DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 231–238

ON FOUR-DIMENSIONAL CONFORMAL STRUCTURES

M. A. AKIVIS

Abstract. The conformal structures CO(4, 0), CO(1, 3) and CO(2, 2) are studied on a real manifold M, dim M = 4. On M isotropic fiber bundles E_{α} and E_{β} are constructed. These bundles are real for the CO(2, 2)-structure, and they satisfy the condition \overline{E}_{α} = E_{β} for the CO(1, 3)-structure, and the conditions \overline{E}_{α} = E_{α}, \overline{E}_{β} = E_{β} for the CO(4)-structure. The tensor C of conformal curvature splits into two subtensors C_{α} and C_{β} which are the curvature tensors of the bundles E_{α} and E_{β}, respectively. These subtensors satisfy the same conditions as the bundles E_{α} and E_{β}. Conformally semiflat and flat structures and their geometrical characteristics are studied. The principal 2-directions are defined, and conditions for their integrability are obtained. These investigations for the CO(1, 3)-structure are connected with Petrov’s classification of Einstein’s spaces. Although the paper is of a servey nature, it also contains some new results.

1. Four-dimensional conformal structures play an important role in general relativity. Space-time in general relativity is a four-dimensional pseudo-Riemannian manifold of signature (1, 3). Many features of general relativity are of a conformally invariant nature. This is a reason that the study of pseudoconformal structures of signature (1, 3) is of special interest. Along with this kind of conformal structures, on a real four-dimensional manifold M, one also can consider conformal structures of signatures (4, 0) and (2, 2). By means of complexification of the manifold M, all these structures can be reduced to one of them, for example, to the structure of signature (4, 0) or (2, 2). This was done in many investigations (see, for example, [AHS 78], [Gi 82], [P 67], and [P 77]).

Unlike the previous investigations, in the present paper, we consider fourdimensional conformal structures on a real manifold M and study their common properties and the differences between them. Moreover, we also apply complexification not of the manifold M itself but only of its tangent spaces T_{x}(M), and consider in these spaces coordinate transformations preserving the real part of these tangent spaces.

1991 Mathematics Subject Classification. 53A30. Key words and phrases. Conformal structure, isotropic fiber bundle, tensor of conformal

curvature, principal distributions, completely isotropic submanifolds, Petrov’s classification. This paper is in final form and no version of it will be submitted for publication elsewhere.

Some of the problems discussed in the current paper were considered earlier in more detail in the papers [A 83] and [AK 93]. In the paper [AZ 95], the apparatus constructed in the current paper is applied to the study of conformal structures of the Schwarzschild metric and the Kerr metric (see, for example, [Ch 83]) arising in general relativity.

2. Let M be a real differentiable manifold of dimension n endowed with a conformal structure of signature (p, q), p + q = n, which is the class {g} of conformally equivalent Riemannian metrics g of signature (p, q). We will denote such a conformal structure by CO(p, q) (see, for example, [K 72]).

A CO(p, q)-structure is a G-structure with the structure group G = O(p, q)×H where SO(p, q) is the special pseudoorthogonal group of signature (p, q) and H = R · Id is the group of homotheties.

Let CT_{x} = T_{x}(M)⊗C be the complexified tangent space T_{x}(M) to the manifold M at a point x. In this space we will allow only complex linear transformations which preserve the real subspace T_{x} of CT_{x} and the symmetry with respect to T_{x}(M). Denote by C_{x} the isotropic cone of the CO(p, q)-structure at the point x, C_{x} = {ξ ∈ CT_{x}|g(ξ, ξ) = 0}.

Let {e_{i}}, i = 1, …, n, be a vectorial frame in the space T_{x}(M) or CT_{x}, and {ω^{i}} be the coframe such that ω^{i}(e_{j}) = δ_{j}^{i}. In the space CT_{x}, the family of these frames must be invariant with respect to admissible transformations of this space.

3. For n = 4, from real point of view, there are three different types of conformal structures:

(i): The structure CO(4, 0) = CO(4), for which by means of a real transformation, its fundamental form g can be reduced to the form:

g = (ω^{1})^{2} + (ω^{2})^{2} + (ω^{3})^{2} + (ω^{4})^{2}. (1)

The isotropic cones C_{x} of this structure are pure imaginary. The CO(4)- structure is called proper conformal. (ii): The CO(1, 3)-structure whose fundamental form g can be reduced to the form:

g = (ω^{1})^{2} − (ω^{2})^{2} − (ω^{3})^{2} − (ω^{4})^{2}. (2)

The isotropic cones C_{x} of this structure are real and have real rectilinear generators. The CO(1, 3)-structure is called a structure of Lorentzian type. (iii): The CO(2, 2)-structure whose form g can be reduced to the form:

g = (ω^{1})^{2} − (ω^{2})^{2} + (ω^{3})^{2} − (ω^{4})^{2}. (3)

The isotropic cones C_{x} of this structure are real and have real twodimensional plane generators. The CO(2, 2)-structure is called a structure of hyperbolic type.

