Twistor Bundle of a Neutral Kähler Surface

Abstract

In this paper, we characterize neutral Kähler surfaces in terms of their positive twistor bundle. We prove that an $O^{+,+}(2,2)$-oriented four-dimensional neutral semi-Riemannian manifold $(M,g)$ admits a complex structure J with ${\Omega }_J\in {\bigwedge }^{-}M$, such that $(M,g,J)$ is a neutral-Kähler manifold if and only if the twistor bundle $(Z^1\left(M\right),g_c)$ admits a vertical Killing vector field.

1. Introduction

A twistor bundle $Z^{\pm }\left(M\right)=\{\omega \in {\bigwedge }^{\pm }\left(M\right):g(\omega ,\omega )=2\}$ of a Riemannian four-manifold $(M,g)$ was first defined and studied by Atiyah, Hitchin and Singer in [1]. We can define $Z^{\pm }\left(M\right)$ as a one-parameter family $g_c$ of Riemannian metrics naturally induced by g and the Levi-Civita connection of g. The twistor bundle $(Z\left(M\right),g_c)$ encodes many important properties of a manifold $(M,g)$. For example, it is proved that $(M,g)$ is a self-dual manifold if and only if the naturally defined almost Hermitian structure on $Z^{-}\left(M\right)$ is integrable. Additionally, it is known that if $(M,g)$ is an Einstein self-dual manifold with a positive scalar curvature then $Z\left(M\right)$ admits a Kähler–Einstein structure. Using this property, a complete classification of compact self-dual Einstein four-manifolds of non-negative scalar curvature was obtained (see [2,3,4]). Twistor bundles are still an object of study (see ex. [5]). We showed in [6] that for any Riemannian four-manifold $(M,g)$ the manifold $(P^{-}\left(M\right),g_c)$ admits a three dimensional Lie algebra $\mathfrak{g}=\mathrm{span}\{V_1,V_2,V_3\}\cong \mathfrak{so}\left(3\right)$ of vertical Killing vector fields $\mathfrak{g}\subset \mathfrak{iso}(P^{-}\left(M\right),g_c)$, and it is a $S^1$-principal bundle over the twistor bundle $Z\left(M\right)$, such that the natural projection is a Riemannian submersion. We also proved in [6] that a Riemannian four-manifold $(M,g)$ is Kähler if and only if its positive twistor bundle admits a non-zero vertical Killing vector field. In the present paper, we introduce an $ϵ$-twistor bundle of a neutral Riemannian four-manifold as a subbundle $Z^{ϵ}M=\{\omega \in {\wedge }^{-}M:g(\omega ,\omega )=2ϵ\}$ of ${\wedge }^{-}M$. We shall write also that $\left|\right|\omega \left|\right|=ϵ\sqrt{2}$ if $\omega \in Z^{ϵ}M$. Our aim here is to give a characterization of a ±- twistor bundle of an $ϵ$- Kähler surface in some sense parallel to the above results. At a workshop in Białowieża, J. Plebański asked the author what was going on in a semi-Riemannian case. This paper is an answer to this question. For an introduction to semi-Riemannian geometry, we refer to [7]. It turns out that the case of neutral metric is similar to the Riemannian case. It is mainly because the properties of the Hodge star operator in both cases are similar, and $Spin\left(4\right)=Spin\left(3\right)\times Spin\left(3\right),Spin(2,2)=SL(2,\mathbb{R})\times SL(2,\mathbb{R})$. In the present paper, we study a twistor bundle of a four-dimensional oriented Riemannian manifold. We prove that the bundle $(P^{-},g_c)$ related to an oriented four-dimensional neutral Riemannian manifold $(M,g)$ admits a vertical Killing field $\xi$ that is invariant with respect to the action of the structural group $SO{(1,2)}_0$ of $P^{-}$ and such that sgn$g(\xi ,\xi )=ϵ\in \{-1,0,1\}$ if and only if g is an $ϵ$-Kähler metric compatible with the orientation. Equivalently, $(M,g)$ is an $ϵ$- Kähler surface compatible with the orientation if and only if the twistor bundle $Z^1\left(M\right)$ admits a vertical Killing vector field $\xi$.

2. Preliminaries

The split-quaternion algebra $\tilde{\mathbb{H}}$ is an $\mathbb{R}$ algebra with a unit 1, generated by the elements $\{1,i,j,k\}$ as a vector space over $\mathbb{R}$:

$$\tilde{\mathbb{H}}=\{x_0+x_{1}i+x_{2}j+x_{3}k:x_i\in \mathbb{R}\},$$

where $i, j, k$ satisfy the following relations:

$$i^2=-1, j^2=k^2=1, ij=-ji=k, jk=-kj=-i, ki=-ik=j.$$

It is well known that $\tilde{\mathbb{H}}$ can be realized as a Clifford algebra $Cliff\left({\mathbb{R}}^2\right)$ associated with the semi-Euclidean space ${\mathbb{R}}_1^2$. The isomorphism $\varphi :\tilde{\mathbb{H}}\to Ma{t}_2\left(\mathbb{R}\right)$ is given by the formula:

$$\varphi (x_0+x_{1}i+x_{2}j+x_{3}k)=\left(\begin{array}{cc}{x}_0+x_2& -x_1+x_3\\ \\ x_1+x_3& x_0-x_2\end{array}\right).$$

