[labelstyle=]
para-Kähler-Einstein 4-manifolds and
non-integrable twistor distributions
Abstract.
We study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.
Key words and phrases:
Para-complex structure, para-Kähler structure, Einstein metric, Cartan geometry, Cartan reduction, Petrov type, (2,3,5)-distribution, conformal structure1991 Mathematics Subject Classification:
53C10, (53C15, 58A30, 58A15, 53C29)1. Introduction and main results
The main purpose of this article is to give a detailed treatment of para-Kähler-Einstein structures in dimension 4 and examine their relation to maximally non-integrable distributions of rank 2 in dimension 5, referred to as (2,3,5)-distributions.
Our initial motivation for this article is twofold. Firstly, it is an extension of the observation made in [BHLN18] where, inspired by the rolling problem of Riemannian surfaces [BH93], a notion of projective rolling was defined which gives rise to (2,3,5)-distributions. Consequently, it was observed that the (2,3,5)-distributions whose algebra of infinitesimal symmetries is maximal i.e. the split real form of can be obtained from such construction with a direct link to the homogeneous para-Kähler-Einstein metric on referred to as the dancing metric. We point out that para-Hermitian structures and their variations naturally appear in various geometric settings since they were first defined in [Lib52]. The reader may consult [CFG96] for a survey.
Our second motivation comes from the twistorial construction of rank 2 distributions from conformal structures of split signature in dimension four, referred to as twistor distributions, described in [AN14]. In open subsets where the self-dual Weyl curvature is non-zero such distributions, which are naturally induced on the -bundle of self-dual null planes, are (2,3,5). We will discuss that, in general, the fundamental invariant of twistor distributions, referred to as the Cartan quartic, depend on the fourth jet of the components of the Weyl curvature of the conformal structure. This poses a basic question: whether the Cartan quartic of twistor distributions can have any root type? In this article we answer this question affirmatively. As a by-product of our construction one naturally obtains explicit examples of para-Sasaki-Einstein metrics.
Before proceeding further, a brief definition of the geometric structures appearing in this article is in order. As will be defined in § 2.1, an almost para-complex structure on a manifold, is defined as an endomorphism satisfying whose -eigenspaces have rank 2. As a result, unlike almost-complex structures, the eigenspaces of split each tangent space of into two transversal distributions. The integrability of these distributions induces a para-complex structure on which results in two transversal foliations of Similarly, an almost-para-Hermitian, para-Hermitian and para-Kähler structure can be defined in terms of and a pseudo-Riemannian metric of split signature satisfying certain compatibility condition, as explained in § 2.2.1 and § 2.3.1.
The first objective of this article, presented in § 2, is to give a unifying treatment of almost para-Hermitian and para-Kähler structures in dimension four via Cartan’s method of equivalence and analyze the curvature decompositions in each case. To our knowledge such presentation has been absent in the literature.
Our main topic of interest, treated in § 3, is para-Kähler-Einstein (pKE) metrics, defined as para-Kähler structures for which the metric is Einstein i.e. its trace-free Ricci tensor is zero. Because of our interest in non-integrable twistor distributions, we restrict our considerations to only those pKE metrics with non-zero scalar curvature. In § 3.1 it is shown that such metrics define a Cartan geometry of type In § 3.2 we investigate five classes of pKE metrics for which the root type, or the Petrov type, of the anti-self-dual Weyl tensor (see § 2.2.4), is non-generic and real. Let us mention briefly that by root type, or the Petrov type, of we refer to the multiplicity pattern of the roots of the 4th order polynomial obtained from the representation of as a binary quartic at each point. For these five classes of pKE metrics we carry out the Cartan reduction procedure case by case. This enables us to find all homogeneous models of pKE metrics in dimension 4. Moreover, we find explicit examples of pKE metrics with non-zero scalar curvature in every special real root type of . In particular, Theorem 3.12 gives explicit examples of pKE structures of Petrov type , Theorem 3.14 gives examples for Petrov type , and Theorem 3.16 gives examples of Petrov type . The pKE structures of real Petrov type are described in Theorem 3.10. This Petrov type is particularly interesting, since we found all possible such pKE metrics with non-zero scalar curvature. The method we have used to derive these examples is known as Cartan’s reduction method which is followed by integrating the reduced structure equations. Moreover, Cartan’s reduction method combined with Cartan-Kähler analysis is used to obtain the local generality of pKE metrics for each Petrov type in the real analytic case, as presented in Table 1. We point out that § 3.2 serves as an example of how effective Cartan’s method of equivalence is in studying geometric structures.
In § 4 we discuss the relationship between pKE metrics and (2,3,5)-distributions. More precisely, in § 4.1 we briefly review the basic facts about (2,3,5)-distributions, including the Cartan connection, the Cartan quartic and associated conformal structure of split signature. In § 4.2.1, we recall the fact that for any pKE structure, and more generally, any indefinite conformal structure in dimension four, a rank 2 distribution, referred to as a twistor distribution, is naturally induced on its 5-dimensional space of null self-dual planes, and that another rank 2 twistor distribution is induced on the space of null anti-self-dual planes, this was originally observed in [AN14]. Moreover, the distribution induced on or is a (2,3,5) distribution at a point if and only if the self-dual Weyl tensor, or the anti-self-dual Weyl tensor, is non-zero at that point, respectively. After necessary coframe adaptations, which are performed at the beginning of § 4.2.1, we state our prime result on the surprising proportionality of the two quartics, the Cartan quartic of the (2,3,5) twistor distribution on , and the anti-self-dual Weyl quartic of the pKE metric, as in the following theorem.
Theorem 4.9.
Given a pKE metric for which the scalar curvature is non-zero, the Cartan quartic for the non-integrable twistor distribution on is a non-zero multiple of the quartic representation of the anti-self-dual Weyl curvature . In particular, the Cartan quartic of the twistor distribution and the anti-self-dual Weyl curvature of the metric have the same root type.
In Remark 4.10 we explain the natural identification that underlies this theorem. As a result of Theorem 4.9 and our examples of pKE metrics of non-generic Petrov types in § 3, we obtain a large class of explicit examples of twistor distributions for each special real algebraic type of the Cartan quartic. Additionally, one obtains that the associated signature conformal structure on an open subset of has an Einstein representative whose conformal holonomy is a subgroup of this was also observed in [SW17]. The main purpose of § 4.3 is to show why the coincidence of the root types of the quartics, explained in Theorem 4.9, is remarkable. This is done by obtaining the Cartan quartic for the twistor distribution on . It is shown that the coefficients of the Cartan quartic on depend on the 4th jet of and there is no obvious relation between the algebraic types of these quartics.
Finally, we also mention that, starting in § 3.1, a number of Cartan geometries are introduced which are naturally associated with pKE metrics in dimension 4. Since these Cartan geometries live on principal bundles over 4-dimensional manifolds with split-signature conformal metrics, one can study the vacuum Yang-Mills equations for the corresponding Cartan connections. As far as we know few papers are concerned with such studies. Here an honorable exception is a paper by S. Merkulov [Mer84], who established in 1984 that the vacuum Yang-Mills equations for the Cartan normal conformal connection of a 4-dimensional conformal structure , are equivalent to the vanishing of its Bach tensor. Also in this vein is the work [NS03], where in particular, the vacuum Yang-Mills equations for Cartan connections associated with 3-dimensional parabolic geometries of type and were considered, in which denotes the Borel subgroup. It turns out that Cartan geometries of various type appear in the process of Cartan reduction performed on a given Cartan geometry. It is also clear that reduction of a Cartan geometry results in a principal bundle over the same base. Therefore, if the vacuum Yang-Mills equations can be defined for a certain Cartan geometry, they can also be defined for all the Cartan geometries obtained from the reduction procedure. We take this approach in § 3.1 and in the subsequent sections, where we reduce the initial -type Cartan geometry to Cartan geometries of various types depending on the Petrov types. In particular, we find in Proposition 3.3, as a consequence of Theorem 3.2, that all pKE 4-dimensional structures for which the Einstein constant is equal to -3 satisfy vacuum Yang-Mills equations for the -valued Cartan connection of the associated -type Cartan geometry. Similarly, in Theorem 3.12 we give examples of pKE metrics satisfying the vacuum Yang-Mills equations for the Cartan connection of a Cartan geometry of type , which can be obtained for pKE metrics of real special Petrov type.
The EDS calculations mentioned in the text are carried out using the Cartan package in Maple written by Jeanne Clelland and an exterior differential package for Mathematica written by Sotirios Bonanos.
Conventions
In this article we will be working in the real smooth category. Since our results are of local nature, the manifolds can be taken to be the maximal open sets over which the assumptions made in each statement is valid.
The manifold is always 4 dimensional equipped with a metric of split signature. The 1-forms represent a coframe on with respect to which where it is understood that the terms such as represent symmetric tensor product of the 1-forms and i.e.
