The Diffiety Institute Preprint Series

Preprint DIPS–3/2005 December 7, 2005

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Ricci Flat Metrics with bidimensional null

Orbits and non-integrable orthogonal Distribution

by

Michael Bächtold

Ricci Flat Metrics with bidimensional null Orbits and non-integrable orthogonal Distribution

MICHAEL BÄCHTOLD

Abstract. We study Ricci flat 4–metrics of any signature under the assumption that they allow a Lie algebra of Killing fields with 2–dimensional orbits along which the metric degenerates and whose orthogonal distribution is not integrable. It turns out that locally there is a unique (up to a sign) metric which satisfies the conditions. This metric is of signature (+ + −−) and, moreover, homogeneous possessing a 6– dimensional symmetry algebra.

Introduction

In this article we find all metrics g in 4 dimensions with vanishing Ricci tensor that satisfy the conditions

(i) g allows a Lie algebra of Killing vector fields with 2–dimensional orbits, (ii) the tensor g degenerates when restricted to the orbits, (iii) the distribution orthogonal to the orbits is not integrable. This is a particular case of the problem of finding all Ricci flat metrics with bidimensional Killing orbits. A systematical study of this problem was started in [6, 3], where the vacuum Einstein equations where solved for metrics of arbitrary signature under the assumption of integrability of the orthogonal to orbits distribution. A class of Lorentzian solutions with two commuting symmetries and a transversal and completely integrable orthogonal to orbits distribution was already found in the 70’s [4, 1, 2]. Here we prove that under conditions (i)-(iii) there exists only one, up to a sign, Ricci flat metric (theorem 2). This outcome differs drastically from the cases considered previously where one finds an abundance of solutions. In particular, our case does not admit any solutions with an abelian Killing algebra, nor of a Lorentzian signature. This fact suggests that condition (iii) is, in a sense, not very compatible with the Einstein equations. We stress that the results of this work and that of [3] give an exhausting exact descriptions of Ricci flat metrics that are subjected to conditions (i) and (ii). To solve the Einstein equations we work in a non-holonomic 𝔤-invariant frame field. This choice is motivated by the observation that 𝔤-invariant Ricci flat metrics are in one to one correspondence with Ricci flat metrics on the Lie algebroid of 𝔤-invariant vector fields, at least, for free actions. This Lie algebroid is associated to any smooth action on a manifold and forms a vector bundle over the space of orbits. Furthermore, the concept of curvature tensor makes sense for metrics on Lie algebroids. Overview. The article consist of two main sections. In the first of them metrics satisfying the conditions (i)-(iii) without the assumption of Ricci flatness are described. Here it is shown that the Killing algebras to be considered as generators of orbits are necessarily bidimensional. Further we describe the orthogonal to orbits distribution and give normal forms for the action of 𝔤 on D^{⊥}. Section 2 is dedicated to solving the Einstein equations. First, it is shown that in the case of an abelian 𝔤 these have no solutions. Then we treat the nonabelian case and give an exact description of all solutions in terms of the chosen frame by proving further that these are locally isometric to each other up to a sign. Finally, we compute the full Killing algebra of these metrics.

2000 Mathematics Subject Classification. 53C25, 53C50, 53C30.

Key words and phrases. Einstein metrics, Ricci flat metrics, Killing Algebra, null orbits, Kleinian metrics.

Notations and Conventions.

• All objects are assumed to be smooth, i.e., C^{∞}, M stands for a generic 4–fold we are working with and U ⊂ M for a generic open subset in M;

• By a metric g we understand a symmetric non-degenerate (0, 2) tensor of arbitrary (but constant) signature;

• The C^{∞}(M) -module of vector fields is denoted by D(M). The Lie derivative along a vector field X is denoted by X(f) on functions and L_{X}g on higher order tensors and i_{X}g is the insertion of X into g. Basic vector fields of a local chart x^{1}, …, x^{n} are denoted by ∂_{1}, …, ∂_{n}.