(1)

Let us denote the conjugate element $x_0-x_{1}i-x_{2}j-x_{3}k$ of $q=x_0+x_{1}i+x_{2}j+x_{3}k$ by $\overline{q}$. On $\tilde{\mathbb{H}}$ we have a quadratic form $Q(q,q)=\langle q,q\rangle =q\overline{q}=x_0^2+x_1^2-x_2^2-x_3^2$. It easily follows from (1) that the group $S=\{q\in \tilde{\mathbb{H}}:Q(q,q)=1\}$ is isomorphic with the group $SL(2,\mathbb{R})$. Note that $\langle q,q\rangle =ϵ$ if $x_0^2+x_1^2-x_2^2-x_3^2=ϵ$. Vector space $Im\tilde{\mathbb{H}}=\{q\in \tilde{\mathbb{H}}:q=-\overline{q}\}$ can be identified with a Lie algebra $\mathfrak{sl}(2,\mathbb{R})$, with the Lie bracket $[p,q]=pq-qp$. The adjoint action of $SL(2,\mathbb{R})$ on $\mathfrak{sl}(2,\mathbb{R})=Im\tilde{\mathbb{H}}$, is $SL(2,\mathbb{R})\ni q\to ad\left(q\right)$, where $ad\left(q\right)p=qp\overline{q}$. Note that $\langle ad\left(q\right)p,ad\left(q\right)p\rangle =\langle p,p\rangle$ and $ad\left(q\right)\in SO(1,2)$. Since $(\tilde{\mathbb{H}},\langle .,.\rangle )={\mathbb{R}}_1^3$, it easily follows that $S{O}^{+}(1,2)=SL(2,\mathbb{R}){\mathbb{Z}}_2$. Note also that $Spin(2,2)=SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ and $SO{(2,2)}_0=SL(2,\mathbb{R}){\times }_{{\mathbb{Z}}_2}SL(2,\mathbb{R})$.

Let $(M,g)$ be an $O^{+,+}(2,2)$-oriented neutral surface. Let us recall here that by $O^{+,+}(2,2)$ we mean the identity component of the group of isometries of $({\mathbb{R}}_{2,2}^4,g_0)$. We shall also write $S{O}^{+}(2,2)=O^{+,+}(2,2)$. By $\mathfrak{X}\left(M\right)$, we denote the Lie algebra of all local vector fields on M. If D is a vector bundle over M then by $\Gamma \left(D\right)$ we denote the set of all local sections of D. We also write ${\mathcal{A}}^k\left(M\right)=\Gamma \left({\wedge }^{k}T{M}^{*}\right)$. If dim$M=4$, then let $P=P(M,S{O}^{+}(2,2))$ be the principal fiber bundle of oriented orthonormal frames $u:{\mathbb{R}}_{2,2}^4\to T_{x}M$ of $TM$. In the sequel, we identify $TM$ with $T{M}^{*}$ by means of g. In particular, we identify ${\wedge }^{2}TM$ with ${\wedge }^{2}T{M}^{*}$.The bundle of 2-forms over M splits as the Whitney sum

$${\wedge }^{2}M={\wedge }^{+}M\oplus {\wedge }^{-}M,$$

(2)

${\wedge }^{+}M,{\wedge }^{-}M$ being the eigenspace bundles of the Hodge star operator $\ast \in$ End $\left({\wedge }^{2}M\right)$ defined by

$$g(\omega ,\ast \eta )vo{l}_g=\omega \wedge \eta .$$

The group $S{O}^{+}(2,2)=O^{+,+}(2,2)$ acts naturally on

$${\wedge }^2{\mathbb{R}}^4={\wedge }^{+}{\mathbb{R}}^4\oplus {\wedge }^{-}{\mathbb{R}}^4.$$

(3)

Define $G_{+}=\{A\in S{O}^{+}(2,2):{\left(A_{*}\right)}_{|{\wedge }^{-}{\mathbb{R}}^4}=i{d}_{|{\wedge }^{-}{\mathbb{R}}^4}\}$, $G_{-}=\{A\in SO(2,2):{\left(A_{*}\right)}_{|{\wedge }^{+}{\mathbb{R}}^4}=i{d}_{|{\wedge }^{+}{\mathbb{R}}^4}\}$, where by $S{O}^{+}(2,2)\ni A\to A_{*}\in Aut\left({\wedge }^2{\mathbb{R}}^4\right)$ we denote the natural representation of $SO(2,2)$ on ${\wedge }^2{\mathbb{R}}^4$. Then, both $G_{\pm }$ are isomorphic to $SL(2,\mathbb{R})$, and there exists a two fold covering $SO{(2,2)}^{+}\to G_{+}{\times }_{{\mathbb{Z}}_2}{G}_{-}$. Let $u=(e_1,e_2,e_3,e_4)\in P$ be an oriented orthonormal frame such that $g(e_1,e_1)=g(e_2,e_2)=1$ and $g(e_3,e_3)=g(e_4,e_4)=-1$. Note that the forms $\omega =-e_1\wedge e_2+e_3\wedge e_4, \eta =e_1\wedge e_3+e_4\wedge e_2, \theta =e_1\wedge e_4+e_2\wedge e_3$ are an orthogonal basis of ${\wedge }_{p\left(u\right)}^{-}M$ satisfying relations: ${\omega }^2=-id, {\eta }^2={\theta }^2=id, \omega \eta =-\eta \omega =\theta$.