Given an -dimensional manifold, equipped with a coframe the corresponding set of frame will be expressed as i.e. Given a function the so-called coframe derivatives of denoted by are defined as
| (1.1) |
When a set of 1-forms on is introduced as it represents the ideal that is algebraically generated by the 1-forms and is called a Pfaffian system. A Pfaffian system is called integrable if it satisfies the Frobenius condition, where is the exterior derivative. The integral manifolds of an integrable Pfaffian system are called its leaves. Locally, around a generic point the leaves of an integrable Pfaffian system induce a smooth foliation which enables one to consider the quotient space of its leaves, referred to as the leaf space of Since our treatment is local, we can always work in sufficiently small neighborhoods which allows one to define the leaf space of an integrable Pfaffian system.
Acknowledgments
This project benefited from the Simons Semester Symmetry and Geometric Structures at IMPAN in 2018. GB was supported by Conacyt grant A1-S-45886. OM would like to thank Kael Dixon and Niky Kamran for their interest and helpful conversations. OM acknowledges the partial support of the grant GA19-06357S from the Czech Science Foundation, GAČR. The research leading to these results has also received funding from the Norwegian Financial Mechanism 2014-2021 with project registration number 2019/34/H/ST1/00636.
2. Almost para-Hermitian and para-Kähler structures
The goal of this section is to fix notation, give necessary definitions and recall some facts that will be needed in subsequent sections. More precisely, in § 2.1 we recall some basic facts about pseudo-Riemannian metrics in dimension four. The notion of almost para-Hermitian structure and the decomposition of its curvature into irreducible components with respect to the action of its structure group is defined in § 2.2. Furthermore, in § 2.2 we define the so-called Petrov type of the Weyl curvature. In § 2.3 we define para-Kähler structures in terms of additional integrability and compatibility conditions imposed on an almost para-Hermitian structure. We derive their structure equations, curvature decomposition and give a local coordinate expression in terms of a potential function. We end the section by giving examples of pKE structures in terms of potential functions which, as will be shown in § 3.2, correspond to the only homogeneous models with non-zero scalar curvature.
2.1. Rudiments of indefinite pseudo-Riemannian metrics in dimension 4
In this section we briefly recall the decomposition of the space of 2-forms into self-dual and anti-self-dual 2-forms using the Hodge star operator. As a result, two 5-dimensional circle bundles of self-dual null planes and anti-self-dual null planes is obtained for any indefinite pseudo-Riemannian metric in dimension four. Subsequently, we recall the structure equations of split signature metrics and their curvature decomposition.
2.1.1. The Hodge star operator.
From now on let be a 4-dimensional real oriented manifold equipped with a split signature metric . Locally, we can always find a real coframe in which the metric takes the form:
| (2.1) |
We denote by the dual frame. The coframe is null i.e. the metric has constant coefficients , with as the only non-vanishing ones, which implies nullity, for . Note that in this paper the coframe/frame on will be denoted by two notations or , and by or , respectively. The reason for this redundancy will be made clear in the next sections, when sometimes one, and sometimes the other notation will be more convenient.
We assume that the coframe is positively oriented, therefore, the volume form on can be expressed as
This enables one to define the Hodge star operator i.e. a linear map such that
Here is a 1-form associated to a vector field such that for all vector fields on
It is easy to see that
and therefore the operator splits into the direct sum
of its eigenspaces , corresponding to its respective eigenvalues. In what follows we will frequently use the basis of the -eignespaces of , expressed in terms of the null coframe as
| (2.2a) | ||||
| (2.2b) | ||||
As a result, in terms of a basis for the self-dual and anti-self-dual bivectors in can be expressed as
At each point of the bivector defines a null plane . Recall that a plane is null if for all . At every point we then have the space of all null planes at .
Let us consider a pair where and The group acts naturally on the space via:
This action decomposes into two orbits
Each of these orbits is diffeomorphic to a circle . Take and assume Note that its defining bivector is self-dual. Since the action of does not change self-duality of null planes, the orbit is called the space of self-dual null planes at Consequently the orbit is comprised of null planes defined by anti-self-dual bivectors and is therefore called the space of anti-self-dual null planes at
More explicitly, at the set of self-dual and anti-self-dual null planes can be parametrized, respectively, by in the following way
| (2.3) |
This parametrization will be used in § 4. The bundles equipped with the projections
| (2.4) |
where at are referred to by various names, including circle twistor bundles ([AN14]) or bundles of real -planes and -planes [AG96]. In what follows we will frequently refer to the circle bundles and as the bundle of self-dual and anti-self-dual null planes, respectively.
2.1.2. Structure equations.
The null coframe uniquely defines the Levi-Civita connection 1-forms via the first structure equations:
As a result, the Riemann curvature of the metric , given by the -valued 2-form is defined via the second structure equations
| (2.5) |
Via the action of the Riemann curvature decomposes into the components known as the traceless Ricci tensor, Weyl curvature and the scalar curvature. The Ricci tensor is defined as and the scalar curvature is , where The trace-free part of the Ricci tensor is defined as . Defining the Schouten tensor as
the Weyl tensor is expressed as
| (2.6) |
Solving the metricity condition for the first structure equations, it follows that the connection 1-forms can be expressed as
| (2.7) |
Consequently, the torsion-free condition yields
| (2.8) | ||||
Passing to the second structure equations, one notes that due to the symmetries of the Riemann tensor, , setting , one obtains a linear map given by
Since , the matrix form of this map can be expressed as
| (2.9) |
Here and are traceless matrices, and is a matrix related to the matrix of trace-free Ricci tensor, via , where . The matrices are symmetric and their components will be denoted by and respectively. Moreover, let us denote the the 9-components of by It follows that the scalar curvature can be written as , as given in (2.21). Since the Ricci tensor and the Schouten tensor are linearly related, we will be using the Schouten tensor in the sequel. As a result, the second structure equations (2.5) read
| (2.10) | ||||
with analogous equations for the ‘unprimed’ objects:
| (2.11) | ||||
Here we used the respective basis of , as defined in (2.2).
Remark 2.1.
The usual way of employing the system of equations (2.8), (2.10)-(2.11), is to think about as a given coframe on , and to use the equations (2.8), (2.10)-(2.11) to uniquely determine the Levi-Civita connection forms , and consequently the curvature of , in terms of this chosen coframe. Alternatively, in the language of -structures, one observes that ’s are ambiguous up to an action of since they were chosen so that (2.1) is satisfied. One says that is the structure group of the pseudo-Riemannian structure. As a result, one can define a principal -bundle , as the bundle of all null coframes with respect to which (2.1) holds. In this language the ’s give rise to a lifted null coframe at each point of and the ’s mimic the Maurer-Cartan forms of ; they are uniquely defined on as a result of the torsion-free condition. Moreover, these 1-forms, together with s, form a basis of 1-forms at every point of . Hence, one obtains a unique coframe at each point of , consisting of 1-forms , which is transformed equivariantly in each fiber of and satisfy the equations (2.8), (2.10)-(2.11) everywhere on . We refer to [Gar89, Olv95] for an overview of this exterior differential system (EDS) viewpoint.
2.2. Almost para-Hermitian metrics
In this section we define almost para-complex structures and almost para-Hermitian metrics. We obtain the structure equations and curvature decomposition. Using the curvature decomposition, we recall the well-known Petrov classification of such structures.
2.2.1. Definitions
An almost para-Hermitian structure on a 4-dimensional manifold with a metric of signature is defined in terms of an endomorphism
such that
whose -eigenvalues have rank 2 and, additionally, satisfies the compatibility condition
An almost para-Hermitian structure distinguishes a pair of rank 2 distributions and on defined as the -eigenspaces of i.e.
| (2.12) |
It follows that
Moreover and are null with respect to and must belong to the same orbit in the space of all null planes.
An almost para-Hermitian structure is called half-para-Hermitian if precisely one of or are integrable i.e. either or If and are both integrable i.e.
then the almost-para-Hermitian structure is called para-Hermitian.
An almost para-Hermitian structure defines a para-Kähler 2-form
| (2.13) |
The fact that is skew symmetric, , follows from the algebraic properties of .
An almost para-Hermitian structure is called almost para-Kähler if and only if the 2-form is closed i.e.
An almost para-Hermitian structure is para-Kähler if it is para-Hermitian and almost para-Kähler i.e. and are integrable and is closed.
2.2.2. Almost para-Hermitian structure in an adapted frame.
A coframe on a 4-dimensional manifold is adapted to an almost para-Hermitian structure if and only if
| (2.14) | ||||
It follows that in such adapted coframes
| (2.15) |
At every point of a 4-dimensional almost para-Hermitian manifold the stabilizer of the pair , i.e.
satisfies
Expressing in the coframe as in (2.14), provides the 4-dimensional reducible representation
of given by
| (2.16) |
As a result the geometry arising from the pair reduces the structure group of from to via representation The irreducible decomposition of as a -module is . It reflects the splitting of , into .
Proposition 2.2.
Every almost-para-Hermitian structure on a 4-dimensional manifold locally admits an adapted coframe. If is a coframe adapted to then the most general adapted coframe is given by
| (2.17) |
where the matrices are as in (2.16).