By means of admissible transformations of coordinates in the space CT_{x}(M), the quadratic forms (1), (2) and (3) can be reduced to the same form:

g = 2(ω^{1}ω^{4} − ω^{2}ω^{3}), (4)

The coordinates, with respect to which the form g has expression (4), are called isotropic.

For the CO(2, 2)-structure, this transformation has the form:

\begin{matrix}ω^{1} + ω^{4} → \sqrt{2} ω^{1}, & ω^{2} + ω^{3} → \sqrt{2}ω^{2}, \\ ω^{1} − ω^{4} → \sqrt{2} ω^{4}, & ω^{2} − ω^{3} → \sqrt{2} ω^{3} \end{matrix} (5)

and is real. The isotropic coordinates ω^{i}, i = 1, 2, 3, 4, are real too. For the CO(1, 3)-structure, this transformation has the form

\begin{matrix}ω^{1} + ω^{4} → \sqrt{2}ω^{1}, & ω^{2} + iω^{3} → \sqrt{2}ω^{2}, \\ ω^{1} − ω^{4} → \sqrt{2}ω^{4}, & ω^{2} − iω^{3} → \sqrt{2}ω^{3} \end{matrix} (6)

and is complex. Moreover, the isotropic coordinates ω^{i} satisfy the conditions

\overline{ω}^{1} = ω^{1}, \overline{ω}^{4} = ω^{4}, \overline{ω}^{2} = ω^{3}. (7)

For the CO(4)-structure, this transformation has the form

\begin{matrix}ω^{1} + iω^{4} → \sqrt{2}ω^{1}, & ω^{2} + iω^{3} → \sqrt{2}ω^{2}, \\ ω^{1} − iω^{4} → \sqrt{2}ω^{4}, & ω^{2} − iω^{3} → −\sqrt{2}ω^{3} \end{matrix} (8)

and is also complex. In this case, the isotropic coordinates ω^{i} satisfy the conditions

ω^{4} = \overline{ω}^{1}, ω^{3} = −\overline{ω}^{2}, (9)

With respect to the new basis, the equation g = 0 of the isotropic cone C_{x} can be written in two different forms:

\frac{ω^{1}}{ω^{3}} = \frac{ω^{2}}{ω^{4}} = −λ and \frac{ω^{1}}{ω^{2}} = \frac{ω^{3}}{ω^{4}} = −μ, (10)

i.e., in the form of two systems of equations:

ω^{1} + λω^{3} = 0, ω^{2} + λω^{4} = 0 (11)

and

ω^{1} + μω^{2} = 0, ω^{3} + μω^{4} = 0. (12)

The systems (11) and (12) determine two-dimensional plane generators α(λ) and β(μ) of the isotropic cone C_{x} of the four-dimensional conformal structure which are called α-planes and β-planes, respectively (see [PR 86]).

Since for the CO(2, 2)-structure the isotropic coordinates ω^{i} are real, the parameters λ and μ in equations (11) and (12) are real nonhomogeneous coordinates on the projective lines RP_{α} and RP_{β}, respectively. The α- and β-generators of the cones C_{x} are also real. On the manifold M, they form the two fiber bundles E_{α} = (M, RP_{α}) and E_{β} = (M, RP_{β}), respectively, which are called the isotropic fiber bundles.

The structure group G of the CO(2, 2)-structure decomposes into the product: G = SL(2) × SL(2) × H.

For the CO(1, 3)- and CO(4)-structures, the isotropic coordinates ω^{i} are complex. Hence the parameters λ and μ in equations (11) and (12) should be considered as complex nonhomogeneous coordinates on the projective lines CP_{α} and CP_{β}, respectively. For these structures, the α- and β-generators of the cones C_{x} are also complex and form the fiber bundles E_{α} = (M, CP_{α}) and E_{β} = (M, CP_{β}) on the manifold M.