We have a homomorphism of principal fiber bundles $F:P⟶P^{-}$, defined by: $F(e_1,e_2,e_3,e_4)=(-e_1\wedge e_2+e_3\wedge e_4,e_1\wedge e_3+e_4\wedge e_2,e_1\wedge e_4+e_2\wedge e_3)$, which is equivariant under homomorphism ${\Phi }_{-}:S{O}^{+}(2,2)\to S{O}^{+}(1,2)$, where ${\Phi }_{\pm }$ are defined as a composition ${\Phi }_{\pm }:S{O}^{+}(2,2)=G_{+}{\times }_{{\mathbb{Z}}_2}{G}_{-}\to G_{\pm }{\mathbb{Z}}_2\simeq S{O}^{+}(1,2)$. Note that $P^{-}=P{G}_{+}$. The homomorphisms ${\Phi }_{\pm }$ induce the Lie algebras homomorphisms ${\varphi }_{\pm }:\mathfrak{so}(2,2)\to \mathfrak{so}(1,2)$, and $\mathfrak{so}(2,2))=ker{\varphi }_{+}\oplus ker{\varphi }_{-}$ is the direct sum of ideals, both isomorphic to $\mathfrak{so}(1,2)$. If we identify the Lie algebra $\mathfrak{so}(2,2)$ with ${\wedge }^2{\mathbb{R}}^4$, then ${\mathfrak{so}}_{+}(2,2)=ker{\varphi }_{-}={\wedge }^{+}{\mathbb{R}}^4$, and ${\mathfrak{so}}_{-}(2,2)=ker{\varphi }_{+}={\wedge }^{-}{\mathbb{R}}^4$ are the Lie algebras of $G_{+},G_{-}$ respectively. So, we have proved:

Lemma 1. 

Let $(M,g)$ be an oriented neutral four-manifold, and let $x\in M$. Then

(a) ${\wedge }_x^{+}M$ and ${\wedge }_x^{-}M$ are mutually commuting ideals, both isomorphic to $\mathfrak{sl}(2,\mathbb{R})=Im\tilde{\mathbb{H}}$ in the Lie algebra ${\wedge }^{2}M=\mathfrak{so}(2,2)$ of skew-adjoint endomorphisms of $(T_{x}M,$$g_x)$

(b) Elements of length $ϵ\sqrt{2}$ in ${\wedge }_x^{\pm }M$ coincide with the orthogonal almost complex structures if $ϵ=-1$, with null orthogonal structures if $ϵ=0$ and with orthogonal product structures if $ϵ=1$

(c) Every oriented orthogonal basis $\omega ,\eta ,\theta$ of ${\wedge }_x^{\pm }M$ such that $\left|\omega \right|=-\sqrt{2},\left|\eta \right|=\left|\theta \right|=\sqrt{2}$, satisfies the condition ${\omega }^2=-id,{\eta }^2={\theta }^2=id,\omega \eta =-\eta \omega =\theta$, so that it defines a split quaternionic structure in $T_{x}M$.

Definition 1. 

Let $(M,g)$ be a neutral oriented four-manifold and let $=ϵ\in \{-1,0,1\}$. Then we call $(M,g)$ an ϵ-Kähler manifold if there exists a skew-symmetric endomorphism $J\ne 0$ of $(TM,g)$ such that $J^2=-ϵI{d}_{TM}$ and the corresponding Kähler form ${\Omega }_J=g(J.,.)$ is self-dual i.e., ${\Omega }_J\in {\wedge }^{-}M$, and parallel i.e., $\nabla {\Omega }_J=0$.

Remark 1. 

Note that if $ϵ=1$, then we obtain a neutral Kähler metric (see [8,9]); if $ϵ=0$, then we get a null Kähler metric. Note that in this case Im $J\subset \{X\in TM:g(X,X)=0\}$ and dim Im $J=2$ (see [10], where, however, this notion is stronger, the author assumes additionally that Kähler form comes from a parallel spinor field $\varphi \in \Gamma \left(S_{-}\right)$), and for $ϵ=-1$ we get a Norden Kähler metric (see [11]).