2.2.3. invariant curvature decomposition
Any coframe adapted to is in particular a null coframe, as in (2.1). Thus to analyze the properties of we can use the structure equations (2.8), (2.10)-(2.11). The stabilizer of the pair is therefore the structure group of the almost para-Hermitian structure . It acts, via the representation , on any adapted coframe as in (2.17). The induced transformation of the Levi-Civita connection (2.7) and its curvature is given by
| (2.18a) | |||
| (2.18b) | |||
| (2.18c) | |||
The transformations (2.18c) gives the action of on the 20-dimensional vector space of the curvature tensors . Using this action one can decompose the curvature tensor into its indecomposable components. First we define 10 vector spaces defined in terms of the curvature components , , as
| (2.19) | ||||
Here the (spinorial) indices , and the equations mean that is totally symmetric in indices . The notation for the spaces and is such that the upper index indicates the dimension of each space, and the lower index enumerates spaces of the same dimension. In particular the spaces and have dimensions 3 as spaces of symmetric matrices, has dimension 3 as the space of traceless matrices, and has dimension 5 as the space of symmetric tensors of degree 4 in dimension 2.
Proposition 2.3.
The invariant decomposition of the 20-dimensional curvature space, of an almost para-Hermitian structure in dimension 4 is
| (2.20) | ||||
Proof.
This decompositions can be obtained similar to the decomposition of into , and via the invariant decomposition of . In this case one decomposes using the invariant decomposition of the tangent space
which is possible for any almost para-Hermitian manifold . The associated decomposition of the cotangent bundle is given by
where
As a result, is decomposed into
It turns out that and are 1-dimensional, and has dimension 4. Choosing an adapted coframe we can write a basis for these spaces in terms of the self-dual and anti-self-dual 2-forms, in (2.2), as follows.
This gives a natural decomposition of into 1-dimensional invariant subspaces
Using this we can further decompose the map from (2.9) as
| (2.21) |
Comparing this with the decomposition (2.9) one obtains
-
•
gets decomposed into five 1-dimensional invariant subspaces denoted by which correspond to the components in (2.21) respectively.
-
•
is decomposed into three invariant subspaces, which correspond to the rows , , and in (2.21) respectively.
-
•
remains indecomposable with its 5-dimensional representation whose components are .
-
•
The Ricci scalar is proportional to the trace of and gives the 1-dimensional invariant subspace .
As a result one obtains the decompositions (2.20). ∎
It is straightforward to find the explicit action of the group on the indecomposable components of the curvature in (2.19).
Proposition 2.4.
The curvature components , , and are ‘tensorial’ with respect to the action of i.e. for given by (2.16) if an adapted coframe is transformed by
then the transformation law for the curvature components , and in (2.19) is
| (2.22a) | |||
| (2.22b) | |||
| (2.22c) | |||
| (2.22d) | |||
The curvature scalars , , , , , , are weighted scalars and transform to
| (2.23) | ||||||
Corollary 2.5.
Every almost para-Hermitian structure in dimension 4 possesses two scalar invariants which are the scalar curvature of the metric given by , and arising from the self-dual Weyl tensor of the metric. Moreover the vanishing of each of the densities, , as well as each of the tensors, , and is an invariant property of almost para-Hermitian structures.
2.2.4. Cartan-Penrose-Petrov classification of the Weyl tensor
One of the basic pointwise invariants of 4-dimensional metrics of split signature is the so-called Petrov type of its self-dual and anti-self-dual Weyl curvatures. To define it, note that the transformation law (2.22d) shows the action of the structure group on the anti-self-dual Weyl tensors, as an 5-dimensional representation. This representation is isomorphic with the standard representation of on i.e. the degree 4 homogeneous polynomials in two variables. Using (2.19), the quartic polynomial is given by
where It turns out that can serve as a homogeneous coordinate for the circle bundle of anti-self-dual planes, More precisely, using the Weyl curvature (2.6), define which can be used to define the multilinear map
Restricting to anti-self-dual null planes, , as in (2.3), one can define the quartic polynomial
| (2.24) | ||||
where
which establishes the relation between the parameters. The quartic (2.24) is a representation of the anti-self-dual Weyl curvature of the metric
Similarly, restricting to self-dual null planes, , and using the affine parametrization (2.3), one can define the quartic polynomial
| (2.25) | ||||
where
The quartic is a representation of the self-dual Weyl curvature of whose coefficients transform according to (2.23).
The Petrov type at each point is the root type of the quartics and at that point, since multiplicity pattern of the roots is invariant under the induced action of the structure group . Note that since the coefficients of the quartics are real and transform under the action of , the root type is closed under complex conjugation. As a result, there are 10 root types for each of the quartics and Following the tradition in General Relativity, where the metric has Lorentzian signature, root types are grouped into the six Petrov types, denoted by , , , , and . In the case of metrics of split signature, due to different reality conditions, one obtains a finer classification of Petrov types given by
-
(1)
type : 4 real simple roots.
-
(2)
type : 2 real simple roots and 2 complex conjugate roots.
-
(3)
type : 2 pairs of complex conjugate roots..
-
(4)
type : 1 double real root, 2 simple real roots.
-
(5)
type : 1 double real root, 2 complex conjugate roots.
-
(6)
type : 1 triple real root and 1 simple real root.
-
(7)
type : 2 double real roots.
-
(8)
type : 2 double complex conjugate roots.
-
(9)
type : 1 quadruple real root.
-
(10)
type : when all the coefficients of the quartic are zero.
The letter stands for general type since, generically, the Petrov type of a quartic is . If the quartic is non-zero, then the 9 root types are listed in Figure 1, which shows the self-conjugate pattern of roots in each type. The horizontal line represents the real line and conjugation of roots is given by reflection with respect to the horizontal line.
In what follows the root types will be referred to as the special real Petrov types. It is clear from our discussion that since the parameters and in the quartics and parametrize and respectively, a choice of real root for these quartics determine a choice of an anti-self-dual and self-dual null plane. This enables one to consider anti-self-dual or self-dual null planes that correspond to a real root of the quartics or
2.3. Para-Kähler (pK) metrics
In this section the para-Kähler condition is used to reduce the structure equations of an almost para-Hermitian structure. After deriving their structure equations and curvature decomposition, we show that para-Kähler structures can be described in terms of a potential function, using which we give two examples of para-Kähler-Einstein metrics. These two examples turn out to be homogeneous as will be explained in § 3.2.
2.3.1. PK structures in an adapted coframe.
A special feature of every almost para-Hermitian geometry is that in addition to the (weighted) tensorial invariants arising from the curvature of the Levi-Civita connection, it has invariants of lower order referred to as the intrinsic torsion. These are defined in terms of the (Grey-Harvella type) decomposition of the covariant derivative of the 2-form (2.13) with respect to . Two of these (relative) invariants are of particular interest in our setting. We will express them in terms of the Levi-Civita connection 1-forms Using (2.18b), one obtains that the transformation of the connection 1-forms and does not involve the inhomogeneous terms , which leads to the following proposition.
Proposition 2.6.
Under the gauge transformation (2.18) of adapted coframes for an almost para-Hermitian structure, the connection 1-forms and transform as
where . As a result, the vanishing of each of the connection 1-forms and is an invariant property of an almost para-Hermitian structure.
We have the following proposition.
Proposition 2.7.
An almost para-Hermitian structure is para-Kähler if and only if
in one (and therefore any) adapted coframe. As a result, the Levi-Civita connection form of is reduced to
in any adapted coframe.
Proof.
By Frobenius theorem the integrability of and in an adapted coframe is equivalent to and respectively. Using the first structure equations (2.8), it follows that the simultaneous integrability of and implies
| (2.26) | ||||
On the other hand, the almost Kähler condition , when written in an adapted coframe reads
Using the first structure equations (2.8) it follows that this condition is equivalent to
| (2.27) |
Proposition 2.7 leads to the following “para” analogue of the well-known fact that the holonomy of Riemannian 4-manifolds which are Kähler is a subgroup of .
Corollary 2.8.
For any 4-dimensional para-Kähler structure the pseudo-Riemannian holonomy of the metric is reduced from to via the representation in (2.17). This holonomy reduction is equivalent to the property that is parallel with respect to the Levi-Civita connection of
Remark 2.9.
The corollary above is a consequence of the so-called holonomy principle in pseudo-Riemannian geometry which establishes a one to one correspondence between the space of parallel sections of tensor bundles and the invariant vectors in under the action of the holonomy group at each point . We refer to [BBI97] for further discussion of the holonomy group of pseudo-Riemannian metrics of split signature.
Let us also point out that in the spirit of Remark 2.1, Proposition 2.7 implies that the bundle of adapted null frames for para-Kähler structures is a principal -bundle obtained from reducing the -bundle with the property that the reduced first order structure equations given by (2.8), (2.10)-(2.11) have no intrinsic torsion.
