT_2{}^1\right\} \\&\quad -\frac{1}{2} x_{2'}\left\{ 8\rho ^D(\partial _{x^1},\partial _{x^2}) -hT_1{}^1 -5hT_2{}^2\right. \\&\quad \left. -2(T_2{}^1\Phi _{11}-T_1{}^2\Phi _{22}+(T_2{}^2-T_1{}^1)\Phi _{12}) \right\} \\&\quad +2 \left\{ \partial _2 h -h({}^D\Gamma _{22}{}^2 + {}^D\Gamma _{12}{}^1) +D\Phi _{12;2} - D\Phi _{22;1} \right\} , \end{aligned}$$

where T is a trace-free parallel (1, 1)-tensor field on $(\Sigma ,D)$ and $\Phi$ is a symmetric (0, 2)-tensor field on $\Sigma$.

Theorem 1.1 follows at once from the following result describing the local structure of anti-self-dual para-Kähler surfaces with constant scalar curvature.

Theorem 4.2

Let $(M,g,J_-)$ be an anti-self-dual para-Kähler surface with constant scalar curvature. Then it is locally isometric to a Riemannian extension of the form $(T^*\Sigma ,{\tilde{g}}=\iota T\circ \iota {\text {Id}}+g_D)$ with paracomplex structure determined by $J_-\mid _{{\text {ker}}\pi _*}={\text {Id}}$, where T is a parallel (1, 1)-tensor field on a flat affine surface $(\Sigma ,D)$. Moreover,

• (i) If $T=0$, then $(M,g,J_-)$ is flat.

• (ii) If $T=c{\text {Id}}$, then $(M,g,J_-)$ has constant paraholomorphic sectional curvature c. Hence, it is locally isometric to a Riemannian extension $(T^*\Sigma ,{{\tilde{g}}}=c\,\iota {\text {Id}}\circ \iota {\text {Id}}+g_D)$ of a flat affine surface.

• (iii) If $T^2=k^2{\text {Id}}$, then $(M,g,J_-)$ is locally isometric to a product of two Lorentzian surfaces of constant opposite curvature. Moreover, (M, g) is locally isometric to a Riemannian extension $(T^*\Sigma ,{{\tilde{g}}}=\iota T\circ \iota {\text {Id}}+g_D)$ of a flat affine surface, where T is an affine para-Kähler structure on $(\Sigma ,D)$.

• (iv) If $T^2=0$, then $(M,g,J_-)$ is locally isometric to a Riemannian extension $(T^*\Sigma ,{{\tilde{g}}}=\iota T\circ \iota {\text {Id}}+g_D)$ of a flat affine surface, where T is an affine nilpotent Kähler structure on $(\Sigma ,D)$ and it is modelled on $({\mathfrak {M}}^-)$.

• (v) If $T^2=-k^2{\text {Id}}$, then $(M,g,J_-)$ is locally isometric to a Riemannian extension $(T^*\Sigma ,{{\tilde{g}}}=\iota T\circ \iota {\text {Id}}+g_D)$ of a flat affine surface, where T is an affine Kähler structure on $(\Sigma ,D)$ and it is modelled on $({\mathfrak {N}}_k)$.

Moreover, in all the cases above the para-Kähler two-form is the canonical symplectic two-form of $T^*\Sigma$.

Proof

Let $(M,g,J_-)$ be an anti-self-dual para-Kähler surface. Then there exists a Walker structure $(M,g,{\mathfrak {D}})$ so that (M, g) is self-dual with respect to the Walker orientation and $(M,g,J_-)$ is locally isometric to the cotangent bundle of an affine surface $(\Sigma ,D)$ with paracomplex structure determined by $J_-\mid _{{\text {ker}}\pi _*}={\text {Id}}$ and metric tensor $g=\iota X(\iota {\text {Id}}\circ \iota {\text {Id}})+\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi$ as in Theorem 3.1.

A para-Kähler surface is anti-self-dual if and only if the Bochner tensor vanishes (cf. [7]), and moreover, it is locally symmetric if and only if the scalar curvature is constant.

