Bochner-Flat Para-Kähler Surfaces

Abstract

We show that Bochner-flat para-Kähler surfaces are self-dual Walker manifolds and therefore they are locally isometric to the cotangent bundle of an affine surface equipped with a modified Riemannian extension. Explicit examples of constant and non-constant scalar curvature are given.

1 Introduction

A para-Kähler manifold is a symplectic manifold $(M^{2n},\Omega )$ that is locally diffeomorphic to a product of Lagrangian submanifolds. This way its tangent bundle decomposes as a Whitney sum of Lagrangian subbundles $TM = L\oplus L'$. Considering $\pi _L$ and $\pi _{L'}$ the projections on each subbundle, the (1, 1)-tensor field defined by $J=\pi _L-\pi _{L'}$ is an almost paracomplex structure on M. Moreover, since L and $L'$ are Lagrangian subspaces one has that $\Omega (JX,JY) = -\Omega (X,Y)$ for all vector fields X, Y on M and so $g(X,Y) = \Omega (JX,Y)$ defines a neutral signature metric on M such that $g(JX,JY) = -g(X,Y)$ and $\nabla J=0$, where $\nabla$ denotes the Levi-Civita connection of (M, g).

Para-Kähler structures, which are also called bi-Lagrangian manifolds in the literature, are relevant for both Physics and Geometry. Para-Kähler geometry plays an important role in the study of several geometric problems such as the non-uniqueness of the metric for the Levi-Civita connection [5], the classification of symplectic connections [7], the spaces of oriented geodesics [3], the study of cones over pseudo-Riemannian manifolds [2] or the classical Monge-Kantorovich mass transport [15] (see also [11] for applications to supersymmetry). Paracomplex geometry is also relevant for understanding Weierstrass and Enneper type representations for Lorentzian surfaces in $\mathbb {R}^{2,1}$ [10, 16].

The Bochner curvature tensor was introduced by S. Bochner in 1949 [4]. It is formally defined as an analogue of the Weyl curvature tensor, so that the curvature tensor of a Bochner-flat manifold is completely determined by its Ricci tensor. Let $(M^{2n},g,J)$ be a para-Kähler manifold. Its Bochner curvature tensor is defined as

$$ B(X,Y)Z = R(X,Y)Z + \frac{\tau }{(2n+2)(2n+4)}R_0(X,Y)Z - \frac{1}{2(n+2)}R_1(X,Y)Z $$

for all vector fields X, Y, Z on M, where

$$ \begin{array}{l} R_0(X,Y)Z = g(X,Z)Y- g(Y,Z)X \\ \phantom {R_0(X,Y)Z =}+ g(JX,Z)JY-g(JY,Z)JX + 2g(JX,Y)JZ ,\\ [.1in] R_1(X,Y)Z = g(X,Z){{\,\mathrm{{\text {Ric}}}\,}}(Y)-g(Y,Z){{\,\mathrm{{\text {Ric}}}\,}}(X)+g(X,JZ){{\,\mathrm{{\text {Ric}}}\,}}(JY) \\ \phantom {R_1(X,Y)Z =} -g(Y,JZ){{\,\mathrm{{\text {Ric}}}\,}}(JX)+2g(X,JY){{\,\mathrm{{\text {Ric}}}\,}}(JZ) + \rho (X,Z)Y \\ \phantom {R_1(X,Y)Z =}-\rho (Y,Z)X + \rho (X,JZ)JY-\rho (Y,JZ)JX + 2\rho (X,JY)JZ\,. \end{array} $$

A para-Kähler manifold is said to be Bochner-flat if its Bochner tensor vanishes identically. A para-Kähler manifold has constant paraholomorphic sectional curvature c if and only if its curvature tensor is of the form $R(X,Y)Z = \frac{c}{4}R_0(X,Y)Z$ (see [14]). This way, any para-Kähler manifold of constant paraholomorphic sectional curvature is Bochner-flat. Moreover, a Bochner-flat para-Kähler manifold has constant paraholomorphic sectional curvature if and only if it is Einstein.

Even though the condition of being Bochner-flat is somehow analogous to that of being locally conformally flat, it is more restrictive since a Bochner-flat para-Kähler manifold has constant scalar curvature if and only if it is locally symmetric [17]. Moreover, if its Ricci operator is diagonalizable then the manifold either has constant paraholomorphic sectional curvature or it is locally isometric to a product of two spaces of constant opposite paraholomorphic sectional curvature.

The anti-self-dual Weyl curvature tensor of a four-dimensional para-Kähler manifold is determined by its scalar curvature as $W^- = \frac{\tau }{12}{\text {diag}}[2,-1,-1]$ and the symplectic form $\Omega$ is an eigenvector for the distinguished eigenvalue. On the other hand, the self-dual Weyl curvature tensor of a para-Kähler manifold is completely determined by the Bochner tensor, so $W^+ = 0$ if and only if the manifold is Bochner-flat (see [6]). An immediate consequence of these facts is that a four-dimensional para-Kähler manifold is locally conformally flat if and only if it is Bochner-flat and its scalar curvature vanishes identically.

Let (M, g, J) be a para-Kähler manifold and denote $\mathfrak {D}_{\pm } = {\text {ker}}(J\mp {\text {Id}})$ the eigenspaces corresponding to the eigenvalues $\pm 1$ of the paracomplex structure J. $\mathfrak {D}_{\pm }$ are parallel degenerate distributions and so any para-Kähler surface has an underlying Walker structure. This fact allows us to study para-Kähler structures through Walker manifolds.