2.3.2. Curvature of pK geometry and pK-Einstein (pKE) condition
In this section we discuss the curvature of para-Kähler structures.
Proposition 2.10.
The curvature of every 4-dimensional para-Kähler structure can be decomposed as
which, compared to (2.19), means .
Proof.
The proof follows from a simple substitution of and into the last two of the second structure equations (2.10). ∎
Remark 2.11.
Recall that a 4-dimensional pseudo-Riemannian manifold where has split signature is called Einstein if and only if its traceless Ricci curvature vanishes, i.e. in (2.9). Therefore, one obtains the following.
Corollary 2.12.
The curvature of a 4-dimensional para-Kähler-Einstein structure decomposes to
When written in an adapted coframe it reads
| (2.28) |
Remark 2.13.
It follows from § 2.3.2 that for para-Kähler-Einstein manifolds the two constant curvature components are related by From now on, we restrict ourselves to para-Kähler-Einstein 4-manifolds with non-vanishing i.e. we always assume
Moreover, following the discussion in § 2.2.4 on the Petrov type of the anti-self-dual Weyl curvature, one obtains that the Petrov type of the quartic representation of as the self-dual Weyl curvature of the metric is if and if
2.3.3. Para-Kähler structure in a coordinate system
One of the features of Kähler metrics is that they can be locally expressed in terms of a function, called the Kähler potential. An analogous feature for the para-Kähler structures in 4 dimensions is described in the following two propositions.
Proposition 2.14.
Let be an open set of , and let be Cartesian coordinates in . Consider a real-valued sufficiently differentiable function on such that
Define
Then the pair defines a para-Kähler structure on with .
The para-Kähler structure is Einstein i.e. if and only if the potential function satisfies
| (2.29) |
for a real number and real-valued functions ,
Proof.
In the adapted coframe
the 1-forms , constituting the part of the Levi-Civita connection, read
It is straightforward to check that , and , as it should be for the Levi-Civita connection in an adapted coframe of a para-Kähler structure.
For the calculation of the Ricci tensor it is more convenient to work in the coordinate frame rather than in the adapted frame . Thus, we need to display the Levi-Civita connection 1-forms in the coordinate frame as well. Let us use the following notation for the coordinates
It follows that
and the connection 1-forms and are given by (A.1) in the Appendix.
Using the expressions for the Levi-Civita connection in (A.1) the curvature can be calculated easily and the Ricci tensor satisfies
and
Since in the notation the metric reads as
the Einstein equations are
which after integration give
∎
There is a converse to this proposition:
Proposition 2.15.
Every para-Kähler structure in dimension four is locally expressible in terms of a para-Kähler potential function as in Proposition 2.14.
Proof.
The integrability of the distributions and in a para-Kähler structure implies that in some neighbourhood there exists a coordinate system such that
for some functions defined in . Since the coframe is defined up to the action (2.17), we can use this transformation to bring the coframe into the form
with new functions on such that . In this new adapted frame we have , . Inserting this into (2.8) with we get
which implies
for some unknown functions on . Inserting this back into the last two of the structure equations (2.8) gives
This in particular means that
As a result, locally, there exist functions and on such that
Thus, one obtains
Since the 2-form is given by
the para-Kähler condition implies
This means that for some function of variables only. But since in the coframe functions and appear only in terms of their and derivatives, they can be chosen so that Hence there exists a differentiable function on such that
Thus, the adapted coframe can be expressed as
Expressing , and in terms of the adapted coframe as in (2.14)-(2.15) gives the Proposition. ∎
2.3.4. Homogeneous models
Two particular solutions of the Einstein condition (2.29) are given by the potentials
-
i)
,
-
ii)
,
where .
Both potentials are solutions of (2.29) with and for , and for . Let the para-Kähler structure correspond to where Then, in an open set of parametrized by one finds
Straightforward computation gives
and
The potential corresponds to the homogeneous para-Kähler-Einstein structure referred to as the dancing metric in [BHLN18] which is the unique homogeneous model that is self-dual, i.e. and not anti-self-dual for which The potential corresponds to the only other homogeneous para-Kähler-Einstein structure. It has the property that the Petrov type of is D. A derivation of these potential functions is outlined in § 3.2.2 and § 3.2.6. Finding explicit examples of pKE structures in terms of potential functions satisfying the PDE (2.29) is not an easy task. In the next section we use an alternative technique to give more explicit examples of pKE structures.
3. Para-Kähler-Einstein (pKE) metrics in dimension 4
This section is the heart of the article, in which we describe pKE structures as Cartan geometries, give an in-depth study when the Petrov type is real and special and provide explicit examples. To be more specific, in § 3.1 pKE structures are interpreted as Cartan geometries of type . If the Einstein constant is then they satisfy the Yang-Mills equations for the associated -valued Cartan connection. In § 3.2 we focus on pKE structures for which has special real Petrov type and give examples of each type. In particular, we find all homogeneous models, give a local normal form for all real Petrov type pKE metrics and present examples of real Petrov type II that satisfy Yang-Mills equations. Moreover, we use Cartan-Kähler machinery to find the local generality of all Petrov types assuming analyticity.
3.1. Cartan geometries of type
In order to view pKE structures as a Cartan geometry, let us specialize the EDS (2.8), (2.10)-(2.11) to the case of para-Kähler-Einstein metrics. We have
| (3.1) | ||||
where we have used from Proposition 2.7
As discussed in Remark 2.9 and 2.1, the EDS (3.1) can be regarded as the structure equations for the coframe on the principal -bundle which is the 8-dimensional bundle of adapted null frames for para-Kähler-Einstein structures. In fact, one can show that para-Kähler-Einstein structures correspond to Cartan geometries of type . First let us define a Cartan geometry [Sha97, ČS09].
Definition 3.1.
A Cartan geometry of type is a principal -bundle equipped with a -valued 1-form which is a Cartan connection, i.e.,
-
(1)
is linear isomorphism for all
-
(2)
is -equivariant, i.e., where denotes the right action by
-
(3)
for every fundamental vector field of
The curvature of the Cartan connection is given by which is horizontal and defines the curvature function
Let us now consider the -valued 1-form
| (3.2) |
where
| (3.3) |
Using , the structure equations (3.1) can be expressed as
| (3.4) |
where, using the 2-forms ’s and ’s in (2.2), one has
| (3.5) | ||||
One can interpret the 1-form as an -valued Cartan connection on the principal -bundle of null frames adapted to a pKE structure. As a result, one obtains the following theorem.
Theorem 3.2.
Every pKE 4-manifold defines a Cartan geometry of type for which the structure bundle
is the bundle of adapted null frames. The curvature vanishes, i.e. the Cartan geometry is flat, if and only if the Einstein constant is -3 and the anti-self-dual Weyl tensor vanishes, i.e.
The flat model, i.e. is locally equivalent to the para-Kähler-Einstein structure induced by the dancing metric discussed in § 2.3.4.
Recall that the exterior derivative of (3.4) gives the Bianchi identity
| (3.6) |
As a result of the theorem above one obtains the following.
Proposition 3.3.
A 4-dimensional para-Kähler-Einstein structure satisfies the Cartan connection Yang-Mills equations, , if and only if
Proof.
By (3.5) and the definition of self-dual and anti-self-dual null planes in (2.2), it follows that
Since the curvature of a Cartan connection is always horizontal (see Definition 3.1 and (3.5)), one can apply the Hodge star to defined on . Applying the Bianchi identity (3.6), one obtains Alternatively, by taking a section computing the curvature and applying the Hodge star one can verify the claim.
Remark 3.4.
Note that pKE structures can also be associated to Cartan geometries of type whose flat model satisfies and . This point of view is however not desirable for the purpose of this article, since we always assume
3.2. Cartan reduction: homogeneous models, examples and local generality
In this section we carry out the Cartan reduction procedure for pKE metrics whose anti-self-dual Weyl curvature has non-generic real Petrov type. Our reduction will not be exhaustive and will omit Petrov type . We will describe the reduction procedure for Petrov type and in detail, give a complete local normal form for type pKE metrics, and use the same method to find examples of Petrov types and . The reduction allows us to find all homogeneous models and use the Cartan-Kähler theory to find the local generality of all Petrov types assuming analyticity.
3.2.1. Reduction for special real Petrov types
Recall from (2.24) the quartic
| (3.7) |
where is the affine parameter for the anti-self-dual null planes in (2.3) for a given choice of adapted coframe. Taking a different choice of coframe, by the action of the structure group, as in (2.17), one obtains a quartic whose coefficients, can be expressed in terms of ’s and the elements of the matrix For instance, one obtains
| (3.8) |
Remark 3.5.
It is clear from (3.8) that if then with respect to the coframe obtained from the action of (2.16) where and , the root would be translated to zero, i.e. in this choice of coframe.