Lemma 3.3 shows that the (1, 1)-tensor field T is parallel and Assertion (ii) corresponds to the case when the scalar curvature is nonzero. Anti-self-dual para-Kähler surfaces with vanishing scalar curvature are locally conformally flat and locally symmetric. Hence, the underlying structure is induced by an affine para-Kähler structure, an affine nilpotent Kähler structure or an affine Kähler structure as discussed in Lemmas 3.4, 3.6, and 3.8, respectively.

Let $(\Sigma ,D,T)$ be an affine surface equipped with a parallel trace-free (1, 1)-tensor field T. It follows from Lemma 4.1 that, if the almost paracomplex structure $J_-^h$ determined by $J_-^h\mid _{{\text {ker}}\pi _*}={\text {Id}}$ is parallel, then it is uniquely determined. If $(\Sigma ,D,T)$ is an affine para-Kähler surface, then the coefficients of $x_{1'}$ and $x_{2'}$ in Lemma 4.1 show that $h=-\frac{2}{k}\rho ^D_{s}(\partial _{x^1},\partial _{x^2})$ for a deformation tensor field $\Phi$ given as in Remark 3.5. If $(\Sigma ,D,T)$ is an affine nilpotent Kähler surface, then Lemma 4.1 shows that $h=-\frac{2}{k}\rho ^D(\partial _{x^2},\partial _{x^2})$ and if $(\Sigma ,D,T)$ is an affine Kähler surface, then $h=\frac{2}{k}\rho ^D(\partial _{x^2},\partial _{x^2})$.

Moreover, a straightforward calculation shows that for any $(\Sigma ,D,T)$ there is an appropriate deformation tensor field $\Phi$ so that $(T^*\Sigma ,\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi , J_-^h)$ is para-Kähler, where $\Phi$ is given as in Remarks 3.5, 3.7 and 3.9. In all these cases $(T^*\Sigma ,\iota T\circ \iota {\text {Id}}+g_D+\pi ^*\Phi , J_-^h)$ is locally isometric to the Riemannian extension $\iota T\circ \iota {\text {Id}}+g_D$ of a flat affine surface $(\Sigma ,D,T)$ which is affine para-Kähler, affine nilpotent Kähler or affine Kähler, and the two-form of the corresponding locally conformally flat para-Kähler manifold is the canonical symplectic form of $T^*\Sigma$, from where Assertions (iii), (iv) and (v) follow.

Finally we consider the case $T=0$ corresponding to Assertion (i). Setting $T=0$ in Lemma 4.1, one has that the nonzero components of $\nabla J_-^h$ are given by

$$\begin{aligned} \left( \nabla J_-^h \right) _{1;1}{}^{2'}= & {} 2 x_{1'}\rho ^D_{21}-2 x_{2'}\rho ^D_{11} + \left\{ \partial _1 h-h({}^D\Gamma _{11}{}^1+{}^D\Gamma _{12}{}^2)+ D\Phi _{11;2}-D\Phi _{12;1} \right\} , \\ (\nabla J_-^h)_{1;2}{}^{2'}= & {} 2 x_{1'}\rho _{22}^D-2 x_{2'}\rho _{12}^D +\left\{ \partial _2 h -h({}^D\Gamma _{22}{}^2 + {}^D\Gamma _{12}{}^1) + D\Phi _{12;2} - D\Phi _{22;1} \right\} . \end{aligned}$$

It now follows from the coefficients of the terms of degree one above that if $\nabla J_-^h=0$, then the Ricci tensor $\rho ^D$ vanishes, and thus, $(\Sigma ,D)$ is flat. Since the Ricci tensor of $g_D+\pi ^*\Phi$ is determined by the symmetric part of $\rho ^D$ one has that $(T^*\Sigma ,{{\tilde{g}}}=g_D+\pi ^*\Phi )$ is Ricci-flat. Therefore, if $(T^*\Sigma ,{{\tilde{g}}}=g_D+\pi ^*\Phi )$ is para-Kähler, then it must be Ricci-flat and thus flat since it is locally conformally flat.