• The module of k-forms is denoted with Λ^{k}(M). Unless specified differently, a set of of forms ω_{1}, …, ω_{j} ∈ Λ^{1}(M) will be called linearly independent if they are independent over C^{∞}(M), i.e. if they are independent at every point. A set of functions x^{1}, …, x^{r} ∈ C^{∞}(U) will be called independent if their differentials are so.

• Abelian and nonabelian bidimensional Lie algebras are denoted by A_{2} and G_{2}, respectively.

• We use the term symmetry as a synonym for infinitesimal symmetry.

• The Lie algebra of all Killing symmetries of g is denoted with Kill(g). A subalgebra 𝔤 ⊂ Kill(g) will be called a Killing algebra of the metric.

• 𝔤^{(1)} denotes the first derived subalgebra of a Lie algebra: 𝔤^{(1)} = Span{[ξ, ζ] | ξ, ζ ∈ 𝔤};

• Distributions are denoted by calligraphic letters P that refer to the module of sections of the distribution. We say that forms ω_{1}, …, ω_{k} ∈ Λ^{1}(M) define or describe a distribution, if they generate the module Ann(P) of one-forms annihilating P;

• D denotes the distribution spanned by the vector fields of the Killing algebra 𝔤, D^{⊥} stands for the g–orthogonal to D distribution and C := D ∩ D^{⊥};

• The algebra of 𝔤-invariant functions is denoted by C^{∞}(M)^{𝔤} = {f ∈ C^{∞}(M) | X(f) = 0 ∀X ∈ 𝔤} and similarly for other invariant objects.

• Since most considerations of this paper are of local nature, we often omit to explicitly state arguments of the form: “after restriction to a possibly smaller open set”, etc.

1. Metrics subject to conditions (i)-(iii)

1.1. Preliminaries. Let M be a 4–dimensional manifold supplied with a metric g. We assume that g allows a Killing algebra 𝔤 ⊂ Kill(g) which spans a 2–dimensional distribution D. Obviously, D is Frobenius and its integral submanifolds are called the Killing leaves or, equivalently, orbits. We further assume that the bidimensional orthogonal distribution D^{⊥} := {X ∈ D(M) | g(X, Y) = 0 ∀Y ∈ D} is not integrable and that the intersection C := D ∩ D^{⊥} is nontrivial. This last condition is equivalent to saying that the metric becomes degenerate when restricted to the Killing leaves. Of course, this can only happen if g is of Lorentzian (− + ++) or Kleinian (+ + −−) signature. Almost everywhere, i.e., in an open dense subset of M, the dimension of C is 1 since an open region where D = D^{⊥} would imply that D^{⊥} is integrable. So, we shall throughout assume that dim C = 1. Finally, note that the distribution C^{⊥} is 3 dimensional and contains all the others: C^{⊥} = D + D^{⊥}. The following proposition shows that under our assumptions 𝔤 can only be 2– dimensional, i.e., 𝔤 is either A_{2} or G_{2}.

Proposition 1. Let g be a metric possessing a Killing algebra 𝔤 with bidimensional orbits. If dim 𝔤 > 2, then D^{⊥} is integrable.

See [6] for a proof.

1.2. The orthogonal distribution. The aim of this section is to bring to normal forms the action of 𝔤 on D^{⊥}. This will allow us to introduce frame fields adapted to the considered situation, which will be used in solving the Einstein equations. Some basic facts we need in this section concerning distributions, can be found in the book [5]. The first simple yet important observation is

Proposition 2. The distribution D^{⊥} is invariant under the action of 𝔤. In other words, the Killing algebra is a symmetry algebra of D^{⊥} as well.

Proof. Take A ∈ D^{⊥}, X, Y ∈ 𝔤 and use the standard formula for the Lie derivative

X(\underbrace{g(A, Y)}_{=0}) = \underbrace{L_{X}(g)}_{=0}(A, Y) + g([X, A], Y) + \underbrace{g(A, [X, Y])}_{=0}

Hence g([X, A], Y) = 0. Since D is spanned by fields Y ∈ 𝔤 this implies that [X, A] ∈ D^{⊥}. □

The next fact we need is that D^{⊥} is contained in a 3–dimensional integrable distribution. Namely, Proposition 3. The distribution C^{⊥} is integrable.