Let $(f_1,f_2,f_3,f_4)$ be the standard orthonormal basis of ${\mathbb{R}}_{2,2}^4$. On $\mathfrak{so}(2,2)$, we consider a metric $\langle X,Y\rangle =-\frac{1}{2}$tr$X\circ Y={\sum }_{i<j}{ϵ}_i{ϵ}_j{X}_j^i{Y}_j^i$. Then, the identification $\mathfrak{so}(2,2)\cong {\wedge }^2{\mathbb{R}}^4$ is an isometry of $\left(\mathfrak{so}\right(2,2),\langle ,\rangle )$ on ${\wedge }^2{\mathbb{R}}^4$ furnished with an extension of the standard neutral metric also denoted by $\langle ,\rangle$ defined on the basis $f_i\wedge f_{j}i<j$ by $\langle f_i\wedge f_j,f_p\wedge f_q\rangle ={ϵ}_j{ϵ}_i{\delta }_p^i{\delta }_q^j$, where ${ϵ}_1={ϵ}_2=-1,{ϵ}_3={ϵ}_4=1$. If $u=(e_1,e_2,e_3,e_4)\in P=S{O}^{+}\left(M\right)$ is an orthonormal frame, then we define an isomorphism ${\alpha }_u:\mathfrak{so}(2,2)\to {\wedge }^2{T}_{\pi \left(u\right)}M$ by ${\alpha }_u\left(A\right)=g(u\circ A\circ u^{-1}.,.)$, where $\pi :S{O}^{+}\left(M\right)\to M$ is a projection on the base of the principal bundle $S{O}^{+}\left(M\right)$. Then. for $A,B\in \mathfrak{so}(2,2)$ we have $\langle A,B\rangle =g({\alpha }_u\left(A\right),{\alpha }_u\left(B\right))$. We denote by $v_{+},v_{-}$ or $v^{+},v^{-}$, the components of v with respect to decomposition (2) or (3). Let $ϵ\in \{-1,0,1\}$. The twistor bundle ${\pi }_Z:Z^{ϵ}\left(M\right)\to M$ over an oriented four-manifold $(M,g)$ is a subbundle of the vector bundle ${\wedge }^{-}TM$: $Z^{ϵ}\left(M\right)=\{\omega \in {\wedge }^{-}TM:\Vert \omega \Vert =ϵ\sqrt{2}\}$, whose fibers are quadrics if $ϵ\in \{-1,1\}$ and cones if $ϵ=0$. These bundles may be identified with the bundle of almost complex structures J on M, if $ϵ=1$ i.e., $J^2=-id, g(JX,JY)=g(X,Y)$; null structures if $ϵ=0$ i.e., $T^2=0, g(TX,Y)=-g(TY,X)$); and product structures if $ϵ=-1$ i.e., $K^2=id, g(KX,KY)=-g(X,Y)$). Let $(P,\overline{g}),(M,g)$ be semi-Riemannian spaces. We denote by V the distribution of vertical vectors ( $X\in V$ if X is tangent to a fiber $p^{-1}\left(x\right)$ for a certain $x\in M$) and by H the horizontal distribution, which is an orthogonal complement of V. A semi-Riemannian submersion $p:P\to M$ (see [7]) is a submersion that preserves lengths of horizontal vectors, i.e., $p:H_x\to T_{p\left(x\right)}M$ is an isometry. Let $\mathcal{H}$ and $\mathcal{V}$ denote the projections of the tangent space $TM$ of M onto the subspaces $H,V$ of horizontal and vertical vectors, respectively. By T and A, we denote the O’Neill’s tensors defined as follows (see [7]): $T_{X}Y=\mathcal{H}\left({\nabla }_{\mathcal{V}X}\mathcal{V}Y\right)+\mathcal{V}\left({\nabla }_{\mathcal{V}X}\mathcal{H}Y\right)$, $A_{X}Y=\mathcal{V}\left({\nabla }_{\mathcal{H}X}\mathcal{H}Y\right)+\mathcal{H}\left({\nabla }_{\mathcal{H}X}\mathcal{V}Y\right).$