Observe that in all cases above the paracomplex structure $J_-^h$ is uniquely determined since $h=0$ if the base surface is flat and $T\ne 0$ (which follows in all cases from the expressions in Lemma 4.1). Moreover, the corresponding Kähler form is just the canonical symplectic two-form of the cotangent bundle. $\square$

Remark 4.3

The Ricci operator of any metric in Assertion (v) of Theorem 4.2 satisfies ${\text {Ric}}^2=-k^2{\text {Id}}$, and hence, since the paracomplex structure $J_-$ commute with the Ricci operator, defining $J_+=\frac{1}{k}{\text {Ric}}\cdot J_-$ one has that $(g,J_+)$ is a locally conformally flat indefinite Kähler structure.

5 Locally symmetric Kähler surfaces

Let $(M,g,J_+)$ be a locally symmetric four-dimensional Kähler manifold. Then the Ricci operator is parallel. In the diagonalizable case the metric is Einstein or locally isometric to a product of two surfaces of constant curvature. The non-diagonalizability of the Ricci operator leads to a Walker structure and hence to the situation in Sect. 3.2.

Lemma 5.1

A Kähler surface $(M,g,J_+)$ with parallel and non-diagonalizable Ricci operator is a Walker manifold.

Proof

Since the Ricci operator ${\text {Ric}}$ commutes with the complex structure $J_+$, then the trace-less Ricci operator ${\text {Ric}}^0={\text {Ric}}-\frac{\tau }{4}{\text {Id}}$ is either two-step nilpotent or complex diagonalizable.

If ${\text {Ric}}^0$ is two-step nilpotent, since ${\text {ker}}{\text {Ric}}^0$ is $J_+$-invariant, then it must be two-dimensional and parallel thus determining a totally degenerate parallel distribution, which shows that (M, g) is Walker.

If ${\text {Ric}}^0$ is complex diagonalizable with eigenvalues $\pm ik$, then $\frac{1}{k}{\text {Ric}}^0$ is a self-adjoint complex structure so that $(M,g,\frac{1}{k}{\text {Ric}}^0)$ is complex Riemannian and the (1, 1)-tensor field $J_-=\frac{1}{k}{\text {Ric}}^0\cdot J_+$ determines a para-Kähler structure $(g,J_-)$ so that $(M,g,J_-)$ is a Walker manifold as well whose parallel distribution is $J_\pm$-invariant. $\square$

Proof of Theorem 1.2

Assertion (ii) in Theorem 1.2 follows at once from Remark 4.3, which shows that any affine Kähler structure T satisfying $T^2=-k^2{\text {Id}}$ on a flat affine surface $(\Sigma ,D)$ induces a Kähler structure $(g,J_+)$ on $T^*\Sigma$ with $g=\iota T\circ \iota {\text {Id}}+g_D$ and $J_+= -\frac{1}{k}{\text {Ric}}\cdot J_-$, where $J_-$ is the para-Kähler structure determined by $J_-\mid _{{\text {ker}}\pi _*}={\text {Id}}$. Moreover, let $g_\Sigma$ be a flat Riemannian metric on $\Sigma$ with Levi–Civita connection $D$. A straightforward calculation shows that the corresponding Kähler two-form is given by $\Omega _+={-}d\iota J_\Sigma$, where $J_\Sigma$ is the complex structure on $(\Sigma ,g_\Sigma )$ induced by the volume element of the flat metric $g_\Sigma$. Choosing local adapted coordinates $(x^1,x^2)$ on $\Sigma$ so that the metric tensor $g_\Sigma = dx^1\otimes dx^1+dx^2\otimes dx^2$, one has that the complex structure $J_+$ on $T^*\Sigma$ is characterized by

$$\begin{aligned} J_+\partial _{x_{1'}}=\partial _{x_{2'}},\qquad J_+\partial _{x_{2'}}=-\partial _{x_{1'}}\,. \end{aligned}$$

Thus, it is a proper complex structure in the sense of [16] with corresponding Kähler form $\Omega _+= dx^1\wedge dx_{2'}-dx^2\wedge dx_{1'}$.