Proof. We can span C^{⊥} by two Killing fields from D and one field from D^{⊥}. Then by the previous proposition all their commutators are either in D or D^{⊥} and hence in C^{⊥}. □

In this connection it is convenient to introduce the following terminology.

Definition 1. A bidimensional non-integrable distribution P on a 4–fold will be called semi-integrable if it is contained in a 3–dimensional integrable distribution.

Remark 1. Such a 3–dimensional integrable distribution containing P is unique. Indeed, it must coincide with the distribution generated by P and all its commutators:

P^{(1)} := Span{X, [X, Y] | X, Y ∈ P}

Now we come to the main results of this section, which allow us to construct the adapted frames for metrics with (i)-(iii).

Proposition 4. Let M be a 4–dimensional manifold supplied with a 2–dimensional semi-integrable distribution P. Suppose A_{2} acts on M as a symmetry algebra of P. If the orbits of the action are bidimensional and run inside the leaves of P^{(1)}, then for a generic point there exists a local chart x, u, p, z ∈ C^{∞}(U) in terms of which P is described by

dz, du − pdx ∈ Ann(P)

while the symmetries are

∂_{u}, ∂_{x} ∈ A_{2}

In the proof we shall use the following simple

Lemma 1. If P is a bidimensional, non-integrable distribution and S ∈ D(M) a nontrivial symmetry of it, then the set of points where S lies outside of P is open and dense.

Proof. Otherwise there would be a region where S is a characteristic symmetry, but a bidimensional distribution with a nontrivial characteristic symmetry is integrable. □

Now we pass to prove proposition 4

Proof. Fix a basis S_{1}, S_{2} ∈ A_{2}. Since P^{(1)} is integrable we can locally choose a function z ∈ C^{∞}(U) so that P^{(1)} is given by dz = 0. This way one finds the required coordinate function z which is unique up to transformation z ↦ f(z). Choose then any 1–form θ′ ∈ Ann(P) which together with dz generates Ann(P) and scale it so that

i_{S_{1}}θ′ = 1.

This scaling is generically possible by lemma 1 and since θ′ is linearly independent of dz. Note that this choice of θ′ comes with the freedom of adding a term f · dz with arbitrary f ∈ C^{∞}(U). Making use of it we change θ′ to a form θ such that

L_{S_{i}}(θ) = 0, i = 1, 2

Indeed, since S_{1}, S_{2} are symmetries of P we have L_{S_{i}}θ′ ∈ Ann(P), i.e.,

L_{S_{i}}θ′ = a_{i}θ′ + b_{i}dz, i = 1, 2

with a_{i}, b_{i} ∈ C^{∞}(U). The standard formula i_{S_{1}} ∘ L_{S_{i}} = L_{S_{i}} ∘ i_{S_{1}} − i_{[S_{i},S_{1}]} applied to θ′ shows that a_{i} = 0 or in other words, L_{S_{i}}θ′ = b_{i}dz. Since S_{1}, S_{2} commute we also have L_{S_{2}}L_{S_{1}}θ′ = L_{S_{1}}L_{S_{2}}θ′, implying the “compatibility condition” S_{1}(b_{2}) = S_{2}(b_{1}) for the overdetermined system

S_{1}(f) = −b_{1}, S_{2}(f) = −b_{2}.

Hence this system admits a solution f and we get the desired result with θ = θ′ + fdz. Since P is non-integrable, dθ ≠ 0. Moreover, it follows from L_{S_{1}}(θ) = 0 and i_{S_{1}}(θ) = 1 that i_{S_{1}}(dθ) = 0. In other words, S_{1} ∈ ker(dθ) and hence ker(dθ) ≠ 0. Since the rank of the 2–form dθ is an even integer ≤ 4 we conclude that it is exactly 2. Put now p := −i_{S_{2}}(θ). Then

0 = L_{S_{2}}(θ) = d(i_{S_{2}}θ) + i_{S_{2}}(dθ) ⇔ i_{S_{2}}(dθ) = dp

and

S_{2}(p) = i_{S_{2}}(dp) = i_{S_{2}}(i_{S_{2}}(dθ)) = 0.