3. Vertical Killing Vector Fields on the $\mathbf{ϵ}$-Twistor Bundle ${\mathbf{Z}}^{\mathbf{ϵ}}\left(\mathbf{M}\right)$

Let $(M,g)$ be a Neutral Semi-Riemannian four-manifold and $\pi :S{O}^{+}\left(M\right)=P\to M$ be the $S{O}^{+}(2,2)$-Principal Bundle of Orthonormal oriented frames on M. By $\omega$, we denote the connection form $\omega \in {\mathcal{A}}^1\left(P\right)\otimes \mathfrak{so}(2,2)$ on P induced by the Levi-Civita connection ∇ of $(M,g)$, and by $\tilde{\Omega }=d\omega +[\omega ,\omega ]$, the curvature form induced by ∇. By V, we denote the vertical bundle $T^{vert}P$ of the principal bundle P. There is a natural parallelism $V\cong \mathfrak{so}(2,2))$ given by $\mathfrak{so}(2,2)\ni A\to A^{*}\in V$, where by $A^{*}$, we denote the fundamental vector field corresponding to A. The connection $\Gamma$ on P given by $\omega$ induces a connection ${\Gamma }^{-}$ on $P^{-}$ with a connection form ${\omega }^{-}$ defined by $F^{*}{\omega }^{-}={\varphi }_{-}\omega$ (see [12]), where ${\varphi }_{-}=d{\Phi }_{-}$ is a homomorphism of Lie algebras. We identify the Lie algebra of the structural group of $P^{-}$ with ${\mathfrak{g}}_{-}=\mathfrak{so}(1,2)=\mathfrak{sl}(2,\mathbb{R})$. Then, ${\varphi }_{-}$ is just a projection ${\varphi }_{-}:\mathfrak{so}(2,2)\to {\mathfrak{g}}_{-}$, with respect to the decomposition $\mathfrak{so}(2,2)=\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{sl}(2,\mathbb{R})$. The curvature form of ${\Gamma }^{-}$ is ${\tilde{\Omega }}^{-}=d{\omega }^{-}+[{\omega }^{-},{\omega }^{-}]$. From [10], it follows that $F^{*}{\tilde{\Omega }}^{-}={\varphi }_{-}\tilde{\Omega }$. We denote by $V^{-}$ the vertical distribution and by H the horizontal distribution on $T{P}^{-}$. Hence, $H=\{X\in T{P}^{-}:{\omega }^{-}\left(X\right)=0\}$. We define the metric $g_c$ on $P^{-}$ by the formula

$$g_c=-\frac{1}{2}{c}^2\mathrm{B}{}_{\mathfrak{so}(1,2)}({\omega }^{-}\left(X\right)\circ {\omega }^{-}\left(Y\right))+p^{*}g(X,Y).$$

(4)

By ${\nabla }^c$, we denote the Levi-Civita connection of $(P^{-},g_c)$. A natural projection $p:P^{-}\to M$ is a semi-Riemannian submersion of $P_c^{-}=(P^{-},g_c)$ onto $(M,g)$, such that $p\circ F=\pi$. The O’Neill’s tensor $T=0$ (the fibers of p are totally geodesic and homothetic to $SO(1,2)$). Note that ${\alpha }_{u|{\mathfrak{g}}_{-}}:{\mathfrak{g}}_{-}={\wedge }^{-}{\mathbb{R}}^4\to {\wedge }^{-}{M}_{\pi \left(u\right)}$ is an isomorphism and if ${\Phi }_{-}\left(g\right)=id$, then ${\alpha }_{ug|{\mathfrak{g}}_{-}}={\alpha }_{u|{\mathfrak{g}}_{-}}$. If $g\in SO(2,2)$ and $X\in \mathfrak{so}(2,2))$, then $A{d}_{g}X=g\circ X\circ g^{-1}$. Under the isomorphism $\mathfrak{so}(2,2)={\wedge }^2{\mathbb{R}}^4$, if $X=\sum A_j^i{f}_i\wedge f_j$ then $A{d}_{g}X=\sum A_j^{i}g{f}_i\wedge g{f}_j$. In particular,

$${\alpha }_{ug}\left(X\right)={\alpha }_u\left(A{d}_{g}X\right),$$

(5)

Note that if $F\left(u\right)=F\left(v\right)$ then $u=vg$ where ${\Phi }_{-}\left(g\right)=id$. It follows that for any $l\in P^{-}$, the isomorphism ${\alpha }_l^{-}{:\mathfrak{so}(2,2))}_{-}\to {\wedge }^{-}{M}_{p\left(l\right)}$, given by ${\alpha }_l^{-}\left(X\right)={\alpha }_u\left(X\right)$ for any $u\in P$ such that $F\left(u\right)=l$ is well defined. There is a natural homomorphism of vector bundles $\widehat{\pi }:V\to {\wedge }^{2}M$ covering the projection $\pi :P\to M$ given by the formula $\widehat{\pi }\left(A_u^{*}\right)={\alpha }_u\left(A\right)$ for $A\in \mathfrak{so}(2,2))$. The homomorphism $\widehat{\pi }$ is an isomorphism on the fibers. It induces the homomorphism $\widehat{p}:T^{vert}{P}^{-}=V^{-}\to {\wedge }^{-}M$ of fiber bundles, which also is an isomorphism on the fibers and covers $p:P^{-}\to M$. The homomorphism $\widehat{p}$ is given explicitly by the formula $\widehat{p}\left(A_l^{*}\right)={\alpha }_l^{-}\left(A\right)$ where ${A\in \mathfrak{so}(2,2))}_{-}$ (we treat here ${\mathfrak{so}(2,2))}_{-}$ as the Lie algebra of the structural Lie group $G_{-}{\mathbb{Z}}_2$ of the principal fiber bundle $P^{-}$). Let $\Omega$ be a Kähler form of a $ϵ$-Kähler surface $(M,g,J)$. Let us define by $\xi$ the unique vertical vector field on $P^{-}$ such that

$$\widehat{p}\left(\xi \right)=\Omega .$$

(6)