The present work is organized as follows. Section 2 is devoted to the description of Walker structures in dimension four, paying special attention to self-dual Walker structures, in order to pave the way for the understanding of Bochner-flat para-Kähler structures in Sect. 3. Note that the para-Kähler and the Walker structures induce distinguished opposite orientations on the manifold, a fact that plays an important role in the theory. The classification of Bochner-flat para-Kähler surfaces of constant scalar curvature is given in Theorem 2, specifying the different curvature models realized in each situation. Finally, some examples of Bochner-flat para-Kähler surfaces of non-constant scalar curvature are provided in Sect. 3.2.

2 Walker Structures

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 [19] the existence of local coordinates $(x^1,x^2,x_{1'},x_{2'})$ so that $\mathfrak {D} = {\text {span}}\{\partial _{x_{1'}},\partial _{x_{2'}}\}$ 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)

The simplest examples of Walker manifolds are given by the so-called Riemannian extensions. We briefly review their construction as follows. Consider a surface $\Sigma$ and let $\pi :T^*\Sigma \rightarrow \Sigma$ be the projection from its cotangent bundle. Let $(p,\omega )\in T^*\Sigma$ denote a point in $T^*\Sigma$, where $p\in \Sigma$ and $\omega \in T_p^*\Sigma$. For each vector field X on $\Sigma$ the evaluation map is the function $\iota X\in \mathcal {C}^\infty (T^*\Sigma )$ defined by $\iota X(p,\omega ) = \omega (X_p)$. Two vector fields $\bar{X}$ and $\bar{Y}$ on $T^*\Sigma$ satisfy $\bar{X}=\bar{Y}$ if and only if they act on evaluation maps as $\bar{X}(\iota Z)=\bar{Y}(\iota Z)$ for any vector field Z on $\Sigma$. Given a vector field X on $\Sigma$, its complete lift $X^C$ is the vector field determined by the identity $X^C(\iota Z) = \iota [X,Z]$. In the same way as vector fields on $T^*\Sigma$ are characterized by their action on evaluation maps, (0, s)-tensor fields on $T^*\Sigma$ are characterized by their action on complete lifts of vector fields. In particular, any (1, 1)-tensor field T on $\Sigma$ induces a 1-form $\iota T$ on $T^*\Sigma$ characterized by $\iota T (X^C) = \iota (TX)$ (see [20] for more details concerning this matter).

Riemannian extensions of torsion-free connections were introduced by Patterson and Walker in [18] as metrics on $T^*\Sigma$ such that $g_D(X^C,Y^C) = -\iota (D_X Y +D_Y X)$, where D is a torsion-free connection on the base manifold $\Sigma$. Deformed Riemannian extensions are neutral signature metrics on $T^*\Sigma$ such that $g_{D,\Phi } = g_D +\pi ^*\Phi$, where $\Phi$ is a symmetric (0, 2)-tensor field on the affine surface. Afifi showed in [1] that a Walker manifold with parallel null distribution $\mathfrak {D}$ is locally isometric to a deformed Riemannian extension of an affine connection if and only if its curvature tensor satisfies $R(\cdot ,\mathfrak {D})\mathfrak {D} = 0$. These metrics were further generalized in [8] as follows. Considering a symmetric (0, 2)-tensor field $\Phi$ and (1, 1)-tensor fields T and S on an affine surface $(\Sigma ,D)$, the modified Riemannian extension is the neutral signature metric on $T^*\Sigma$ defined by $g_{D,\Phi ,T,S}=\iota T \circ \iota S + g_D +\pi ^*\Phi$, where ‘$\circ$’ denotes the symmetric product. Considering local coordinates $(x^1,x^2)$ on a neighbourhood $\mathcal {U}$ in $\Sigma$ and induced coordinates $(x^1,x^2,x_{1'},x_{2'})$ on $\pi ^{-1}(\mathcal {U})$, one has

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

where $T=T_i\!{}^j dx^i\otimes \partial _{x^j}$, $S = S_i\!{}^jdx^i\otimes \partial _{x^j}$, $\Phi = \Phi _{ij} dx^i\otimes dx^j$ and ${}^D\!\Gamma _{ij}\!{}^k$ are the Christoffel symbols of the affine connection D. Moreover, the Walker distribution is given by $\mathfrak {D}={\text {ker}}\pi _*$. Furthermore, a Walker metric corresponds to the modified Riemannian extension of an affine connection if and only if $(\nabla _{\mathfrak {D}}R)(\mathfrak {D},\cdot )\mathfrak {D} = 0$.

2.1 Self-Dual Walker Manifolds

The existence of a parallel degenerate 2-dimensional distribution $\mathfrak {D}$ on a neutral signature manifold (M, g) of dimension four naturally induces an orientation. We recall the discussion in [12]. 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 $u^*,v^*\in T_p^*M$ denote the corresponding dual forms. This way, any four-dimensional Walker manifold is naturally oriented by the self-duality of $u^*\wedge v^*$. Let $(x^1,x^2,x_{1'},x_{2'})$ be local coordinates on a four-dimensional Walker manifold 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 manifolds were described in [8] as follows.

Theorem 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

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

(2)

where $g_D$ denotes de Riemannian extension of the affine connection, 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.