If has a repeated root then by our discussion above, there is a coframe adaptation with respect to which the double root is translated to zero. If the root has multiplicity , in the newly adapted coframe we have
| (3.9) |
Using the group action on ’s or, equivalently the Bianchi identities (A.2), it follows that the bundle of adapted coframes that preserves the condition (3.9) gives rise to a 7-dimensional principal bundle More precisely, the new adapted coframes were determined by a choice of in (2.16) as a result of which the structure group is reduced to defined as
| (3.10) |
Using the gauge transformations (2.18b) arising from the structure group for adapted coframes with respect to which (3.9) holds one obtains that the transformation of the connection 1-form does not involve the inhomogeneous terms and therefore the following proposition holds.
Proposition 3.6.
Given a pKE metric, if the anti-self-dual Weyl curvature has special real Petrov type, i.e. it has a repeated root whose multiplicity is at least 2, then the bundle of adapted coframes which preserves the condition (3.9) is given by a principal -bundle Adapted coframes arising as sections of satisfy (3.1) where
| (3.11) |
for some functions and on .
Proof.
Remark 3.7.
A second coframe adaptation will result in the following proposition.
Proposition 3.8.
Every 4-dimensional pKE structure whose anti-self-dual Weyl tensor is of special real Petrov type defines a Cartan geometry of type where is the maximal torus in The structure equations are given by (3.1) where
| (3.12) | ||||
for some functions on The -valued Cartan connection can be represented as
| (3.13) |
The flatness condition for the resulting Cartan geometry, is equivalent to , and and implies that has Petrov type .
Proof.
Assuming that the Petrov type is or , the adaptation (3.9) reads
| (3.14) |
If the conditions (3.14) are to be preserved, by Proposition 3.6, one has (3.11) and, using the Bianchi identities (A.2), one obtains the differential relations
| (3.15) | ||||
for some functions on These relations are obtained by inserting (3.11) in the structure equations (3.1) and requiring that the exterior derivative of the right hand side of the equations are zero.
Let the quantities and be the quantities appearing in the structure equations for two choices of such adapted coframe related via . Using the structure group as in (3.10) and its induced gauge transformation (2.18), one obtains
| (3.16) | ||||||
As was mentioned in Remark 3.5, the infinitesimal version of the transformations above are given by the Bianchi identities (3.15).
Consequently, using the transformation law for given in (3.16), we can further restrict to the bundle of adapted coframes with respect to which
| (3.17) |
This can be seen explicitly from (3.16) by setting
As a result, when the Petrov type of is or the bundle of adapted coframes preserving (3.17) gives rise to a principal -bundle where
| (3.18) |
Since for such coframes the gauge transformations (2.18b) do not affect and by the inhomogeneous term , one obtains the expressions (3.12) for some functions satisfying the Bianchi identities (A.3).
It follows that the set of coframes adapted to the condition (3.17) gives a principal -bundle which is equipped by the Cartan connection The flatness condition follows from a straightforward computation.
For real Petrov types and one needs to find the appropriate reduction of the structure bundle for pKE metrics and proceeds analogously to find the Cartan connection (3.13). If the Petrov type of is the desired principal -bundle is given by the set of adapted null coframes with respect to which
If the Petrov type is by Proposition 3.6 one can consider the bundle of adapted coframes with respect to which
For such adapted coframes the 1-forms is reduced as in (3.11). Subsequently, one can use identities (A.2) to obtain
where is the coframe derivative of with respect to . Therefore the principal -bundle is given by adapted null coframes with respect to which one additionally has After these reductions and can be expressed as (3.12) for some functions on We will not express the Bianchi identities for ’s when the Petrov type is or It is a matter of straightforward computation to show that for real Petrov types and implies which is not being considered in this article.
∎
3.2.2. Petrov type
PKE metrics of real Petrov type are particularly interesting because we find an explicit local normal form for them as shown below. Suppose that has Petrov type i.e. the quartic (3.7) has two real roots with double multiplicity. Although Proposition 3.8 describes such pKE metrics as a Cartan geometry of type , here we carry out the reduction procedure further and describe all such pKE metrics in some normal coordinate system. First, we have the following.
Proposition 3.9.
Given a pKE metric whose anti-self-dual Weyl curvature has real Petrov type everywhere, the Cartan connection on satisfies the structure equations (3.1) where
| (3.19) |
The EDS obtained from the reduced structure equations together with the differential relations among the functions given by
| (3.20) | ||||
for some function is closed under the exterior derivative operator . As a result, the local moduli space of type pKE metrics is 5-dimensional.
Proof.
The differential identities (3.20) are obtained via straightforward computations discussed previously. The fact that the space of such pKE metrics is 5-dimensional follows from the Frobenius theorem applied to the resulting closed EDS. More precisely, define the 13-dimensional bundle whose fibers are parametrized by Since the Pfaffian system (3.20) is integrable, its leaf space is 7-dimensional parametrized by As the infinitesimal group actions in (3.20) suggests, one obtains that the action of the structure group transforms the quantities and by
To find the local generality one considers generic pKE metrics of type As a result, it can be assumed that after restricting to sufficiently small neighborhoods. Hence, setting and one can normalize and obtain a canonical coframe at each point of such pKE metrics. Consequently, the remaining 5 parameters of can be used to distinguish every generic type pKE metric up to isomorphism. Therefore, the local generality of type pKE metrics depends on 5 constants. ∎
It turns out that the pKE metrics of type D locally belong to one of four branches which can be characterized according to the vanishing of the quantities and Each branch can be locally expressed using normal coordinates given below.
Theorem 3.10.
In sufficiently small open sets, every pKE metric
whose anti-self-dual Weyl curvature has real Petrov type , belongs to one of the following four branches. The first branch is characterized by the property that and are non-vanishing and is comprised of 5-parameter family of pKE metrics for which a choice of normalized coframe can be expressed as
| (3.21) | ||||
for some constants and The second branch is characterized by the condition that and non-vanishing and is comprised of a 3-parameter family of pKE metrics for which a choice of coframe can be expressed as
| (3.22) |
where and are constants. Similarly, the third branch is characterized by and non-vanishing, which is comprised of 3-parameter family of pKE metrics and can be expressed as in (3.22) after switching and
Lastly, the fourth branch, characterized by is comprised of the only homogeneous pKE metrics of type They form a 1-parameter family parametrized by for which a choice of coframe is given by
| (3.23) |
Proof.
Let us first work in a neighborhood in which are nowhere vanishing. Using the action of and whose infinitesimal version is given in (3.20), there is a unique coframe with respect to which and therefore
| (3.24) |
To find our normal coordinate system we make use of the orbits of the Killing vector fields of these pKE metrics. Let be a Killing vector field i.e. where denotes the Lie derivative. It is straightforward to use the structure equations (3.1) and reductions (3.19) and (3.24) when to obtain for . Subsequently, the identities imply that the isometry group for such pKE metrics is two dimensional since
where
| (3.25) |
Because of the relation we restrict ourselves to an open set where and are non-zero. As a result, one can assume and define
From (3.25) it follows that
| (3.26) |
Using the reduced 1-forms (3.24) and normalized values in the Bianchi identities (3.20) together with the expressions (3.26), it follows that and are functions of and For instance, one obtains which implies for a constant Similarly, elementary calculations can be carried out to show
| (3.27) | ||||
for constants and To express and in a local normal form, one makes use of the structure equations (3.1) and the reductions (3.19) and (3.24) when to obtain
By Darboux’ theorem, locally one obtains for a local coordinate Lastly, the relation
implies that for a local coordinate This proves the local normal form presented in (3.21). Furthermore, it is straightforward to verify that and are the Killing vector fields for these pKE metrics.
For the second branch we restrict ourselves to open sets in which and is nowhere vanishing. Over one obtains Using the relations (3.20), consider the set of coframes with respect to which and
| (3.28) |
As described before, it is straightforward to express the Killing vector fields of such pKE metrics which are of the form
| (3.29) |
and, as a result, have to satisfy the differential relations
| (3.30) | ||||
Using the structure equations (3.1) and the reduced 1-forms (3.19) and (3.28) when and together with the Bianchi identities (3.20), it follows that
| (3.31) |
for a local coordinate It follows from (3.29) that the orbits of the isometry group for such pKE metrics are level sets of This implies that such pKE metrics have cohomogeneity one. Using the Bianchi identities (3.20) and the reduction (3.28), one obtains that and are given by
It remains to express in a local normal form which will be done using (3.30). Restricting to open sets where the relations (3.30) imply that
| (3.32) | ||||
Using (3.31), (3.32) in (3.30), it is a matter of elementary calculation to show
Lastly, the reduced structure equations imply that
Using Darboux’ theorem, one can find local coordinate system with respect to which
As a result, we have a local coordinate system where and with respect to which we have expressed ’s and . It is clear that acts by scaling on and and therefore, corresponds to the element in the reduced structure group (3.18). The coframe (3.22) is obtained by setting Finally, note that in terms of the local coordinates , the trajectories of the Killing vector fields are given by the level sets of and and The third branch characterized by and nowhere vanishing can be treated similarly.