Similarly,

S_{1}(p) = i_{S_{1}}(i_{S_{2}}(dθ)) = −i_{S_{2}}(i_{S_{1}}(dθ)) = 0.

Thus S_{i}(p) = 0, i = 1, 2. Now consider a (local) function x ∈ C^{∞}(U) such that S_{2}(x) = 1, and which is constant along the 2–dimensional leaves of ker(dθ). To find such a function note that S_{2} is a symmetry of the integrable distribution ker(dθ) and hence, is related to a vector field S̃_{2} on the quotient manifold U/ ker(dθ). We may then take x as the pullback of a function x̃ on U/ ker(dθ) with S̃_{2}(x̃) = 1. For ω := −pdx it follows that ker(dω) = ker(dθ) and therefore, dω = λdθ, λ ∈ C^{∞}(U). Using previous relations it is further easily seen that

i_{S_{2}}(dω) = dp = i_{S_{2}}(dθ)

which implies λ = 1. Thus dω = dθ and, so, θ = ω + du for a function u ∈ C^{∞}(U). Functions x, u, p, z constructed above form, obviously, a local chart on M and it remains to show that S_{1} = ∂_{u} and S_{2} = ∂_{x}. But 1 = i_{S_{1}}(θ) = S_{1}(u) + i_{S_{1}}(ω) = S_{1}(u). This says that the u–component of S_{1} is equal to 1. Other components vanish because of already established relations S_{1}(x) = S_{1}(p) = S_{1}(z) = 0. This proves that S_{1} = ∂_{u} and, similarly, that S_{2} = ∂_{x}. □

The analog of proposition 4 in the nonabelian case is

Proposition 5. Let M be a 4 dimensional manifold supplied with a 2–dimensional semi-integrable distribution P. Suppose G_{2} acts on M as a symmetry algebra of P. If the orbits of the action are bidimensional and run inside the leaves of P^{(1)}, then for a generic point there exists a local chart x, u, p, z ∈ C^{∞}(U) in terms of which P is described by

dz, du − pdx ∈ Ann(P)

while the symmetries are

∂_{u}, u∂_{u} + p∂_{p} ∈ G_{2}

Proof. Fix a basis S_{1}, S_{2} of G_{2} with the property [S_{1}, S_{2}] = S_{1}. The coordinate z and an 1–form θ′ such that i_{S_{1}}(θ′) = 1 and L_{S_{1}}(θ′) = 0 are constructed exactly as in the previous case. Since dz and θ′ generate Ann(P)

L_{S_{2}}(θ′) = aθ′ + bdz, a, b ∈ C^{∞}(U),

The identity i_{S_{1}}∘L_{S_{2}} = L_{S_{2}}∘i_{S_{1}}+i_{[S_{1},S_{2}]} applied to θ′ shows that a = 1 while [S_{1}, S_{2}] = S_{1} gives S_{1}(b) = 0. Now construct a form θ = θ′ + fdz such that

L_{S_{1}}(θ) = 0, L_{S_{2}}(θ) = θ.

This is, obviously, equivalent to

S_{1}(f) = 0, S_{2}(f) = f − b.

As it is easy to see, the compatibility condition for this system is exactly S_{1}(b) = 0 and, so, it possesses solutions. Hence the required form θ exists. Also note that by construction i_{S_{1}}θ = 1. Put

u := i_{S_{2}}θ

The identities i_{S_{2}} ∘ L_{S_{1}} = L_{S_{1}} ∘ i_{S_{2}} + i_{[S_{1},S_{2}]} and i_{S_{2}} ∘ L_{S_{2}} = L_{S_{2}} ∘ i_{S_{2}} applied to θ give

S_{1}(u) = 1, S_{2}(u) = u ⇔ i_{S_{1}}du = 1, i_{S_{2}}du = u.