Let $\Sigma$ be a surface with local coordinates $(x^1,x^2)$ and consider $(x^1,x^2,x_{1'},x_{2'})$ the induced local coordinates on $T^*\Sigma$. The canonical symplectic structure of the cotangent bundle determined by the tautological 1-form $\theta = x_{k'}dx^k$ induces an orientation determined by the volume form $d\theta \wedge d\theta = -dx^1\wedge dx^2\wedge dx_{1'}\wedge dx_{2'}$, which is the opposite of the orientation induced by the Walker structure given by $\mathfrak {D}={\text {ker}}\pi _*$.

3 Bochner-Flat Para-Kähler Surfaces

Let (M, g, J) be a para-Kähler surface and denote $\mathfrak {D}_\pm = {\text {ker}}(J\mp {\text {Id}})$. We consider Walker coordinates $(x^1,x^2,x_{1'},x_{2'})$ as in (1) and set the Walker distribution to be $\mathfrak {D}=\mathfrak {D}_+$ so that $J\vert _{\mathfrak {D}}={\text {Id}}$. We point out that para-Kähler surfaces are Walker manifolds but the converse is not true, since the parallelizability of $\mathfrak {D} = \mathfrak {D}_+$ does not ensure the integrability of the complementary distribution $\mathfrak {D}_-$. The almost para-Hermitian structures satisfying $J\vert _{\mathfrak {D}}={\text {Id}}$ are locally parametrized by a real-valued function $f(x^1,x^2,x_{1'},x_{2'})$ so that

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

(3)

Their associated Kähler 2-forms $\Omega _f(X,Y)=g(J_fX,Y)$ are given by $\Omega _f = (f-g_{12})dx^1\wedge dx^2 + dx_{1'}\wedge dx^1 + dx_{2'}\wedge dx^2$, thus

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

Therefore, $d\Omega _f = 0$ 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

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

(4)

Considering an almost para-Hermitian structure given by (1) and (4), the associated Kähler 2-form is given by $\Omega _h = h dx^1\wedge dx^2 + dx_{1'}\wedge dx^1 + dx_{2'}\wedge dx^2$. It is important to emphasize that the para-Kähler and Walker orientations are opposite. Indeed, the 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.

In order to describe Bochner-flat para-Kähler surfaces we consider the cotangent bundle $T^*\Sigma$ of an affine surface $(\Sigma ,D)$ with metric $g = \iota X(\iota {\text {Id}}\circ \iota {\text {Id}}) + \iota T \circ \iota {\text {Id}} + g_D + \pi ^*\Phi$ as in (2) and set the paracomplex structure satisfying the condition $J\vert _{{\text {ker}}\pi _*} = {\text {Id}}$. The almost para-Hermitian structures defined by (1) and (4) are not para-Kähler in general. We use the notation $(\nabla _{\partial _{x^\alpha }}J_h)\partial _{x^\beta } = (\nabla J_h)_{\beta ;\alpha }{}^\gamma \partial _{x^\gamma }$ to denote the components of $\nabla J_h$ on $T^*\Sigma$ and $(D_{\partial _{x^i}}T)\partial _{x^j}=DT_{j;i}{}^k\partial _{x^k}$, $(D_{\partial _{x^i}}\Phi )(\partial _{x^j},\partial _{x^k})= D\Phi _{jk;i}$ to represent the covariant derivatives of the (1, 1)-tensor field T and the symmetric (0, 2)-tensor field $\Phi$ on $\Sigma$, respectively. Using the notation in Theorem 1, long but straightforward calculations show that:

Lemma 1

Let (M, g) be a self-dual Walker manifold of dimension four. Let $J_h$ be an almost paracomplex structure given by $J_h\vert _{{\text {ker}}\pi _*}={\text {Id}}$ so that $(g,J_h)$ is an almost para-Hermitian structure on M. Then the nonzero components of the covariant derivative $\nabla J_h$ are given by