Finally, when straightforward computation shows that all ’s vanish and As a result, such metrics are homogeneous. The structure equations are given by
| (3.33a) | |||
| (3.33b) | |||
Hence, it is sufficient to find a normal form for (3.33a). Using Darboux’s theorem, (3.33a) imply that there are coordinates with respect to which
for arbitrary functions and which satisfy
| (3.34) |
The equation above is known as Liouville’s equation [Hen86] whose solutions can be expressed as
| (3.35) |
for two arbitrary functions and Setting
one obtains the expression (3.23) for and The expressions of and are obtained similarly.
∎
It is straightforward to use the coframe (3.23) in order to arrive at the potential function in § 2.3.4. Moreover, one can characterize the branchings in Theorem 3.10 in terms of the vanishing of and
Remark 3.11.
Theorem 3.10 is yet another instance of explicit local normal form for certain classes of (pseudo-)Riemannian metrics whose Weyl curvature has algebraic type which includes the Plebański-Demiański metrics [Kin69, Deb71, PD76] in the Lorentzian signature (see [GP06, KK09] for a survey of all the results), and ambitoric metrics in Riemannian signature [ACG16].
3.2.3. Petrov type
Now we proceed to pKE metrics whose anti-self-dual Weyl curvature has Petrov type .
Theorem 3.12.
Given a pKE metric of Petrov type , the -valued Cartan connection on the principal bundle , as defined in Proposition 3.8, satisfies the Yang-Mills equations , where , if and only if
| (3.36) |
in (3.12). Examples of such pKE structure are given by where
| (3.37) | ||||
for some arbitrary functions where are nowhere vanishing.
Proof.
Using the expression of in (A.4), one immediately obtains (3.36) (see Remark 3.13). One can integrate the structure equations for pKE metrics of Petrov type that are Yang-Mills assuming some simplifying conditions. The integration procedure can be carried out similarly to what was explained in Theorem 3.9 and will not be explained here. It turns out that pKE metrics arising from the coframe (3.37) satisfy
Note that if one obtains homogeneous pKE metrics of type .
The -valued Cartan connection , as defined in (3.13), has curvature
which is anti-self-dual, and therefore satisfies the Yang-Mills equations . ∎
Remark 3.13.
Note that if the conditions (3.36) for in (3.12), which arise from the Yang-Mills equation, are replaced by then the of the pKE metric has type as discussed in Theorem 3.9. Furthermore, as shown in Proposition 3.8, one can always associate a Cartan geometry of type , with a canonical Cartan connection, to pKE metrics of any Petrov type by appropriately reducing the structure group. However, except for type , it can be shown that the set of Yang-Mills solutions among other types is empty.
3.2.4. Petrov type
Assume that the quartic in (3.7) has a repeated root of multiplicity three. As we did in § 3.2.2, by coframe adaptation one can translate the multiple root to zero, which is equivalent to finding an adapted coframe with respect to which
In this case Proposition 3.6 still remains valid and the following differential relations hold
| (3.38) | ||||
for some functions
Using the action of and , one can find the set of adapted coframes with respect to which the quantities and are normalized to constants. The set of such adapted coframes give rise to a line bundle with the group parameter in (3.10) as the fiber coordinate.
Proposition 3.14.
Given a pKE metric whose anti-self-dual Weyl curvature has Petrov type III and for which , the bundle of adapted coframes preserving
is a line bundle whose sections satisfy the structure equations (3.1) wherein
Examples of such pKE metrics for which , are given by where
where are arbitrary functions.
Remark 3.15.
We point out that by the action of the structure group, infinitesimally given in (3.38), one can reduce the structure group to identity by translating to zero and obtain a unique choice of coframe at each point. However, to obtain examples above one does not need to carry out full reduction. Some solutions satisfying these conditions where obtained earlier by A. Chudecki in [Chu17].
3.2.5. Petrov type
Assume that the quartic in (3.7) has a repeated root of multiplicity four. As we did in § 3.2.2, by coframe adaptation one can translate the multiple root to zero, which is equivalent to finding an adapted coframe with respect to which
In this case Proposition 3.6 still remains valid and the following differential relations hold
| (3.39) | ||||
As a result, by normalizing and to non-zero constants we can reduce the parameters and in the structure group (3.10) which reduces the bundle of adapted coframes to a line bundle with the element in (3.10) as the fiber coordinate. It follows that
| (3.40) |
for some functions It is straightforward to obtain
Because the quantity is normalized to a non-zero constant, the above differential relation can be interpreted as the infinitesimal action of the 1-dimensional structure group on (see Remark 3.5). Hence, by choosing appropriately, one can translate to zero which would reduce the structure group to identity. In other words there is a unique coframe at each point with respect to which one has the relations equations (3.11), (3.40) and
| (3.41) |
for some functions and on As a result we obtain the following.
Theorem 3.16.
3.2.6. Petrov type
The Petrov type O corresponds to pKE metrics for which Since the only non-zero quantity in the structure equations (3.1) is the constant it follows that such metrics are homogeneous and therefore no reduction of the structure bundle is possible. Nevertheless, one can follow the procedure explained before and integrate the structure equations from which the following choice of coframe is obtained
| (3.42) |
Using the coframe above one can recover the potential function given in § 2.3.4 for the so-called dancing metric.
3.2.7. Homogeneous models and local generality of various Petrov types
The structure equations of pKE metrics in dimension four and reduced structure equations obtained for non-generic Petrov types enable one to use the Cartan-Kähler theory and obtain the local generality of analytic pKE metrics of each Petrov type. We will not give the details of how Cartan-Kähler theory is implemented and refer the reader to [Bry14] for details.
Assuming analyticity for pKE metrics, the following table gives the local generality of various Petrov types.
| Petrov type | Local generality |
|---|---|
| 2 functions of 3 variables | |
| 4 functions of 2 variables | |
| 3 functions of 2 variables | |
| 2 functions of 2 variables | |
| and Yang-Mills | 2 functions of 2 variables |
| 5 constants | |
| 1 constant |
Furthermore, the reduced structure equations for each Petrov types allows one to look for homogeneous models. Finding homogeneous models involves a straightforward inspection of structure equations considering all possible normalizations which can be carried out algorithmically. We will not present all the necessary computation here. It turns out that the only homogeneous models of pKE metrics satisfying are the 1-parameter families of pKE metrics of type and which correspond to the coframes (3.23) and (3.42). In particular, there is no homogeneous pKE metric of type and for which
4. (2,3,5)-distributions arising from pKE metrics
This section contains the highlight of the article. In § 4.1 we give a brief review of the geometry of (2,3,5)-distributions. In § 4.2 we show that the naturally induced rank 2 twistor distribution on the space of self-dual null planes of any pKE metric is (2,3,5) in an open subset if Furthermore, the root type of the Cartan quartic of this twistor distribution agrees with the root type of the quartic representation of This remarkable and surprising coincidence is contrasted with the case of twistor distribution naturally arising on the space of anti-self-dual null planes of pKE metrics satisfying which is considered in § 4.3. In the latter case, the coefficients of the Cartan quartic depend on the fourth jet of the coefficients of and there is no further simplification from the larger context of twistor distributions arising from indefinite conformal structures in dimension four satisfying In other words, a priori, no relation between the type of the Cartan quartic and the Petrov type of or can be made (see Remark 4.15). Moreover, our construction in § 4.2 gives rise to 5-dimensional para-Sasaki-Einstein structures and conformal structures with holonomy, as studied in [SW17]. Consequently, our explicit examples of pKE metrics of special real Petrov type in § 3.2, provide examples of 5-dimensional para-Sasaki-Einstein metrics.
4.1. A primer on distributions
In this section we recall the basic definitions and theorems about the local geometry of a generic 2-plane field on a 5-dimensional manifold, which involves a Cartan connection and a naturally induced conformal structure of signature (3,2).
Given a 5-dimensional manifold, with a rank 2 distribution let denote its first derived system defined as the distribution whose sections are given by where denotes the sheaf of sections of the distribution Moreover, define
Definition 4.1.
A rank 2 distribution in dimension 5, is called a (2,3,5)-distribution if
Locally, a generic rank 2 distribution is a (2,3,5)-distribution. Given a (2,3,5)-distribution, locally, one can find a frame for such that
where
As a result, the corresponding coframe satisfies
| (4.1) | ||||
Cartan in his famous ’five-variables’ paper [Car10] solved the equivalence problem for (2,3,5)-distributions and explicitly introduced the distribution whose algebra of infinitesimal symmetries is given by the split real form of the exceptional Lie algebra Recall that the noncompact exceptional simple Lie group of dimension 14, acts transitively on the projective quadric defined by the (3,2)-signature diagonal matrix. Let be the parabolic subgroup of that preserves a null line in . Using his method of equivalence, Cartan associated an -structure on a 14-dimensional -principal bundle to any (2,3,5)-distribution.
Using the appropriate transformation Cartan’s original construction results in the following theorem.
Theorem 4.2 ([Car10, Nur05]).