This shows that du and dz are independent. Consider now the non-closed form

α := du − θ = di_{S_{2}}(θ) − L_{S_{2}}(θ) = −i_{S_{2}}(dθ).

By the same arguments as in the proof of the previous proposition one can see that dθ has rank 2. Hence dθ ∧ dθ = 0 and

dα ∧ α = dθ ∧ i_{S_{2}}dθ = \frac{1}{2}i_{S_{2}}(dθ ∧ dθ) = 0

which means that ker(α) is integrable. Hence there exist (locally) two independent functions x, p such that α = pdx and, so, α = du − pdx. Functions x, u, p restricted to a generic hypersurface z = const are independent. Indeed, otherwise the distribution P would be integrable. This also shows that functions x, u, p, z are (locally) independent. Finally, observe that

i_{S_{i}}(α) = 0 ⇔ S_{i}(x) = 0, i = 1, 2.

Now the relations i_{S_{1}}dα = 0 and i_{S_{2}}dα = α together with dα = dp∧dx and the previous equation yield

S_{1}(p) = 0, S_{2}(p) = p

Together with previous relations this shows that S_{1} = ∂_{u} and S_{2} = u∂_{u} + p∂_{p}. □

2. Solving the Einstein equations

2.1. The abelian case. In this section we show that under assumptions (i)-(iii) there are no Ricci flat metrics in the case 𝔤 = A_{2}. Consider the chart x, u, p, z constructed in proposition 4. In terms of this chart Killing fields in question are ∂_{x} and ∂_{u}, while the orthogonal distribution D^{⊥} is spanned by ∂_{p} and ∂_{x} + p∂_{u}. In the following non-holonomic frame field:

e_{1} := ∂_{u} ∈ D
e_{2} := ∂_{p} ∈ D^{⊥}
e_{3} := ∂_{x} + p∂_{u} ∈ C
e_{4} := ∂_{z}

the matrix g_{ij} := g(e_{i}, e_{j}) is of the form

( \begin{matrix}a & 0 & 0 & s_{1} \\ 0 & b & 0 & s_{2} \\ 0 & 0 & 0 & s_{3} \\ s_{1} & s_{2} & s_{3} & s_{4} \end{matrix} )  (1)

with a, b, s_{1}, s_{2}, s_{3}, s_{4} ∈ C^{∞}(U). Since the chosen frame consists of 𝔤–invariant fields all the components are 𝔤–invariant as well and so, depend only on (p, z). Since

det(g_{ij}) = −ab(s_{3})^{2},

non-degeneracy of the metric requires that functions a, b and s_{3} be everywhere nonzero. Now we compute the component ric(e_{3}, e_{3}) of the Ricci tensor. Recall that for a frame e_{1}, …, e_{4} ∈ D(U) the corresponding components of the Ricci tensor are given by:

ric(e_{i}, e_{j}) = e_{j}(γ_{hi}^{h}) − e_{h}(γ_{ji}^{h}) + γ_{jk}^{h}γ_{hi}^{k} − γ_{hk}^{h}γ_{ji}^{k} − c_{jh}^{k}γ_{ki}^{h}

(summation over crossed indices is understood). Here γ_{ij}^{l}’s are the Christoffel symbols defined by ∇_{e_{i}}e_{j} = γ_{ij}^{l}e_{l} and as it is well-known

γ_{ij}^{l} = \frac{1}{2}g^{lh}(−e_{h}(g_{ij}) + e_{i}(g_{hj}) + e_{j}(g_{hi}))

−\frac{1}{2}(c_{ji}^{l} + g^{kl}g_{hi}c_{jk}^{h} + g^{kl}g_{hj}c_{ik}^{h})

with the c_{ij}^{h}’s being structure “constants”, i.e., [e_{i}, e_{j}] = c_{ij}^{h}e_{h}.