$$\begin{aligned} \begin{array}{rcl} 8\left( \nabla J_h \right) _{1;1}{}^4 &{}=&{} x_{1'}^3\left\{ T_2\!{}^1 {\text {tr}}(T) +8 \mathcal {S}_2\!{}^1 \right\} + x_{1'}^2x_{2'}\left\{ (T_2\!{}^2)^2-(T_1\!{}^1)^2 +8 (\mathcal {S}_2\!{}^2-\mathcal {S}_1\!{}^1) \right\} \\ &{}&{} + x_{1'}x_{2'}^2 \left\{ - T_1\!{}^2{\text {tr}}(T) -8 \mathcal {S}_1\!{}^2\right\} \\ &{}&{} + x_{1'}^2\left\{ 8DT_{1;2}{}^1-4 DT_{2;1}{}^1+2X^1(8h-4\Phi _{12})-4X^2\Phi _{22} \right\} \\ &{}&{} + x_{1'}x_{2'}\left\{ 8DT_{1;2}{}^2-4DT_{2;1}{}^2-4DT_{1;1}{}^1+{16} hX^2 +{ 8}X^1\Phi _{11} \right\} \\ &{}&{} + x_{2'}^2 \left\{ -4DT_{1;1}{}^2+4 X^2 \Phi _{11}\right\} \\ &{}&{} + x_{1'}\left\{ 16\rho ^D_{21}+ 10hT_1\!{}^1 + 2hT_2\!{}^2 - 2{\text {tr}}(T)\Phi _{12}+4(\hat{\Phi }_{21}-\hat{\Phi }_{12}) \right\} \\ &{}&{} + x_{2'}\left\{ - 16\rho ^D_{11} + 8hT_1\!{}^2 + 2{\text {tr}}(T)\Phi _{11} \right\} \\ &{}&{} +8 \left\{ \partial _1 h-h({}^D\!\Gamma _{11}\!{}^1+{}^D\!\Gamma _{12}\!{}^2)+ D\Phi _{11;2}-D\Phi _{12;1} \right\} , \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{rcl} 8(\nabla J_h)_{1;2}{}^4 &{}=&{} x_{2'}^3\left\{ -T_1\!{}^2 {\text {tr}}(T) - 8 \mathcal {S}_1\!{}^2 \right\} + x_{1'}^2x_{2'}\left\{ T_2\!{}^1{{\,\mathrm{\mathrm tr}\,}}(T) + 8 \mathcal {S}_2\!{}^1 \right\} \\ &{}&{} + x_{1'}x_{2'}^2 \left\{ (T_2\!{}^2)^2 - (T_1\!{}^1)^2 + 8(\mathcal {S}_2\!{}^2 - \mathcal {S}_1\!{}^1) \right\} \\ &{}&{} + x_{1'}^2\left\{ 4DT_{2;2}{}^1 - 4 X^1\Phi _{22} \right\} \\ &{}&{} + x_{1'}x_{2'}\left\{ -8D T_{2;1}{}^1 + 4DT_{1;2}{}^1 + 4DT_{2;2}{}^2 + 16hX^1 - 8X^2\Phi _{22} \right\} \\ &{}&{} + x_{2'}^2 \left\{ - 8DT_{2;1}{}^2 + 4DT_{1;2}{}^2 + 2X^2(8h + 4\Phi _{12}) + 4X^1\Phi _{11} \right\} \\ &{}&{} + x_{1'}\left\{ 16 \rho _{22}^D + 8hT_2\!{}^1 - 2{{\,\mathrm{\mathrm tr}\,}}(T)\Phi _{22} \right\} \\ &{}&{} + x_{2'}\left\{ -16\rho _{12}^D + 2hT_1\!{}^1 + 10hT_2^2 + 2{{\,\mathrm{\mathrm tr}\,}}(T)\Phi _{12} + 4(\hat{\Phi }_{21}-\hat{\Phi }_{12}) \right\} \\ &{}&{} +8 \left\{ \partial _2 h -h({}^D\!\Gamma _{22}\!{}^2 + {}^D\!\Gamma _{12}\!{}^1) + D\Phi _{12;2} - D\Phi _{22;1} \right\} , \end{array} \end{aligned}$$

where $X = X^i\partial _i$, $T = T_j\!{}^i \partial _{x^i}\otimes dx^j$ and $\Phi = \Phi _{ij} dx^i\otimes dx^j$ are the vector field, the (1, 1)-tensor field and the symmetric (0, 2)-tensor field on $\Sigma$ given in Theorem 1, $\mathcal {S}$ is the (1, 1)-tensor field on $\Sigma$ defined as $\mathcal {S}(Z):= D_Z X$, $\hat{\Phi }(X,Y):=\Phi (TX, Y)$ and ${}^D\!\Gamma _{ij}\!{}^k$ are the Christoffel symbols of the affine connection.

Notice that the expressions in Lemma 1 are polynomials on the fiber coordinates $x_{1'}$ and $x_{2'}$ whose coefficients are functions of the base coordinates $x^1$ and $x^2$.

3.1 Bochner-Flat Para-Kähler Surfaces of Constant Scalar Curvature

It follows from Theorem 1 that the scalar curvature of a Bochner-flat para-Kähler surface is given by $\tau = 12 \iota X + 3{{\,\mathrm{\mathrm tr}\,}}(T)$, where $\iota X$ is the evaluation map of the vector field X. Therefore if a Bochner-flat para-Kähler surface has constant scalar curvature then the vector field X vanishes and T must have constant trace. If $\tau \ne 0$ there exist local coordinates in which the (1, 1)-tensor field $T = c {\text {Id}}$ with $c\in \mathbb {R}$. In this situation, a Bochner-flat para-Kähler surface has constant paraholomorphic sectional curvature and so it is locally isometric to the cotangent bundle of a flat affine surface $(\Sigma ,D)$ endowed with a modified Riemannian extension $g = c\iota {\text {Id}}\circ \iota {\text {Id}} + g_D$ (see [9, Theorem 2.2]).

Bochner-flat para-Kähler surfaces with $\tau = 0$ are locally conformally flat. Working at a purely algebraic level, we consider $(V,\langle \cdot ,\cdot \rangle ,J)$ a para-Hermitian inner product space and a para-Kähler algebraic curvature tensor $\mathcal {A}: V\times V \times V \times V \rightarrow \mathbb {R}$ so that $\mathcal {A}(X,Y)\cdot J = J\cdot \mathcal {A}(X,Y)$. There are three non-flat locally conformally flat algebraic curvature models $(V,\langle \cdot ,\cdot \rangle , \mathcal {A})$ as follows.

$(\mathfrak {M})$: $((V,\langle \,\cdot \,,\,\cdot \,\rangle , \mathcal {A})$ given by

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

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

($\mathfrak {N}_k$): $((V,\langle \,\cdot \,,\,\cdot \,\rangle , \mathcal {A})$ given by

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

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.

($\mathfrak {P}_k$): $((V,\langle \,\cdot \,,\,\cdot \,\rangle , \mathcal {A})$ given by

$$ \mathcal {A}_{1212}=\mathcal {A}_{4334}=k $$

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.