Any (2,3,5)-distribution, defines a Cartan geometry of type Expressing the -valued Cartan connection as
| (4.2) |
the distribution is the projection of from to
Remark 4.3.
Cartan realized that the curvature can be interpreted as a ternary quartic form and expressed as
| (4.3) | ||||
where the coefficients are components of the curvature (see [Nur05]). Moreover, the fundamental curvature tensor is a binary quartic form referred to as the Cartan quartic, given by the first 5 terms in (4.3). If is identically zero it follows that i.e. the (2,3,5)-distribution is flat. It is convenient to express the Cartan quartic in 1-variable as follows
| (4.4) | ||||
Using Cartan’s result and the embedding the following non-trivial link between (2,3,5)-distributions and conformal structures of signature can be obtained.
Theorem 4.4 ([Nur05]).
Any (2,3,5)-distribution defines a conformal structure of signature (3,2) on which can be expressed as for any section where
| (4.5) |
The conformal holonomy of this conformal structure takes value in and its Weyl curvature can be expressed in terms of
Remark 4.5.
Using Theorem 4.4, we give another interpretation of the Cartan quartic (4.4) which will be important for analyzing non-integrable twistor distributions. At each point consider the -family of planes
| (4.6) |
where Such planes are null with respect to the conformal structure defined in (4.5), and intersect the distribution along the lines The bundle
| (4.7) |
where is the circle bundle of such null planes. Denote the components of the Weyl curvature for the conformal structure by where Following our discussion in § 2.2.4, let and define the multilinear map
Restricting to one obtains the quartic polynomial
where
Let us point out that the circle bundle is not preserved by the action of the full structure group for the geometry of (2,3,5)-distributions. In order to remedy this issue and define one can make a choice of splitting for given by Such splitting will reduce the structure group and allows one to define invariantly. We will see in the next section that twistor distributions arising from pKE metrics are naturally equipped with such splitting therefore enable one to define
Remark 4.6.
Using Theorem 4.4, the rank 2 distribution is null with respect to the conformal structure . In fact, it has been shown [HS11] that induces a parallel spin tractor. Conversely, it has been shown that conformal structures of signature (3,2) which are equipped with a parallel spin-tractor arise from the construction of Theorem 4.4. The existence of such parallel objects implies that the conformal holonomy of the conformal structure is a subgroup of (see [HS09, HS11] for more details.) This is an instance of an extension of the holonomy principle in pseudo-Riemannian geometry, as explained in Remark 2.9, to the context of Cartan geometries (see [ČS09].)
As was mentioned in Remark 2.1, in order to find the Cartan connection (4.2), one can either work with a lifted coframe defined on the bundle or start with a choice of coframe on the manifold and impose the structure equations to find the Cartan connection in terms of the coframe, which, if needed, can consequently be equivariantly lifted to In this article we will follow the latter approach, as we did for the pKE structures.
4.2. Null self-dual planes and a remarkable coincidence
In this section we show the main result of this article by finding the Cartan connection of the twistor distribution on the space of self-dual null planes and showing that the root type of its Cartan quartic is the same as the root type of Furthermore, as a by-product of our construction, one obtains para-Sasaki-Einstein metrics in dimension five and 5-dimensional conformal structures with -holonomy.
4.2.1. Twistor distribution on
It was observed in [AN14] that for a 4-dimensional conformal structure of split signature the circle bundles of self-dual and anti-self-dual null planes, and are each equipped with a naturally defined rank 2 distribution which is referred to as the twistor distribution. The twistor distribution on and is (2,3,5) in an open set if the self-dual and anti-self-dual Weyl curvature of the conformal structure is nowhere vanishing in .
Given a pKE structure, in order to define the twistor distribution on we make use of the parametrization (2.3), where . As a result, on one obtains the 0-adapted coframe
where the subscript refers to the adaptation with respect to which the 2-distribution will be defined.
A coframe defines a (2,3,5) distribution if
| (4.8a) | ||||
| (4.8b) | ||||
| (4.8c) | ||||
To define the twistor distribution on we further adapt the coframe so that it satisfies (4.8). Using the structure equations (3.1), one obtains
modulo In order to obtain the relations (4.8a),(4.8b) define the 1-adapted coframe as
| (4.9) |
Using (3.1), the 1-adapted coframe satisfies
| (4.10) | ||||||
Using the fact that and the 2-adapted coframe defined by
| (4.11) |
satisfies (4.8), therefore, defines a (2,3,5)-distribution,
| (4.12) |
on for . It is straightforward to show that is invariant under the induced action of the structure group . We have the following theorem.
Theorem 4.7.
The -bundle of self-dual null planes of a pKE metric, is naturally equipped with a rank 2 distribution If the scalar curvature of the pKE metric is non-zero, the twistor distribution is (2,3,5) on the open subset where are the -eigenspaces of the para-complex structure as defined in (2.12). In terms of the affine parameter in (2.3), the open subset corresponds to and the conformal structure of signature associated to the twistor distribution, as in Theorem 4.4, is given by where
| (4.13) |
Proof.
Using the coframe (4.9), the relations (4.10) imply that the 2-plane field is integrable on the hypersurfaces corresponding to and which by (2.3) are given by and i.e. 2-plane fields and . For and the twistor distribution is (2,3,5) by (4.12). Consequently, using the adapted coframe (4.11), the metric (4.5) gives (4.13). ∎
Remark 4.8.
The simple expression (4.13) for the metric is the key to what follows in Theorem 4.9. For general twistor distributions the expression for involves the second jet of the self-dual Weyl curvature of whose components are denoted by ’s. However, in pKE metrics the only non-zero component of is the constant This point is explained further in § 4.3, in particular Remark 4.15.
Using the theorem above, one obtains the following theorem which is the main result of this section.
Theorem 4.9.
Given a pKE metric with non-zero scalar curvature, the Cartan quartic for the twistor distribution on is a non-zero multiple of the quartic representation of the anti-self-dual Weyl curvature . More explicitly, one obtains
In particular, the root types of the Cartan quartic and the anti-self-dual Weyl curvature coincide.
Remark 4.10.
We point out that there is an underlying bundle map behind Theorem 4.9 which allows one to express as a non-zero multiple of . As discussed in § 2.2.4, is defined on and, by Remark 4.5, is defined on However, using the twistorial nature of one can naturally identify and in the following way. First note that given a pKE structure, the derived system is equipped with a splitting which is invariant under the induced action of Therefore, by Remark 4.5, the circle bundle as defined in (4.7), is well-defined on . Consequently, via the bundle map it is elementary to check that where for all using definitions (2.3) and (4.6). Therefore, the property of and being proportional everywhere is well-defined.
Proof of Theorem 4.9.
Having the twistor distribution defined by (4.12) on , it is straightforward to find the explicit Cartan connection (4.2) for starting with the 2-adapted coframe (4.11). Consequently, one obtains that the Cartan connection (4.2) for is given by
| (4.14) | ||||||||
Using the Cartan connection, one computes the curvature
Therefore, the coefficients of the Cartan quartic (4.4) are found to be
| (4.15) |
By the formulas for and as given in (4.15) and (3.7), it follows that
∎
Remark 4.11.
As was mentioned in the introduction, it was not known whether (2,3,5) distributions that arise as a twistor distribution of split signature metrics can have any fixed (Petrov) root type. Theorem 4.9 shows that any root type can be achieved via pKE metrics with non-zero scalar curvature for which has the same root type. Moreover, our examples in § 3.2 provide explicit metrics whose twistor distributions have Cartan quartics of real root type and
4.2.2. An invariant description
To have an invariant understanding of we define another space equipped with a rank 2 distribution which will be shown to be isomorphic to the twistor distribution on Consider the principal -bundle of adapted null coframes for pKE metrics, equipped with the Cartan connection as in (3.2). Using the structure equations (3.1), one can define the 5-dimensional leaf space, of the Pfaffian system
Note that by the structure equations (3.1) the Pfaffian system is integrable and its 5-dimensional leaf space is the quotient of by the orbits of which would give an -bundle over . As a result, is a cone. Similar to our previous discussion, if after necessary adaptation, one obtains the following coframe on
| (4.16) |
which defines a rank 2 distribution by The distribution is invariant under the induced action of the structure group
The Cartan connection for the is obtainable from (4.14) by setting In the language of parabolic geometries [ČS09], this is the explicit form of the extension functor from a pKE structure to the corresponding (2,3,5)-distribution.
To relate our discussion to note that the structure group of a pKE metric can be decomposed as
The -action transforms the pKE coframe in the following way
by setting and in the structure group (2.16). This action results in the following change of coframe (4.11) on
| (4.17) |
Using the expressions (4.17) one finds the bundle isomorphism in terms of the fiber coordinates and given by for and for via which This gives the equivalence of the (2,3,5) distributions induced on and
4.2.3. para-Sasaki-Einstein structures and holonomy
To state the last result of this section we give the following definition of a para-Sasaki-Einstein structure.
Definition 4.12.