For the considered frame the only non-vanishing commutator is [e_{2}, e_{3}] = e_{1} and a straightforward computation gives

ric(e_{3}, e_{3}) = \frac{a}{2b}

But since the function a does not vanish we have

Theorem 1. There are no Ricci flat metrics satisfying conditions (i)-(iii) for an abelian Lie algebra 𝔤.

2.2. The nonabelian case. In the local chart x, u, p, z of proposition 5 the symmetries are S_{1} = ∂_{u}, S_{2} = u∂_{u} + p∂_{p} and the orthogonal distribution is spanned by ∂_{x} + p∂_{u} and ∂_{p}. As before it is convenient to pass to a 𝔤–invariant frame:

e_{1} := p∂_{u} ∈ D
e_{2} := ∂_{x} + p∂_{u} ∈ D^{⊥}
e_{3} := p∂_{p} ∈ C
e_{4} := ∂_{z}

As before the metric matrix in this frame is of the form (1) with functions a, b, s_{1}, s_{2}, s_{3}, s_{4} ∈ C^{∞}(M)^{G_{2}}, depending on (x, z) only. In the nonabelian case there is, additionally, an 1–dimensional distribution, namely, that generated by 𝔤^{(1)}, i.e., by ∂_{u} in the considered chart. Denote it by \overline{D}. A peculiarity of this frame is that e_{1} ∈ \overline{D}. The only nonzero commutators are:

[e_{3}, e_{2}] = e_{1}, [e_{3}, e_{1}] = e_{1}  (2)

Summing up we have

Corollary 1. In the nonabelian case there exists a G_{2}-invariant frame e_{1}, e_{2}, e_{3}, e_{4} ∈ D(M) with e_{1} ∈ \overline{D}, e_{2} ∈ D^{⊥} and e_{3} ∈ C, that satisfies commutation relations (2)

Such a frame is not unique as the following proposition shows.

Proposition 6. Frames described in corollary (1) are related one to another by a ”gauge transformation”

e_{1} ↦ f · e_{1}
e_{2} ↦ f · e_{2} − e_{2}(f) · e_{3}
e_{3} ↦ e_{3}
e_{4} ↦ α · e_{4} + β · (e_{1} − e_{2}) + e_{2}(β) · e_{3}

where f is a nowhere vanishing G_{2}-invariant function, α is a nowhere vanishing function which is constant along the 3–dimensional leaves of C^{⊥} and β ∈ C^{∞}(M)^{G_{2}} is a solution of the first order differential equation

e_{2}(β) = \frac{1}{f} (βe_{2}(f) − αe_{4}(f)).

Proof. The proof is straightforward if one proceeds by passing from one frame to another in the order: e_{1}, e_{3}, e_{2}, e_{4} and taking into account that any G_{2}-invariant frame consists of linear combinations of e_{1}, …, e_{4} with coefficients in C^{∞}(M)^{G_{2}}. □

We shall now solve the Einstein equations in various steps. Step 1. First it is convenient, using the gauge freedom, to pass to a new frame, say X_{1}, X_{2}, X_{3}, X_{4}, such that g(X_{1}, X_{1}) = ±1. This is possible by proposition 6, since a is G_{2}-invariant and nowhere zero. In such a frame the metric components take the form

( \begin{matrix}ϵ & 0 & 0 & t_{1} \\ 0 & c & 0 & t_{2} \\ 0 & 0 & 0 & t_{3} \\ t_{1} & t_{2} & t_{3} & t_{4} \end{matrix} )

with ϵ = ±1 and some new components c, t_{1}, …, t_{4} which are G_{2}-invariant. Step 2. Put R_{(i)(j)} := ric(X_{i}, X_{j}). A direct computation then gives:

R_{(3)(3)} = \frac{2c + ϵ}{2c}.