Moreover, for the CO(1, 3)-structure, by equations (7), the α- and β-planes are related by the condition \overline{α(λ)} = β(\overline{λ}). Thus, the isotropic bundles E_{α} and E_{β} are complex conjugates: \overline{E}_{α} = E_{β}. The generators α(λ) and β(\overline{λ}) of the cone C_{x} have the common rectilinear generator defined by the vector ξ = λ\overline{λ}e_{1} − λe_{2} − \overline{λ}e_{3} + e_{4}. The structure group G of the CO(1, 3)-structure admits two isomorphic representations on the fiber bundles E_{α} and E_{β}: G ≅ SL(2, C) × H ≅ \overline{SL(2, C)} × H, where the groups SL(2, C) and \overline{SL(2, C)} act concordantly on the fiber bundles E_{α} and E_{β}.

For the CO(4)-structure, by equations (9), the α-planes and β-planes are related by the conditions \overline{α(λ)} = α(−\frac{1}{\overline{λ}}) and \overline{β(μ)} = β(−\frac{1}{\overline{μ}}). Thus, the fiber bundles E_{α} and E_{β} are self-conjugate: \overline{E_{α}} = E_{α}, \overline{E_{β}} = E_{β}, and the structure group G of this structure decomposes into the product G = SU(2) × SU(2) × H.

Since on the fiber bundle E_{α}, an α-plane is determined by a point x and a value of the parameter λ, sometimes we will denote it by (x, λ). By a similar reason, we will denote a β-plane by (x, μ).

4. Denote by C the tensor of conformal curvature of a four-dimensional conformal structure which is called also the Weyl tensor (see, for example, [SS 38] and [A 83]), and by C_{α} and C_{β} the curvature tensors of the isotropic fiber bundles E_{α} and E_{β}. The vanishing of the tensor C characterizes the conformally flat structures, and the vanishing of one of the tensors C_{α} and C_{β} characterizes the conformally semiflat structures.

A Riemannian manifold endowed with the conformally flat structure can be mapped into the conformal space C^{4} or the pseudoconformal spaces C_{1}^{4} and C_{2}^{4} for which the quadratic form g has signature (4, 0), (1, 3) and (2, 2), respectively. Note that the conformal space C_{q}^{4} is a compactified Euclidean space R_{q}^{4}, i.e. it is R_{q}^{4} enlarged by the point at infinity and the isotropic cone C_{∞} with vertex at this point: C_{q}^{4} = R_{q}^{4} ∪ C_{∞}.

The connection between the tensors C, C_{α} and C_{β} can be described as follows. For the CO(2, 2)-structure, the tensor C decomposes into the sum of the tensors

C_{α} and C_{β}: C = C_{α} ˙+C_{β}, and each of the tensors C_{α} and C_{β} has five independent components:

C_{α} = {a_{0}, a_{1}, a_{2}, a_{3}, a_{4}}, C_{β} = {b_{0}, b_{1}, b_{2}, b_{3}, b_{4}}

(see [A 83]).

For the CO(1, 3)- and CO(4)-structures, the tensors C_{α} and C_{β} have complex coordinates a_{u} and b_{u}, u = 0, 1, 2, 3, 4. Moreover, the relation \overline{E}_{α} = E_{β} between the isotropic fiber bundles of the CO(1, 3)-structure implies a similar relation \overline{C}_{α} = C_{β} for their curvature tensors.

Similarly the relations \overline{E}_{α} = E_{α} and \overline{E}_{β} = E_{β} between the isotropic fiber bundles of the CO(4)-structure imply similar relations \overline{C}_{α} = C_{α} and \overline{C}_{β} = C_{β} for their curvature tensors. One can prove that these relations can be written in the following more detailed form:

\begin{matrix}\overline{a}_{0} = a_{4}, & \overline{a}_{1} = −a_{3}, & \overline{a}_{2} = a_{2}, \\ \overline{b}_{0} = b_{4}, & \overline{b}_{1} = −b_{3}, & \overline{b}_{2} = b_{2}. \end{matrix} (13)

It follows from these considerations that only the CO(2, 2)- and CO(4)-structures can be conformally semiflat, and the CO(1, 3)-structure cannot be conformally semiflat without being conformally flat. It is not difficult to show that the conformally semiflat CO(4)-structures coincide with the self-dual and anti-self-dual structures considered in [AHS 78].

5. Denote by V_{x}^{2} = {p} the bivector space formed by the bivectors p = ξ ∧ η of the tangent space T_{x}(M) to a manifold M endowed with CO(p, q)-structure. Since here ξ = ξ^{i}e_{i} and η = η^{i}e_{i} are vectors of the space T_{x}, we have p = p^{ij}e_{i} ∧ e_{j} where p^{ij} = \frac{1}{2}(ξ^{i}η^{j} − ξ^{j}η^{i}), i, j = 1, 2, …, n. The quadratic form

C(p) = C_{ijkl}p^{ij}p^{kl},

where C_{ijkl} are components of the Weyl tensor in an arbitrary coordinate system is called the relative conformal curvature of the bivector p.