A para-Sasaki-Einstein structure on is where is an endomorphism, is a vector field, is a 1-form, and is a split signature metric. The quadruple satisfies
| (4.18) |
Additionally, the -eigenspaces of define rank 2 integrable sub-distributions of the metric is Einstein, and the compatibility conditions
| (4.19) | |||||
hold.
We have the following proposition.
Proposition 4.13.
The cone is equipped with a para-Sasaki-Einstein structure arising from the para-Kähler-Einstein structure on
Proof.
Remark 4.14.
The proposition above is an analogue of the well-known construction of Sasakian structures from Kähler metrics [GKN00]. Moreover, by the same analogy, following [GKN00], it can be checked that using the potential function of a para-Kähler-Einstein metric and the coordinate on the cone, as introduced in § 4.2.2, one obtains a potential function for the resulting para-Sasaki-Einstein metric on
Finally one can directly verify the well-known relation between the existence of an Einstein representative in the conformal structure of the metric (4.13) defined by the (2,3,5) distribution, and the holonomy reduction of the Cartan conformal connection. This relation goes back to works of many people, including [Sas43] (see Section 5.2 in [ČS09] for an overview.) One can check that the Cartan connection (4.2) given by (4.14) takes value in for any non-zero value of This implies that the conformal holonomy of the conformal structure is reduced and the conformal class must contain at least one Einstein metric. We refer the reader to [SW17] for a more detailed study of conformally Einstein structures arising from (2,3,5) distributions and their corresponding conformal holonomy reductions.
4.3. Null anti-self-dual planes
The purpose of this short section is to justify the remarkable nature of Theorem 4.9 by examining the twistor distribution induced on the -bundle of anti-self-dual null planes of a pKE metric, Following the description in § 4.2.2, one can identify as the leaf space of the Pfaffian system
Consequently, if one can follow the discussion in § 4.2.1 to find an adapted coframe. An adapted coframe for the twistor distribution satisfying (4.8) on the open set of where is given by
Note that if one can always find a coframe with respect to which
Consequently, the expressions for the coefficients of the Cartan quartic, , are found to be very complicated and depend on the 4th jet of ’s. For instance, in the coframe introduced above, one obtains
| (4.21) |
The equation arising from the vanishing of is referred to as Noth’s equation and its general solution can be presented by a certain family of rational sextics [AN18, DS11].
We could not identify any relation between the root types of and the root type of the Cartan quartic of the twistor distribution on with the exception of the homogeneous pKE metrics of Petrov type where they coincide.
Remark 4.15.
Following our approach in this section, one can find the Cartan quartic for the twistor distribution on and for any conformal structure of split signature provided that and respectively. The computations are extremely tedious. Restricting to , if one obtains that the expression for the metric as defined in (4.5), involves the second jets of Furthermore, in an appropriate coframe, ’s can be expressed in terms of the fourth jets of and zeroth jets of at each point. For instance, consider conformal structures where with and denote the components of and by ’s and ’s respectively, as we did in § 2.1.2. Then on the open subset of where there is an adapted coframe for the twistor distribution with respect to which
Comparing the expression above to 4.21, it is clear that the case of twistor distribution induced on for pKE metrics is nearly as complicated as in the case of general conformal structures. As a result, one cannot expect any relation between the root types of for a conformal structure and the Cartan quartic of the twistor distribution on
Appendix
The connection 1-form referred to in § 2.3.3 are given by
| (A.1) | ||||
If is a para-Kähler-Einstein structure written in a null adapted frame satisfying (3.1) then the derivatives of the non-vanishing curvature coefficients are given by
| (A.2a) | |||
| (A.2b) | |||
| (A.2c) | |||
| (A.2d) | |||
| (A.2e) | |||
| (A.2f) | |||
for some functions on which represent the coframe derivatives of ’s.
References
- [ACG16] V. Apostolov, D. M. J. Calderbank, and P. Gauduchon. Ambitoric geometry I: Einstein metrics and extremal ambikähler structures. J. Reine Angew. Math., 721:109–147, 2016.
- [AG96] M. A. Akivis and V. V. Goldberg. Conformal differential geometry and its generalizations. Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1996. A Wiley-Interscience Publication.
- [AN14] D. An and P. Nurowski. Twistor space for rolling bodies. Comm. Math. Phys., 326(2):393–414, 2014.
- [AN18] D. An and P. Nurowski. Symmetric distributions, an interesting ODE of 7th order and Plebański metric. J. Geom. Phys., 126:93–100, 2018.
- [BBI97] L. Bérard Bergery and A. Ikemakhen. Sur l’holonomie des variétés pseudo-riemanniennes de signature . Bull. Soc. Math. France, 125(1):93–114, 1997.
- [BH93] R. L. Bryant and L. Hsu. Rigidity of integral curves of rank distributions. Invent. Math., 114(2):435–461, 1993.
- [BHLN18] G. Bor, L. Hernández Lamoneda, and P. Nurowski. The dancing metric, -symmetry and projective rolling. Trans. Amer. Math. Soc., 370(6):4433–4481, 2018.
- [Bry14] R. L. Bryant. Notes on exterior differential systems. arXiv preprint, 2014.
- [Car10] E. Cartan. Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre. Ann. Sci. École Norm. Sup. (3), 27:109–192, 1910.
- [CFG96] V. Cruceanu, P. Fortuny, and P. M. Gadea. A survey on paracomplex geometry. Rocky Mountain J. Math., 26(1):83–115, 1996.
- [Chu17] A. Chudecki. On some examples of para-Hermite and para-Kähler Einstein spaces with . J. Geom. Phys., 112:175–196, 2017.
- [ČS09] A. Čap and J. Slovák. Parabolic geometries. I, volume 154 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2009. Background and general theory.
- [Deb71] R. Debever. On type expanding solutions of Einstein-Maxwell equations. Bull. Soc. Math. Belg., 23:360–376, 1971.
- [DS11] M. Dunajski and V. Sokolov. On the 7th order ODE with submaximal symmetry. J. Geom. Phys., 61(8):1258–1262, 2011.
- [Gar89] R. B. Gardner. The method of equivalence and its applications, volume 58 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
- [GKN00] M. Godliński, W. Kopczyński, and P. Nurowski. Locally Sasakian manifolds. Classical Quantum Gravity, 17(18):L105–L115, 2000.
- [GP06] J. B Griffiths and J. Podolskỳ. A new look at the Plebański–Demiański family of solutions. International Journal of Modern Physics D, 15(03):335–369, 2006.
- [Hen86] P. Henrici. Applied and computational complex analysis. Vol. 3. Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, 1986. Discrete Fourier analysis—Cauchy integrals—construction of conformal maps—univalent functions, A Wiley-Interscience Publication.
- [HS09] M. Hammerl and K. Sagerschnig. Conformal structures associated to generic rank 2 distributions on 5-manifolds—characterization and Killing-field decomposition. SIGMA Symmetry Integrability Geom. Methods Appl., 5:Paper 081, 29, 2009.
- [HS11] M. Hammerl and K. Sagerschnig. The twistor spinors of generic 2- and 3-distributions. Ann. Global Anal. Geom., 39(4):403–425, 2011.
- [Kin69] W. Kinnersley. Type D vacuum metrics. Journal of Mathematical Physics, 10(7):1195–1203, 1969.
- [KK09] N. Kamran and A. Krasiński. Editorial Note to: Brandon Carter, Black hole equilibrium states part I. Analytic and geometric properties of the Kerr solutions. General Relativity and Gravitation, 41(12):2867, 2009.
- [Lib52] P. Libermann. Sur les structures presque paracomplexes. C. R. Acad. Sci. Paris, 234:2517–2519, 1952.
- [Mer84] S A Merkulov. A conformally invariant theory of gravitation and electromagnetism. Classical and Quantum Gravity, 1(4):349–354, jul 1984.
- [NS03] P. Nurowski and G. A J Sparling. Three-dimensional cauchy–riemann structures and second-order ordinary differential equations. Classical and Quantum Gravity, 20(23):4995–5016, oct 2003.
- [Nur05] P. Nurowski. Differential equations and conformal structures. J. Geom. Phys., 55(1):19–49, 2005.
- [Olv95] P. J. Olver. Equivalence, invariants, and symmetry. Cambridge University Press, Cambridge, 1995.
- [PD76] J. F. Plebański and M. Demiański. Rotating, charged, and uniformly accelerating mass in general relativity. Ann. Physics, 98(1):98–127, 1976.
- [Sas43] S. Sasaki. On the spaces with normal conformal connexions whose groups of holonomy fix a point or a hypersphere. I,II,III. Jpn. J. Math., 18:615–622, 1943.
- [Sha97] R. W. Sharpe. Differential geometry, volume 166 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. Cartan’s generalization of Klein’s Erlangen program, With a foreword by S. S. Chern.
- [SW17] K. Sagerschnig and T. Willse. The geometry of almost Einstein distributions. SIGMA Symmetry Integrability Geom. Methods Appl., 13:Paper No. 004, 56, 2017.