It follows from Lemma 1 that if $\tau = 0$, then the (1, 1)-tensor field T must be parallel and so the classification of Bochner-flat para-Kähler surfaces of constant scalar curvature is summarized as follows.

Theorem 2

([13, Theorem 4.2]) Let (M, g, J) be a Bochner-flat para-Kähler surface of constant scalar curvature. Then it is locally isometric to a Riemannian extension of the form $(T^*\Sigma ,g=\iota T \circ \iota {\text {Id}}+g_D)$ with paracomplex structure determined by $J\vert _{{\text {ker}}\pi _*}={\text {Id}}$, where T is a parallel (1, 1)-tensor field on a flat affine surface $(\Sigma ,D)$. Moreover, one of the following holds:

• (i) $T=0$ and (M, g, J) is flat.

• (ii) $T = c{\text {Id}}$ and (M, g, J) has constant paraholomorphic sectional curvature $H=c$.

• (iii) $T^2=\kappa ^2{\text {Id}}$ and (M, g, J) is isometric to a product of two Lorentzian surfaces of constant opposite curvature, thus modelled on $(\mathfrak {P}_k)$.

• (iv) $T^2 = 0$ and (M, g, J) is modelled on $(\mathfrak {M})$.

• (v) $T^2=-\kappa ^2{\text {Id}}$ and (M, g, J) is modelled on $(\mathfrak {N}_k)$.

3.2 Some Examples of Bochner-Flat Para-Kähler Structures of Non-constant Scalar Curvature

Consider an affine surface $(\Sigma ,D)$ and let $(g,J_h)$ be an almost para-Hermitian structure on $T^*\Sigma$ given by $J_h\vert _{{\text {ker}}\pi _*} = {\text {Id}}$, where the metric $g = \iota X (\iota {\text {Id}}\circ \iota {\text {Id}}) + \iota T\circ \iota {\text {Id}} + g_D + \pi ^*\Phi$ is given as in Theorem 1. Aimed to construct examples of Bochner-flat para-Kähler surfaces of non-constant scalar curvature we analyze the case where the (1, 1)-tensor field T is parallel. In this situation, the nonzero components of the covariant derivative of $J_h$ reduce to

$$\begin{aligned} \begin{array}{rcl} 8\left( \nabla J_h \right) _{1;1}{}^4 &{}=&{} x_{1'}^3\left\{ T_2\!{}^1 {\text {tr}}(T) +8 \mathcal {S}_2\!{}^1 \right\} + x_{1'}^2x_{2'}\left\{ (T_2\!{}^2)^2-(T_1\!{}^1)^2 +8 (\mathcal {S}_2\!{}^2-\mathcal {S}_1\!{}^1) \right\} \\ &{}&{} + x_{1'}x_{2'}^2 \left\{ - T_1^2{\text {tr}}(T) -8 \mathcal {S}_1\!{}^2\right\} + x_{1'}^2\left\{ 2X^1(8h-4\Phi _{12})-4X^2\Phi _{22} \right\} \\ &{}&{} + x_{1'}x_{2'}\left\{ 16 hX^2 +{ 8}X^1\Phi _{11} \right\} + x_{2'}^2 \left\{ 4 X^2 \Phi _{11}\right\} \\ &{}&{} + x_{1'}\left\{ 16\rho ^D_{21}+ 10hT_1\!{}^1 + 2hT_2\!{}^2 - 2{\text {tr}}(T)\Phi _{12}+4(\hat{\Phi }_{21}-\hat{\Phi }_{12}) \right\} \\ &{}&{} + x_{2'}\left\{ - 16\rho ^D_{11} + 8hT_1\!{}^2 + 2{\text {tr}}(T)\Phi _{11} \right\} \\ &{}&{} +8 \left\{ \partial _1 h-h({}^D\!\Gamma _{11}\!{}^1+{}^D\!\Gamma _{12}\!{}^2)+ D\Phi _{11;2}-D\Phi _{12;1} \right\} , \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{rcl} 8(\nabla J_h)_{1;2}{}^4 &{}=&{} x_{2'}^3\left\{ -T_1\!{}^2 {\text {tr}}(T) - 8 \mathcal {S}_1\!{}^2 \right\} + x_{1'}^2x_{2'}\left\{ T_2\!{}^1{{\,\mathrm{\mathrm tr}\,}}(T) + 8 \mathcal {S}_2\!{}^1 \right\} \\ &{}&{} + x_{1'}x_{2'}^2 \left\{ (T_2\!{}^2)^2 - (T_1\!{}^1)^2 + 8(\mathcal {S}_2\!{}^2 - \mathcal {S}_1\!{}^1) \right\} + x_{1'}^2\left\{ - 4 X^1\Phi _{22} \right\} \\ &{}&{} + x_{1'}x_{2'}\left\{ 16hX^1 - 8X^2\Phi _{22} \right\} + x_{2'}^2 \left\{ 2X^2(8h + 4\Phi _{12}) + 4X^1\Phi _{11} \right\} \\ &{}&{} + x_{1'}\left\{ 16 \rho _{22}^D + 8hT_2\!{}^1 - 2{{\,\mathrm{\mathrm tr}\,}}(T)\Phi _{22} \right\} \\ &{}&{} + x_{2'}\left\{ -16\rho _{12}^D + 2hT_1\!{}^1 + 10hT_2\!{}^2 + 2{{\,\mathrm{\mathrm tr}\,}}(T)\Phi _{12} + 4(\hat{\Phi }_{21}-\hat{\Phi }_{12}) \right\} \\ &{}&{} +8 \left\{ \partial _2 h -h({}^D\!\Gamma _{22}\!{}^2 + {}^D\!\Gamma _{12}\!{}^1) + D\Phi _{12;2} - D\Phi _{22;1} \right\} . \end{array} \end{aligned}$$

Since the tensor field T is parallel, it has constant trace and X must be nonzero so that $\tau = 12 \iota X + 3{{\,\mathrm{\mathrm tr}\,}}(T)$ is non-constant.