Consider the CO(2, 2)-structure, and let p_{λ} = ξ_{λ} ∧ η_{λ} be an isotropic bivector belonging to the fiber bundle E_{α}. The coordinates of the vectors ξ_{λ} and η_{λ} in the isotropic frame satisfy equations (11), and this implies that

ξ_{λ} = e_{3} − λe_{1}, η_{λ} = e_{4} − λe_{2}.

In [A 83] it was proved that the relative conformal curvature of the bivector p_{λ} is expressed by the formula

\frac{1}{4}C(p_{λ}) = a_{0}λ^{4} − 4a_{1}λ^{3} + 6a_{2}λ^{2} − 4a_{3}λ + a_{4} := C_{α}(λ). (14)

Similarly, the relative conformal curvature of the bivector p_{μ} = ξ_{μ} ∧ η_{μ} has the form

\frac{1}{4}C(p_{μ}) = b_{0}μ^{4} − 4b_{1}μ^{3} + 6b_{2}μ^{2} − 4b_{3}μ + b_{4} := C_{β}(μ). (15)

The coefficients of the polynomials (14) and (15) are components of the tensors C_{α} and C_{β} which are real for the CO(2, 2)-structure.

The isotropic bivector p_{λ} or p_{μ} for which relative conformal curvature vanishes, i.e. for which C_{α}(λ) = 0 or C_{β}(μ) = 0, is called the principal isotropic bivector.

It follows from formulas (14) and (15) that the CO(2, 2)-structure is α-semiflat ( β-semiflat) if and only if all isotropic bivectors belonging to the fiber bundle E_{α} (respectively, to E_{α}) are principal. This structure is flat if and only if all isotropic bivectors are principal.

A two-dimensional submanifold V of the manifold M endowed with a pseudoconformal CO(2, 2)-structure is called completely isotropic if for any x ∈ V, the tangent 2-plane T_{x}(V) is isotropic. If T_{x}(V) ⊂ E_{α}, then the submanifold is called α-isotropic and denoted by V_{α}; and if T_{x}(V) ⊂ E_{β}, then V = V_{β} is a β-isotropic submanifold.

The submanifolds V_{α} are determined on M by the system of equations (11). Exterior differentiation of equations (11) leads to the equations

θ_{λ} ∧ ω^{3} = 0, θ_{λ} ∧ ω^{4} = 0, (16)

where

θ_{λ} := dλ + λ(θ_{1}^{1} + θ_{2}^{2}) − θ_{3}^{1} + λ^{2}θ_{1}^{3},

and θ_{1}^{1}+θ_{2}^{2}, θ_{3}^{1}, θ_{1}^{3} are invariant forms of the group GL(2) acting in the fiber bundle E_{α}. Since the forms ω^{3} and ω^{4} are linearly independent on the submanifold V_{α}, it follows from equation (16) that

θ_{λ} = 0. (17)

By taking the exterior derivative of this equation, we obtain C_{α}(λ) = 0. Thus, all tangent 2-planes T_{x}(V_{α}) to an α-isotropic submanifold are α-principal.

In exactly the same way, we can prove that all tangent 2-planes T_{x}(V_{β}) to a β-isotropic submanifold are β-principal.

A CO(2, 2)-structure is called α-semiintegrable if for any α-plane (x, α), there exists a submanifold V_{α} such that T_{x}(V_{α}) = (x, α). In this case, the system of equations (11) and (17) is completely integrable and defines a three-parameter family of completely isotropic submanifolds V_{α} on M.

The following theorem is valid: A CO(2, 2)-structure is α-semiintegrable if and only if it is α-semiflat.

In a similar way, one can define β-semiintegrable CO(2, 2)-structures and prove that the class of such structures coincides with the class of β-semiflat structures. Finally, a CO(2, 2)-structure is simultaneously α- and β-semiintegrable if and only if it is conformally flat.

Suppose that the CO(2, 2)-structure is not conformally semiflat. Then each of the equations

C_{α}(λ) = 0 (18)

and

C_{β}(μ) = 0 (19)

has four roots if we count each of these roots as many times as its multiplicity. Denote these roots by λ_{p} and β_{p}, p = 1, 2, 3, 4, respectively. Each of these roots determines an isotropic distribution which is called principal and denoted by Δ_{p}(α) and Δ_{p}(α), respectively.