Next we construct a locally conformally flat Kähler surface showing the geometric realizability of the model $({\mathfrak {M}}^+)$ and thus proving Assertion (i) in Theorem 1.2. Let $(\Sigma ,g_\Sigma )$ be a Riemannian surface of nonzero constant Gauss curvature and let $D$ be its Levi–Civita connection. The Riemannian extension $(T^*\Sigma ,g_D)$ is a locally symmetric four-manifold with curvature tensor modelled on $({\mathfrak {M}}^+)$. Let $\omega _\Sigma$ be the Riemannian volume form of $(\Sigma ,g_\Sigma )$ and let $J_\Sigma$ be the complex structure $g_\Sigma (J_\Sigma X,Y)=\omega _\Sigma (X,Y)$. Then $\Omega _+=-d\iota J_\Sigma$ is a symplectic structure on $(T^*\Sigma ,g=g_D)$ which induces a Kähler structure with complex structure $g_D(J_+\xi ,\eta )=\Omega _+(\xi ,\eta )$ for all vector fields $\xi ,\eta$ on $T^*\Sigma$.

Let $(x^1,x^2)$ be local coordinates on $\Sigma$ so that the metric $g_\Sigma =\Psi (x^1,x^2)(dx^1\otimes dx^1+dx^2\otimes dx^2)$ and $J_\Sigma \partial _{x^1}=\partial _{x^2}$, $J_\Sigma \partial _{x^2}=-\partial _{x^1}$. Then the complex structure $J_+$ is determined, on induced coordinates $(x^1,x^2,x_{1'},x_{2'})$, by

$$\begin{aligned} J_+\partial _{x_{1'}}=\partial _{x_{2'}},\qquad J_+\partial _{x_{2'}}=-\partial _{x_{1'}}\,. \end{aligned}$$

It corresponds to a proper Kähler structure whose corresponding Kähler form is given by $\Omega _+= dx^1\wedge dx_{2'}-dx^2\wedge dx_{1'}$. $\square$

Remark 5.2

Let $\Sigma ={\mathbb {H}}^2_1$ be the Lorentzian hyperbolic plane and let $(T^*\Sigma ,g_D)$ be the Riemannian extension of its Levi–Civita connection. Let $J_\Sigma$ be the paracomplex structure on $(\Sigma ,g_\Sigma )$ determined by the Lorentzian volume form. Further, let $\Omega _-=-d\iota J_\Sigma$. Then it determines a para-Kähler structure $(g_D,J_-)$ by $g_D(J_-\xi ,\eta )=\Omega _-(\xi ,\eta )$ which is locally conformally flat with curvature tensor modelled on $({\mathfrak {M}}^-)$ and thus locally isometric to that of Theorem 1.1 -(i).

Furthermore, let $(x^1,x^2)$ be local coordinates on $\Sigma$ so that the metric tensor $g_\Sigma =\frac{1}{(x^1)^2}(dx^1\otimes dx^1-dx^2\otimes dx^2)$ and $J_\Sigma \partial _{x^1}=\partial _{x^2}$, $J_\Sigma \partial _{x^2}=\partial _{x^1}$. Let $(x^1,x^2,x_{1'},x_{2'})$ be the induced coordinates on $T^*\Sigma$. It now follows that the paracomplex structure $J_-$ is given by

$$\begin{aligned} \begin{array}{ll} J_-\partial _{x_{1'}}=-\partial _{x_{2'}}, &{}J_-\partial _{x^1}=\partial _{x^2}-2 \frac{x_{2'}}{x^1}\partial _{x_{1'}} -2 \frac{x_{1'}}{x^1}\partial _{x_{2'}}, \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{ll} J_-\partial _{x_{2'}}=-\partial _{x_{1'}}, &{} J_-\partial _{x^2}=\partial _{x^1}-2 \frac{x_{1'}}{x^1}\partial _{x_{1'}} -2 \frac{x_{2'}}{x^1}\partial _{x_{2'}}, \end{array} \end{aligned}$$

and the corresponding para-Kähler two-form $\Omega _-= - d\iota J_\Sigma =dx^1\wedge dx_{2'}+dx^2\wedge dx_{1'}$.