So, for a Ricci flat metric c = −\frac{ϵ}{2}. Step 3. Taking into account the previous result we obtain

R_{(3)(1)} = \frac{t_{1} − 2t_{2}}{t_{3}}

and t_{1} = 2t_{2} for Ricci flat metrics. Step 4. From

R_{(3)(2)} = \frac{X_{2}(t_{3})}{2t_{3}}

we subsequently get X_{2}(t_{3}) = 0 which means that t_{3} is constant along the 3–dimensional leaves of C^{⊥}. Step 5. Next

R_{(3)(4)} = −ϵX_{2}(t_{2})

which means that t_{2} is also constant along C^{⊥}. Step 6. Finally,

R_{(1)(2)} = \frac{2t_{2}^{2} − ϵt_{4}}{t_{3}^{2}}

allows us to eliminate t_{4}. Namely, t_{4} = 2ϵt_{2}^{2}. Up to this point the metric is of the form

( \begin{matrix}ϵ & 0 & 0 & 2t_{2} \\ 0 & −\frac{ϵ}{2} & 0 & t_{2} \\ 0 & 0 & 0 & t_{3} \\ 2t_{2} & t_{2} & t_{3} & ϵ2t_{2}^{2} \end{matrix} )

with two arbitrary functions t_{2}, t_{3} which are constant along C^{⊥} and t_{3} ≠ 0. Any metric of this form is Ricci flat, but we can still use a remaining gauge freedom in X_{4} to eliminate t_{2}, t_{3}. This is done by changing X_{4} to

X̃_{4} = \frac{ϵ}{t_{3}} (X_{4} − 2ϵt_{2}(X_{1} − X_{2}))

and leaving X_{1}, X_{2}, X_{3} intact. It is easily seen that this field commutes with the first three, and in this final frame the metric assumes the form

ϵ · ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −\frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ) , ϵ = ±1  (3)

The frame is now completely fixed up to a sign of X_{1}, X_{2}. Thus we have proven

Theorem 2. Let g be a Ricci flat metric that possesses a symmetry algebra 𝔤 ⊂ Kill(g) satisfying properties (i)-(iii). Then 𝔤 is the 2–dimensional nonabelian Lie algebra G_{2} and there exist (locally) a non-holonomic frame field X_{1}, X_{2}, X_{3}, X_{4} satisfying conditions of corollary 1 in which the components of the metric are that of (3). Such a frame is unique up to the signs of X_{1} and X_{2}.

Corollary 2. Two Ricci flat metrics g_{1}, g_{2} satisfying conditions (i)-(iii), are either locally isomorphic or g_{1} ≃ −g_{2}.

This is an immediate consequence of theorem 2 and

Lemma 2. Given two frame fields e_{1}, …, e_{4} and ẽ_{1}, …, ẽ_{4} satisfying commutation relations (2), then there is a local diffeomorphism of the underlying manifold which sends one of them to another.

Proof. A straightforward application of Lie’s third theorem. □

2.2.1. The full Killing algebra of the solutions. Henceforth we shall denote a frame in which g assumes the form (3) by X_{1}, …, X_{4}. We may introduce coordinates x^{1}, …, x^{4} in which

\begin{matrix}X_{1} = exp(x^{3})∂_{1} & X_{3} = ∂_{3} \\ X_{2} = ∂_{2} + exp(x^{3})∂_{1} & X_{4} = ∂_{4} \end{matrix}

Note also that in this chart the vector fields:

\begin{matrix}V_{1} := ∂_{1} & V_{3} := ∂_{3} + x^{1}∂_{1} \\ V_{2} := ∂_{2} & V_{4} := ∂_{4} \end{matrix}

are symmetries of the frame X_{1}, …, X_{4}, and since the metric has constant coefficients in that frame the fields V_{1}, …, V_{4} are also symmetries of g. Moreover, V_{1}, …, V_{4} form an transitively acting Lie algebra which is isomorphic to G_{2} ⊕ A_{2}. So, we have proven

Corollary 3. The solutions described in theorem 2 are homogeneous metrics.