A pseudoconformal CO(2, 2)-structure is called α-semirecurrent if ∇C_{α} = σ_{1}C; it is called β-semirecurrent if ∇C_{β} = σ_{2}C; and it is called recurrent if ∇C = σC. Here ∇ is symbol of the covariant differential with respect to the invariant conformal connection defined on M, and σ, σ_{1} and σ_{2} are 1-forms (cf. [AM 67]).

The following theorem is valid.

Theorem ([A 83])

a) If the CO(2, 2)-structure is α-semirecurrent (respectively, β-semirecurrent), then all principal distributions Δ_{p}(α) (respectively, Δ_{p}(β)) are integrable. If this structure is recurrent, then all principal distributions are integrable.

b) If λ_{p} (respectively, μ_{p}) is a multiple root of equation (18) (respectively, (19)), then the principal distribution Δ_{p}(α) (respectively, Δ_{p}(β)) determined by this root is integrable.

The theory developed in this section for the CO(2, 2)-structure can be immediately extended to the CO(1, 3)- and CO(4)-structures. In addition, a few more results following from the properties of their tensors C_{α} and C_{β} can be added.

Since for the CO(1, 3)-structure, we have \overline{C}_{α} = C_{β}, equations (18) and (19) have complex conjugate roots: μ_{p} = \overline{λ}_{p}. Hence the principal α- and β-bivectors are also complex conjugate, and for the principal distributions, the relation \overline{Δ_{p}(β)} = Δ_{p}(α) holds. Thus, the distribution Δ_{p}(β) is integrable if and only if the distribution Δ_{p}(α) is integrable. The intersection of the principal bivectors α_{p} and β_{p} determines a real vector field ξ_{p} = α_{p} ∧ β_{p}, whose integral curves are isotropic geodesics with respect to any Riemannian connection compatible with the CO(1, 3)-structure. In the general case, a manifold M endowed with the CO(1, 3)-structure carries four isotropic geodesic congruences.

For the CO(4)-structure, the components of the tensor C_{α} satisfy the equation

C_{α}(λ) = a_{0}λ^{4} − 4a_{1}λ^{3} + 6a_{2}λ^{2} + 4\overline{a}_{1}λ + \overline{a}_{0} = 0,

where \overline{a}_{2} = a_{2}. Thus, the roots of this equation are connected by the relations λ_{4} = −(\overline{λ}_{1})^{−1} and λ_{3} = −(\overline{λ}_{2})^{−1}. It follows that the roots λ_{1} and λ_{4} as well as λ_{2} and λ_{3} cannot coincide. Furthermore, if λ_{1} = λ_{2}, then λ_{3} = λ_{4}. Thus, the above equation has either four distict roots or two pairs of double roots; in the later case, the isotropic fiber bundle E_{α} carries two double principal α-distributions. However, since these distributions are complex, they do not define foliations on the real manifold M.

6. Space-time in general relativity is a four-dimensional pseudo-Riemannian manifold of signature (1, 3) of special type. Such manifolds are called Einstein’s spaces (see, for example, [Ch 83]) and are defined by the equation

R_{jki}^{i} = αg_{jk} + βT_{jk},

where R_{jkl}^{i} is the tensor of conformal curvature, and T_{jk} is the energy-momentum tensor.

Many features of general relativity are related to the conformal structure of space-time since its light cones coincide with the isotropic cones of the CO(1, 3)- structure defined by its Riemannian metric. In particular, Petrov’s classification of Einstein’s spaces is of conformally invariant nature. This classification is based on the structure of roots of equations (18) and (19). Since for the CO(1, 3)-structure, these equations are complex conjugate, we will consider only equation (18).

In terms of the present paper, Petrov’s classification is represented in the form of the following table:

Petrov’s Roots of the  Characterization Characterization of
type  equation  of principal  isotropic geodesic
C_{α}(λ) = 0  distributions  congruences
I  λ_{p} ≠ λ_{q}, p ≠ q,  4 different of  4 simple
p, q = 1, 2, 3, 4  general type
II λ_{1} = λ_{2} ≠ λ_{3}, λ_{4}; 1 double and 1 double and
λ_{3} ≠ λ_{4}  2 of general type 2 simple
D λ_{1} = λ_{2} ≠ λ_{3} = λ_{4} 2 double 2 double
III λ_{1} = λ_{2} = λ_{3} ≠ λ_{4} 1 triple and  1 triple and
1 of general type 1 simple
N  λ_{1} = λ_{2} = λ_{3} = λ_{4} 1 quadruple  1 quadruple

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Department of Mathematics and Computer Science Ben-Gurion University of the Negev P.O. Box 653, Beer-Sheva 84105, Israel