Let ${\omega }^{-}$ be the induced connection form on $P^{-}$. It follows that for any $u\in P$ we have

$${\alpha }_u\left({\omega }_{F\left(u\right)}^{-}\left(\xi \right)\right)={\Omega }_{\pi \left(u\right)}.$$

(7)

The field $\xi$ is invariant with respect to the action of the group $SO(1,2)=G_{-}{\mathbb{Z}}_2$ on $P^{-}$. In fact, we have

$$\widehat{p}\circ R_g=\widehat{p}$$

(8)

for any $g\in SO(1,2)$, where by $R_g$ we denote the right action of $SO(1,2)$ on $T{P}^{-}$. Let $v_1=-f_1\wedge f_2+f_3\wedge f_4, v_2=f_1\wedge f_3+f_4\wedge f_2, v_3=f_1\wedge f_4+f_2\wedge f_3$, where $(f_1,f_2,f_3,f_4)$ is the standard basis of ${\mathbb{R}}_2^4$. Then, $(v_1,v_2,v_3)$ is an orthogonal basis of $\mathfrak{sl}(2,\mathbb{R})=\mathfrak{so}{(2,2)}_{-}$, and let $\{V_1,V_2,V_3\}$ be Killing vector fields corresponding to $v_i$. Since $\xi$ is invariant with respect to the action of $SO(1,2)$ on $P^{-}$, it follows that $[V_i,\xi ]=0$.

Next, we shall prove

Theorem 1. 

Let an oriented neutral semi-Riemannian four-manifold $(M,g)$ be an ϵ-Kähler surface, and Ω be its Kähler form. Then, the vertical vector field $\xi \in \mathfrak{X}\left(P^{-}\right)$, such that $\widehat{p}\left(\xi \right)=\Omega$, is an $SO(1,2)$-invariant Killing vector field on $(P^{-},g_c)$, which is time-like if $ϵ=-1$, null if $ϵ=0$ and space-like if $ϵ=1$. On the other hand, if $(P^{-},g_c)$ admits a non-zero SO(1,2)-invariant vertical Killing vector field ξ then ξ has constant length and, if sgn $g(\xi ,\xi )=ϵ$, then $(M,g)$ is an ϵ- Kähler surface.

Proof. 

We start by showing that if $(M,g)$ is an $ϵ$-Kähler manifold then $\xi$ is the Killing vector field of $(P^{-},g_c)$. We have to show that $g_c({\nabla }_A^c\xi ,B)=-g_c({\nabla }_B^c\xi ,A)$ for all $A,B\in T{P}^{-}$. Let $l\in P^{-}$, $H_l=\{X\in T_l{P}^{-}:{\omega }_l^{-}\left(X\right)=0\}$, and let $X^{*}\in H_l$ be a horizontal lift to $T_l{P}^{-}$ of a vector $X\in T_{p\left(l\right)}M$. Let us take a geodesic $\gamma$ on M such that $\dot{\gamma }\left(0\right)=X$. Let $l_t$ be a horizontal lift to $P^{-}$ of the geodesic ${\gamma }_t$. Then, $X_t^{*}=\dot{l_t}$ is a horizontal vector field along $l_t$ and ${\nabla }_{X_t^{*}}^c{X}_t^{*}=0$. We have $g_c(\xi ,X_t^{*})=0$. Consequently, $g_c({\nabla }_{X^{*}}^c\xi ,X^{*})=0$. It follows that $g_c({\nabla }_{X^{*}}^c\xi ,Y^{*})+g_c({\nabla }_{Y^{*}}^c\xi ,X^{*})=0$ for all $X^{*},Y^{*}\in H$. Note also that

$$\begin{array}{ll}{g}_c({\nabla }_{V_i}^c\xi ,V_j)& =g_c({\nabla }_{\xi }^c{V}_i,V_j)=-g_c({\nabla }_{V_j}^c{V}_i,\xi )=g_c({\nabla }_{V_i}^c{V}_j,\xi )\\ & =-g_c({\nabla }_{\xi }^c{V}_j,V_i)=-g_c({\nabla }_{V_j}^c\xi ,V_i)\end{array},$$

(9)

where we used $[\xi ,V_i]=0,{\nabla }_{V_i}^c{V}_j=-{\nabla }_{V_j}^c{V}_i$, which follows from the fact that $V_i$ are left invariant vector fields on the fibers $SO{(1,2)}^{+}$ with bi-invariant standard metric, thus ${\nabla }_{V_i}^c{V}_j=\frac{1}{2}[V_i,V_j]=-\frac{1}{2}[V_j,V_i]=-{\nabla }_{V_j}^c{V}_i$ (note that fibers are totally geodesic).