If $X^1 \ne 0$ it follows immediately from the expression of the coefficient of $x_{1'}^2$ in $(\nabla J_h)_{1;2}{}^4$ above that $\Phi _{22} = 0$. Knowing this, the expression of the coefficient of $x_{1'}x_{2'}$ in $(\nabla J_h)_{1;2}{}^4$ shows that $h=0$. The same coefficient in $(\nabla J_h)_{1;1}{}^4$ shows that $\Phi _{11} = 0$ and now, focusing on the coefficient of $x_{1'}^2$ in $(\nabla J_h)_{1;1}{}^4$ we see that $\Phi _{12} = 0$. If $X^2\ne 0$, proceeding analogously it follows that h and $\Phi$ vanish identically.

Since both the function h and the symmetric (0, 2)-tensor field $\Phi$ vanish, the components of the covariant derivative $\nabla J_h$ reduce to

$$\begin{aligned} \begin{array}{rcl} 8\left( \nabla J_h \right) _{1;1}{}^4 &{}=&{} x_{1'}^3\left\{ T_2\!{}^1 {\text {tr}}(T) +8 \mathcal {S}_2\!{}^1 \right\} + x_{1'}^2x_{2'}\left\{ (T_2\!{}^2)^2-(T_1\!{}^1)^2 +8 (\mathcal {S}_2\!{}^2-\mathcal {S}_1\!{}^1) \right\} \\ &{}&{} + x_{1'}x_{2'}^2 \left\{ - T_1\!{}^2{\text {tr}}(T) -8 \mathcal {S}_1\!{}^2\right\} +16 x_{1'} \rho ^D_{21} - 16 x_{2'}\rho ^D_{11}, \\ 8(\nabla J_h)_{1;2}{}^4 &{}=&{} x_{2'}^3\left\{ -T_1\!{}^2 {\text {tr}}(T) - 8 \mathcal {S}_1\!{}^2 \right\} + x_{1'}^2x_{2'}\left\{ T_2\!{}^1{{\,\mathrm{\mathrm tr}\,}}(T) + 8 \mathcal {S}_2\!{}^1 \right\} + 16 x_{1'}\rho _{22}^D \\ &{}&{} + x_{1'}x_{2'}^2 \left\{ (T_2\!{}^2)^2 - (T_1\!{}^1)^2 + 8(\mathcal {S}_2\!{}^2 - \mathcal {S}_1\!{}^1) \right\} -16 x_{2'}\rho _{12}^D . \end{array} \end{aligned}$$

The linear terms in these two expressions show that the Ricci tensor of the affine surface must vanish identically. Therefore the affine connection is necessarily flat. Assume that T is trace-free. At this point, the components of $\nabla J_h$ take the form

$$\begin{aligned} \begin{array}{rcl} 8\left( \nabla J_h \right) _{1;1}{}^4 &{}=&{} 8x_{1'}^3\mathcal {S}_2\!{}^1 \!+\! x_{1'}^2x_{2'}\left\{ (T_2\!{}^2)^2 \!-\! (T_1\!{}^1)^2 \!+\! 8 (\mathcal {S}_2\!{}^2 \!-\! \mathcal {S}_1\!{}^1) \right\} -8 x_{1'}x_{2'}^2 S_1\!{}^2, \\ 8(\nabla J_h)_{1;2}{}^4 &{}=&{} 8x_{1'}^2x_{2'}\mathcal {S}_2\!{}^1 \!+\! x_{1'}x_{2'}^2 \left\{ (T_2\!{}^2)^2 \!-\! (T_1\!{}^1)^2 \!+\! 8(\mathcal {S}_2\!{}^2 \!-\! \mathcal {S}_1\!{}^1) \right\} - 8 x_{2'}^3 \mathcal {S}_1\!{}^2. \end{array} \end{aligned}$$

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

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

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

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

Straightforward calculations now show that, for any case described above, there exist local coordinates in which the (1, 1)-tensor filed $\mathcal {S} = D X$ takes the form $\mathcal {S} = \lambda {\text {Id}}$ for some function $\lambda \in \mathcal {C}^\infty (\Sigma )$ and the scalar curvature is given by $\tau = \iota X$.

We summarize the discussion above in the following

Theorem 3

Let $(\Sigma ,D)$ be an affine surface and let $(g,J_h)$ be an almost para-Hermitian structure on $T^*\Sigma$ such that

$$J_h\vert _{{\text {ker}}\pi _*}={\text {Id}} \quad \text { and }\quad g = \iota X (\iota {\text {Id}}\circ \iota {\text {Id}}) + \iota T\circ \iota {\text {Id}} + g_D + \pi ^*\Phi .$$

If T is parallel and ${{\,\mathrm{\mathrm tr}\,}}(T)=0$ then $(T^*\Sigma ,g,J_h)$ is a Bochner-flat para-Kähler surface if and only if $h= 0$, $\Phi =0$, the affine connection D is flat and $\mathcal {S} = \lambda {\text {Id}}$ for some $\lambda \in \mathcal {C}^\infty (\Sigma )$, being $\mathcal {S}(Z) = D_Z X$.