Furthermore, we have

Proposition 7. The full Killing algebra of the metrics (3) is 6–dimensional. Its commutation relations are given below.

Proof. To find further symmetries of g we take V_{1}, …, V_{4} as a frame and present vector fields on M in the form Z = f^{i}V_{i} with f^{i} ∈ C^{∞}(M), i = 1, …, 4. Using the fact that V_{i}’s are symmetries of g and the formula

L_{fX}(g) = fL_{X}(g) + i_{X}(g)df

where the last term is a symmetric product of one forms, the condition that Z is a symmetry becomes:

L_{Z}(g) = i_{V_{i}}(g) · df^{i} = 0.

This equality is equivalent to

df^{i} = Λ^{ij} · i_{V_{j}}(g)  (4)

with some functions Λ^{ij}, i, j = 1, …, 4, that are skew symmetric in the indices, Λ^{ij} = −Λ^{ji}. It follows from equation (4) that

dΛ^{ij} ∧ ω_{j} + Λ^{ij}dω_{j} = 0  (5)

with ω_{j} := i_{V_{j}}(g). Equations (5) form a system of 24 first order linear differential equations for the 6 unknown functions Λ^{ij}, i < j. The trivial solution Λ^{ij} = 0 corresponds to the already known symmetries V_{j}. In the chart (x^{1}, …, x^{4}) as above, these equations can be easily solved and the general solution is

Λ^{1,4} = c_{1}e^{x^{3}} + c_{2}e^{2x^{3}}, Λ^{2,4} = c_{1} + 2c_{2}e^{x^{3}}

with vanishing remaining components. Here c_{1}, c_{2} are arbitrary constants corresponding to two “new” symmetries:

V_{5} := e^{x^{3}}∂_{1} + x^{3}∂_{2} + \frac{1}{2}x^{2}∂_{4}, c_{1} = 1, c_{2} = 0,

and

V_{6} := \frac{1}{2}e^{2x^{3}}∂_{1} + 2e^{x^{3}}∂_{2} + x_{1}∂_{4}, c_{1} = 0, c_{2} = 1

Thus the Killing algebra of the metrics (3) is spanned by V_{1}, … V_{6} whose commutation relations are

[V_{1}, V_{3}] = V_{1}, [V_{2}, V_{5}] = \frac{1}{2}V_{4}, [V_{3}, V_{5}] = V_{2}, [V_{1}, V_{6}] = V_{4}, [V_{3}, V_{6}] = V_{6}

with all other commutators vanishing. The center of this algebra is spanned by V_{4}, the first derived subalgebra is spanned by V_{1}, V_{2}, V_{4}, V_{6} and the second one coincides with the center. □

Acknowledgement. I would like to thank A.M. Vinogradov for many useful discussions and suggestions.

References

[1] V.A. Belinsky, V.E. Zakharov, Integration of the Einstein equations by means of the inverse scattering technique and construction of exact solutions, Soviet Phys. JETP 48 (6) (1978).

[2] V.A. Belinsky, V.E. Zakharov, Stationary gravitational solitons with axial symmetry, Soviet Phys. JETP 50 (1) (1979).

[3] D. Catalano Ferraioli, A.M. Vinogradov, Ricci flat 4-metrics with bidimensional null orbits Parts I and II, Diffiety Institute Preprint Archive, DIPS 7/2004.

http://diffiety.ac.ru/preprint/preprint.htm

[4] R. Geroch, A method for generating new solutions of Einsteins equation Parts I and II J. Math. Phys. 12 and 13. (1971).

[5] I. S. Krasil’shchik, A.M. Vinogradov et al., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Translations of Mathematical Monographs, Vol. 182. Amer. Math. Soc., Providence, RI. 1999

[6] G. Sparano, G. Vilasi, A.M. Vinogradov, Vacuum Einstein metrics with bidimensional Killing leaves Parts I and II, Differential Geom. Appl. 16 and 17. (2002).

Mathematik Departement, ETH, Zürich, Switzerland

E-mail address: bamichae@student.ethz.ch