The last case is to prove that for every horizontal field $X^{*}\in \Gamma \left(H\right)$, we have

$$g_c({\nabla }_{X^{*}}^c\xi ,V_i)=-g_c({\nabla }_{V_i}^c\xi ,X^{*}).$$

(10)

Note that $g_c({\nabla }_{V_i}^c\xi ,X^{*})=0$ since the distribution $V^{-}$ is totally geodesic. Consequently, we have to prove that $g_c({\nabla }_{X^{*}}^c\xi ,V_i)=0$. Since $X^{*}{g}_c(\xi ,V_i)=g_c({\nabla }_{X^{*}}^c\xi ,V_i)+g_c(\xi ,{\nabla }_{X^{*}}^c{V}_i)=g_c({\nabla }_{X^{*}}^c\xi ,V_i)$, the last equation is equivalent to $X^{*}{g}_c(\xi ,V_i)=0$. It is clear that for $l=F\left(u\right)\in P^{-}$, we have $g_c{(\xi ,V_i)}_l=c^{2}g({\alpha }_u({\omega }_l^{-}\left(\xi \right),{\alpha }_u\left(v_i\right))=c^{2}g(\Omega ,{\alpha }_u\left(v_i\right)).$ Let $x_t$ be a smooth path on M such that $\dot{x}\left(0\right)=p\left(X^{*}\right)$. Let $l_t$ be a horizontal lift of $x_t$ such that $l_0=l=F\left(u\right)$. It is obvious that we can take $l_t=F\left(u_t\right)$ where $u_t$ is the horizontal lift of $x_t$ in $S{O}^{+}\left(M\right)$. Thus,

$$X^{*}{g}_c(\xi ,V_i)=\frac{d}{dt}{c}^{2}g({\Omega }_{x_t},{\alpha }_{u_t}\left(v_i\right))=0$$

(11)

since $\nabla \Omega =0$ and ${\alpha }_{u_t}\left(v_i\right)$ is parallel along $x_t$. Now let us assume that $\xi$ is an $SO(1,2)$-invariant vertical Killing vector field on $(P^{-},g_c)$. Since the fibers $F_x$ of $p:P^{-}\to M$ are totally geodesic and isometric to $SO(1,2)$ with bi-invariant semi-Riemannian metric, it follows that ${\xi }_{|F_x}$ is a right-invariant Killing vector field on each $F_x$. In particular, since $\left|\xi \right|$ is constant on $F_x$ it follows that ${\nabla }_{\xi }^c\xi =0$ on $F_x$, and hence ${\nabla }_{\xi }^c\xi =0$ on $P^{-}$. This means that $\left|\xi \right|$ is constant on the whole of $P^{-}$. We can assume that $g_c(\xi ,\xi )=2c^2ϵ$. Let us define the Kähler form on $(M,g)$ by the formula (7). From (10), it follows that $g_c({\nabla }_{X^{*}}^c\xi ,V_i)=0$ (the fibers are totally geodesic); thus, (11) holds and, consequently, $\nabla \Omega =0$ (note that ${\alpha }_u\left(v_i\right):i=1,2,3$ span ${\wedge }^{-}{M}_{\pi \left(u\right)}$). □

Twistor bundle ${\pi }_Z:Z^{ϵ}\left(M\right)\to M$ is the image of $P^{-}$ by a projection ${\Phi }_i:P^{-}\to Z^{ϵ}\left(M\right)$ given by ${\Phi }_i({\omega }_1,{\omega }_2,{\omega }_3)={\omega }_i$, where $i=1$ if $ϵ=1$ and $i=2$ if $ϵ=-1$. Note that the fibers of the bundle ${\pi }_Z:Z^{ϵ}\to M$ are hyperboloids $\{X^2-Y^2-Z^2=ϵ\}$, so in the case $ϵ=1$ the fibers have two connected components. Hence, $P^{-}$ is the principal $S^1$-bundle if $ϵ=-1$ and $(\mathbb{R},+)$ bundle over $Z^{ϵ}\left(M\right)$ if $ϵ=1$, and $V_i$ is a Killing vector field on $(P^{-},g_c)$, which is the fundamental vector field of the action of $S^1$ or $(\mathbb{R},+)$ on $P^{-}$. Consequently, there exists a metric $g_{*c}$ induced on $Z^{ϵ}\left(M\right)$ by $g_c$ and $V_i$, such that $\pi :(P^{-},g_c)\to (Z^{ϵ}\left(M\right),g_{*c})$ is a semi-Riemannian submersion. We have

Theorem 2. 

Let $(Z^{ϵ}\left(M\right),g_{*c}),ϵ\in \{-1,1\}$ be a positive ϵ-twistor bundle of an oriented semi-Riemannian four-manifold $(M,g)$ with the standard metric $g_{*c}$. Then, M admits a self-dual structure J compatible with the metric g such that $(M,g,J)$ is an η-Kähler surface for some $\eta \in \{-1,0,1\}$ if and only if $(Z^{ϵ}\left(M\right),g_{*c})$ admits a non-zero vertical Killing vector field ${\xi }_{*}$.

Proof. 