Remark 1

If the affine connection D is flat, there exist local coordinates on $\Sigma$ so that all the Christoffel symbols are zero. After a suitable linear transformation on the coordinates one can set T being of one of the forms described above. Straightforward calculations show that there exist suitable adapted coordinates in which Bochner-flat para-Kähler structures determined by a trace-free parallel (1, 1)-tensor field T are given by the vector field $X = a x^i\partial _{x^i}$, where $a\in \mathbb {R}$ and thus $\mathcal {S} = a {\text {Id}}$. We subsequently examine the different possibilities.

 

• (a) Let T be an affine para-Kähler structure on a flat affine surface $(\Sigma ,D)$ and take local coordinates so that ${}^D\!\Gamma _{ij}\!{}^k=0$, $T = \kappa (\partial _{x^1}\otimes dx^1-\partial _{x^2}\otimes dx^2)$ and $X = a x^i\partial _{x^i}$. Then the Bochner-flat metric induced on $T^*\Sigma$ is given by

  $$ \begin{array}{l} g = x_{1'}^2(ax_{1'}x^1 \!+\! ax_{2'}x^2 \!+\! \kappa ) dx^1\!\otimes dx^1 \!+\! x_{2'}^2(ax_{1'}x^1 \!+\! ax_{2'}x^2-\kappa )dx^2\!\otimes dx^2 \\ [.05in] \phantom {g=} \!+\! ax_{1'}x_{2'}(x_{1'}x^1 \!+\! x_{2'}x^2)(dx^1\!\otimes dx^2 \!+\! dx^2\!\otimes dx^1) \!+\! dx^1\!\otimes \! dx_{1'} \!+\! dx^2\!\otimes \! dx_{2'}. \end{array} $$

  A straightforward calculation shows that the Ricci curvatures are given by

  $$ \lambda _\pm = 2\iota X \pm \sqrt{(\iota X)^2 + 2\iota (TX) + \kappa ^2}, $$

  and the Ricci operator diagonalizes with real eigenvalues on the zero section of $T^*\Sigma$. The curvature on the zero section of the cotangent bundle corresponds to that of a locally conformally flat para-Kähler surface determined by an affine para-Kähler structure, thus it is modelled on $(\mathfrak {P}_k)$.

• (b) Let T be an affine nilpotent Kähler structure on a flat affine surface $(\Sigma ,D)$ and take local coordinates so that ${}^D\!\Gamma _{ij}\!{}^k=0$, $T= \kappa \partial _{x^1}\otimes dx^2$ and $X = a x^i\partial _{x^i}$. Then the Bochner-flat metric induced on $T^*\Sigma$ is given by

  $$ \begin{array}{l} g = ax_{1'}^2(x_{1'}x^1\!+\! x_{2'}x^2) dx^1\otimes dx^1 \!+\! x_{2'}(ax_{2'}(x_{1'}x^1\!+\!x_{2'}x^2) + \kappa x_{1'})dx^2\otimes dx^2 \\ \phantom {g=} \!+\!\tfrac{1}{2}x_{1'}(2ax_{2'}(x_{1'}x^1 \!+\! x_{2'}x^2)\!+\! \kappa x_{1'})(dx^1\otimes dx^2 \!+\! dx^2\otimes dx^1) \\ \phantom {g=}\!+\! dx^1\otimes dx_{1'} \!+\! dx^2\otimes dx_{2'}. \end{array} $$

  A straightforward calculation shows that the Ricci curvatures are given by

  $$ \lambda _\pm = 2\iota X \pm \sqrt{(\iota X)^2 + 2\iota (TX) }, $$

  and the Ricci operator is two-step nilpotent on the zero section of $T^*\Sigma$. The curvature on the zero section of the cotangent bundle corresponds to that of a locally conformally flat para-Kähler surface determined by an affine nilpotent Kähler structure, thus it is modelled on $(\mathfrak {M})$.

• (c) Let T be an affine Kähler structure on a flat affine surface $(\Sigma ,D)$ and take local coordinates so that ${}^D\!\Gamma _{ij}\!{}^k=0$, $T=\kappa (\partial _{x^2}\otimes dx^1 - \partial _{x^1}\otimes dx^2)$ and $X = a x^i\partial _{x^i}$. Then the Bochner-flat metric induced on $T^*\Sigma$ is given by

  $$ \begin{array}{l} g = x_{1'}(a x_{1'}(x_{1'}x^1+x_{2'}x^2)+\kappa x_{2'}) dx^1\otimes dx^1 \\ \phantom {g=} \!+\! x_{2'}(ax_{2'}(x_{1'}x^1+x_{2'}x^2)-\kappa x_{1'})dx^2\otimes dx^2 \\ \phantom {g=} \!+\! \left( ax_{1'}x_{2'}(x_{1'}x^1 \!+\! x_{2'}x^2)\!+\!\tfrac{1}{2}(x_{2'}^2\!-\! x_{1'}^2)\kappa \right) (dx^1\otimes dx^2 \!+\! dx^2\otimes dx^1) \\ \phantom {g=}\!+\! dx^1\otimes dx_{1'} \!+\! dx^2\otimes dx_{2'}. \end{array} $$

  A straightforward calculation shows that the Ricci curvatures are given by

  $$ \lambda _\pm = 2\iota X \pm \sqrt{(\iota X)^2 + 2\iota (TX) - \kappa ^2}, $$

  and the Ricci operator has complex eigenvalues on the zero section of $T^*\Sigma$. The curvature on the zero section of the cotangent bundle corresponds to that of a locally conformally flat para-Kähler surface determined by an affine Kähler structure, thus it is modelled on $(\mathfrak {N}_k)$.

Remark 2

In the case where T is a parallel (1, 1)-tensor field with nonzero trace on $(\Sigma ,D)$, we denote $T^0 = T - \frac{{{\,\mathrm{\mathrm tr}\,}}(T)}{2}{\text {Id}}$ the traceless part of T so that $T = T^0 + \mu {\text {Id}}$, being ${{\,\mathrm{\mathrm tr}\,}}(T) = 2\mu$. Since the affine connection D is flat, there exist local coordinates in which all the Christoffel symbols ${}^D\!\Gamma _{ij}\!{}^k$ are zero. After a suitable linear transformation on the coordinates we can set $T^0$ being of one of the forms described in (a), (b) and (c) above. Straightforward calculations show that there exist suitable adapted coordinates in which Bochner-flat para-Kähler structures determined by a parallel (1, 1)-tensor field T with ${{\,\mathrm{\mathrm tr}\,}}(T)\ne 0$ correspond to one of the following situations.

• (a) If $T^0$ is an affine para-Kähler structure then the (1, 1)-tensor fields T and $\mathcal {S}$ commute and there exist local coordinates in which $T = (\mu + \kappa )\partial _{x^1}\otimes dx^1 + (\mu -\kappa )\partial _{x^2}\otimes dx^2$ and $X = ax^1 \partial _{x^1} + (a + \frac{\kappa \mu }{2})\partial _{x^2}$. The Bochner-flat metric induced on $T^*\Sigma$ has Ricci curvatures given by

  $$ \lambda _{\pm } = \tfrac{3}{2}\mu + 3\iota X \pm \sqrt{(\iota X)^2 + \iota (TX) + \iota (KX) + \kappa ^2}, $$

  where $K = T - {{\,\mathrm{\mathrm tr}\,}}(T){\text {Id}}$. The Ricci operator diagonalizes with real eigenvalues $\lambda _\pm = \frac{3}{2}\mu \pm \kappa$ on the zero section of the cotangent bundle $T^*\Sigma$.

• (b) If $T^0$ is an affine nilpotent Kähler structure then the (1, 1)-tensor fields T and $\mathcal {S}$ commute and there exist local coordinates in which $T = \mu (\partial _{x^1}\otimes dx^1 + \partial _{x^2}\otimes dx^2) + \kappa \partial _{x^1}\otimes dx^2$ and $X = \left( a x^1 -\frac{\kappa \mu }{4}x^2\right) \partial _{x^1} + ax^2\partial _{x^2}$. The Bochner-flat metric induced on $T^*\Sigma$ has Ricci curvatures given by

  $$ \lambda _\pm = \tfrac{3}{2}\mu + 3\iota X \pm \sqrt{(\iota X)^2 + \iota (TX) + \iota (KX)}, $$

  where $K = T - {{\,\mathrm{\mathrm tr}\,}}(T){\text {Id}}$. On the zero section of the cotangent bundle the Ricci operator has a single eigenvalue $\lambda = \frac{3}{2}\mu$ that is a double root of its minimal polynomial.

• (c) If $T^0$ is an affine Kähler structure then the (1, 1)-tensor fields T and $\mathcal {S}$ commute and there exist local coordinates in which $T = \mu (\partial _{x^1}\otimes dx^1 + \partial _{x^2}\otimes dx^2) + \kappa (\partial _{x^2}\otimes dx^1 - \partial _{x^1}\otimes dx^2)$ and $X = \left( a x^1 + \frac{\kappa \mu }{4}x^2\right) \partial _{x^1} + \left( a x^2 - \frac{\kappa \mu }{4}x^1\right) \partial _{x^2}$. The Bochner-flat metric induced on $T^*\Sigma$ has Ricci curvatures given by

  $$ \lambda _\pm = \tfrac{3}{2}\mu + 3\iota X \pm \sqrt{(\iota X)^2 + \iota (TX) + \iota (KX)-\kappa ^2}, $$

  where $K = T - {{\,\mathrm{\mathrm tr}\,}}(T){\text {Id}}$. The Ricci operator is complex-diagonalizable with eigenvalues $\lambda _\pm = \frac{3}{2}\mu + \sqrt{-\kappa ^2}$ on the zero section of $T^*\Sigma$.

• (d) If $T^0 = 0$ there exist local coordinates in which the (1, 1)-tensor field T is a multiple of the identity $T = \mu {\text {Id}}$ and $X = ax^i\partial _{x^i}$. The Bochner-flat metric induced on $T^*\Sigma$ has Ricci curvatures given by

  $$ \lambda _\pm = \tfrac{3}{2}\mu + 3\iota X \pm \sqrt{(\iota X)^2 + \iota (TX) + \iota (KX)}, $$

  where $K = T - {{\,\mathrm{\mathrm tr}\,}}(T){\text {Id}}$. On the zero section of the cotangent bundle, the Ricci operator is diagonalizable with a single real eigenvalue $\lambda = \frac{3}{2}\mu$. Therefore the paraholomorphic sectional curvature is constant on the zero section of $T^*\Sigma$.