Let ${\xi }_{*}$ be a vertical Killing vector field on $Z^{ϵ}\left(M\right)$. Note that on each fiber ${\xi }_{*}$ is ${\Phi }_i$-related with a Killing right-invariant vector field on $SO(1,2)=SL(2,\mathbb{R}){\mathbb{Z}}_2$ with a bi-invariant metric ($V_i$ corresponds to the left-invariant Killing fields). It follows from the fact that the Lie algebra of Killing vector fields on each fiber of $p:Z^{ϵ}\to M$ coincides with the Lie algebra of right invariant vector fields on $SO(1,2)$. ${\Phi }_1$ is a semi-Riemannian submersion. Thus, we can choose a lift ${\xi }^{*}$ of ${\xi }_{*}$ on each fiber, which is $SO(1,2)$-invariant and hence uniquely determined. Note that these lifts glue together to the global lift ${\xi }^{*}$, which is a $SO(1,2)$-invariant non-zero global vector field. Thus, ${\xi }_{*}$ lifts to a vector field ${\xi }^{*}$ on $P^{-}$. Let $V_1,V_2,V_3$ be a basis of a Lie algebra $\mathfrak{sl}(2,\mathbb{R})$ of the structural group of $P^{-}$. From the construction of ${\xi }^{*}$, we have $[{\xi }^{*},V_i]=0$. We shall show that ${\xi }^{*}$ is a Killing vector field. It is enough to show that $L_{{\xi }^{*}}\theta =0$, where $\theta \left(X\right)=g_c(V_1,X)$. Since the fibers are totally geodesic, we have $L_{{\xi }^{*}}\theta \left(V_i\right)=0$. We shall show that $L_{{\xi }^{*}}\theta \left(X_{P^{-}}^{*}\right)=0$, where $X_{P^{-}}^{*}$ is a horizontal lift of a field $X\in \mathfrak{X}\left(M\right)$ with respect to the semi-Riemannian submersion $p:P^{-}\to M$. Analogously, we shall denote by $X_Z^{*}$ a horizontal lift of a field $X\in \mathfrak{X}\left(M\right)$ with respect to the semi-Riemannian submersion ${\pi }_Z:Z^{ϵ}\left(M\right)\to M$. Note that the fields ${\xi }^{*},X_{P^{-}}^{*}$ are ${\Phi }_i$-related with, respectively, ${\xi }_{*},X_Z^{*}$. However, since ${\xi }_{*}$ is a vertical Killing vector field with respect to ${\pi }_Z:Z^{ϵ}\left(M\right)\to M$, it is clear that $[{\xi }_{*},X_Z^{*}]=0$. It follows that the field $[{\xi }^{*},X_{P^{-}}^{*}]$ is ${\Phi }_1$-related with $[{\xi }_{*},X_Z^{*}]=0$ and consequently is parallel to $V_1$. Note that the field $U=[{\xi }^{*},X_{P^{-}}^{*}]$ is $SO(1,2)$ invariant, i.e., $[V_i,U]=0$, which is a consequence of $[V_i,{\xi }^{*}]=0$ and $[V_i,X_{P^{-}}^{*}]=0$. A field that is $SO(1,2)$ invariant and parallel to $V_1$ must to be 0. Consequently, $[{\xi }^{*},X_{P^{-}}^{*}]=0$. Note that $L_{{\xi }^{*}}\theta \left(X_{P^{-}}^{*}\right)=-\theta \left([{\xi }^{*},X_{P^{-}}^{*}]\right)=0$. It follows that the lift is a Killing vector field on every fiber $SO(1,2)$ with a bi-invariant metric. Consequently, it is a right-invariant vector field on $SO(1,2)$. It follows that it is a field of constant length on every fiber and ${\nabla }_{{\xi }_{*}}^{SO(1,2)}{\xi }_{*}=0$ on every fiber. Since the fibers are totally geodesic, it follows that ${\nabla }_{{\xi }_{*}}{\xi }_{*}=0$ everywhere and ${\xi }_{*}$ has a constant length. Thus, our result is a consequence of Theorem 1. Let us note that to determine $\eta$, we have to find the length of the lift ${\xi }^{*}$ at one point. □

We say that two $ϵ$-Kähler structures $I,J$ are essentially different if their Kähler forms ${\Omega }_I,{\Omega }_J$ satisfy the following condition: ${\Omega }_I\ne \pm {\Omega }_J$. (ex. 0-Kähler and 1-Kähler structures, or two null Kähler structures $I,J$, such that ${\Omega }_I\ne \pm {\Omega }_J$.)

Corollary 1. 

Let us assume that a neutral four-manifold $(M,g)$ admits two essentially different ϵ-Kähler structures $I,J$. Then, $(M,g)$ is a neutral hyperKähler surface.

Proof. 

The structures $I,J$ determine two vertical, linearly independent, $SO(1,2)$-invariant vector Killing fields ${\xi }_I,{\xi }_J$ on $P^{-}$. The field $\eta =[{\xi }_I,{\xi }_J]$ is the third such field, and consequently the algebra of vertical Killing vector field is three-dimensional. This means that $(M,g)$ is neutral hyper-Kähler. □