We construct new examples of four-dimensional Einstein metrics with neutral signature and two-dimensional holonomy Lie algebra.

1. Introduction

The holonomy group of a metric g at a point p of a manifold M is the group of all linear transformations in the tangent space of p defined by parallel translation along all possible loops starting at p [1]. It is obvious that a connection can only be the Levi-Civita connection of a metric g if the holonomy group is a subgroup of the generalized orthogonal group corresponding to the signature of g [1–3]. At any point p∈M, and in some coordinate system about p, the set of matrices of the form

RjkliXkYl,Rjkl;miXkYlZm,Rjkl;mniXkYlZmWn,…,
where X,Y,Z,W∈TpM and semicolon denotes covariant derivative, forms a Lie subalgebra of the Lie algebra of Mn(ℝ) of GL(n,ℝ) called the infinitesimal holonomy algebra of M at p. Up to isomorphism the latter is independent of the coordinate system chosen. The corresponding uniquely determined connected subgroup of GL(n,ℝ) is called the infinitesimal holonomy group of M at p.

A metric tensor g is a nondegenerate symmetric bilinear form on each tangent space TpM for all p∈M. The signature of a metric g is the number of positive and negative eigenvalues of the metric g. The signature is denoted by an ordered pair of positive integers (p,q), where p is the number of positive eigenvalues and q is the number of negative eigenvalues. If p=q, we say that the metric is of neutral signature. In this article, we are interested in four-dimensional metrics with neutral signature.

If the metric g satisfies the condition

Rij=R4gij,
where Rij are the components of the Ricci tensor and R is the scalar curvature, then we say that g is an Einstein metric and the pair (M,g) is an Einstein space.

In [4], Ghanam and Thompson studied and classified the holonomy Lie subalgebras of neutral metrics in dimension four. In this paper, we will focus on one of the subalgebras presented in [4], namely, A17. For this subalgebra we will show that the metric presented in [4] will lead us to the construction of Einstein metrics. In Section 3, we will give the metrics explicitly, and in Section 4, we will show that these Einstein metrics produce A17 at their two-dimensional holonomy.

As a final remark regarding our notation, we will use subscripts for partial derivatives. For example, the partial derivative of a function a with respect to x will be denoted by ax.

2. The Subalgebra <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M40"><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>17</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> As a Holonomy

In this section we will consider the Lie algebra A17; it is a 2-dimensional Lie subalgebra of the Lie algebra of the generalized orthogonal group O(2,2) of neutral signature [4, 5]. A basis for A17 is given by

e1=[-JJ-JJ],e2=[JLLJ],
where

J=[01-10],L=[0110].

We turn now to a theorem of Walker [6] that will be a key to the existence of a metric g that produces A17 as a two-dimensional holonomy.

Let (M,g) be a pseudo-Riemannian manifold of class C∞. If g admits a parallel, null r-distribution, then there is a system of coordinates (xi) relative to which g assumes the following form:
gij=[00I0AHIHtB],
where I is the r×r identity matrix and A,B,H, and Ht are matrix functions of the same class as M, satisfying the following conditions but otherwise arbitrary.

A and B are symmetric; A is of order (n-2r)×(n-2r) and nonsingular, B is of order r×r, H is of order (n-2r)×r, and Ht is the transpose of H.

A and H are independent of the coordinates x1,x2,…,xr.

Now we show that A17 is a holonomy Lie algebra of a four-dimensional neutral metric.

Proposition 2.2.

A17 is a holonomy algebra.

Proof.

In this case, we have an invariant null 2-distribution, and so by Walker's theorem, there exists a coordinate system, say (x,y,z,w), such that the metric g is of the form
g=[0010000110ac01cb],
where a,b,c are smooth functions in (x,y,z,w). Since the invariant distribution contains a parallel null vector field, we must have
ax=bx=cx=0.

It was shown in [4] that, in order for g to produce A17 as its holonomy algebra, the functions a,b, and c must satisfy the following conditions:
byy=0,cyy=0,cyw-byz=0.

3. New Einstein Metrics

In Section 2, we obtained a metric of the form

g=[0010000110ac01cb],
where a,b, and c are smooth functions in y,z,w and they satisfy (2.5) and (2.6). We solve these conditions to obtain

b(y,z,w)=m(z,w)y+n(z,w),c(y,z,w)=r(z,w)y+s(z,w)
with

rw=mz,
where m,n,r,s are smooth functions in z,w.

The nonzero components of the Ricci tensor for g are

R33=-y2mayy-12nayy+ayw-rz-12may+12r2,R34=12(rw-mz).
The Ricci scalar is

R=0.
Because of (3.3), we obtain

R34=0.
In this case, the Einstein condition Rij=(R/4)gij becomes

Rij=0.
Hence, in order to have an Einstein metric, we must have

R33=0,
and so we obtain the following partial differential equation (PDE):

-y2mayy-12nayy+ayw-rz-12may+12r2=0.
Since we are interested in finding at least one solution, we take the following special values in (3.9):

m=1,n=0,r=0,s=0
to obtain the following PDE:

yayy-2ayw+ay=0.
To solve (3.11), we use the method of separation. For example, assume that a(y,z,w) is of the form

a(y,z,w)=f(y)g(z)h(w),
where f,g, and h are smooth functions in y,z, and w, respectively. We substitute (3.12) in (3.11) to obtain

yf′′(y)g(z)h(w)-2f′(y)g(z)h′(w)-f′(y)g(z)h(w)=0.
We assume that g(z) is nowhere zero to obtain

yf′′(y)h(w)-2f′(y)h′(w)-f′(y)h(w)=0,
and so

yf′′(y)h(w)-f′(y)(2h′(w)-h(w))=0.
Hence

yf′′(y)h(w)=f′(y)(2h′(w)-h(w)).
Dividing both sides by h(w)f'(y) gives

yf′′(y)f′(y)=2h′(w)-h(w)h(w)=c,
where c is a constant.

We now solve (3.17) and for that we will consider three cases.

If c=0, then f is a linear function given by
f(y)=c1y+c2,
and the condition on h becomes
2h'-h=0,
which gives
h(w)=c3ew/2.
The solution a(y,z,w) to the PDE equation (3.11) is
a(y,z,w)=(c1y+c2)c4ew/2g(z)=(c1y+c2)ew/2g(z),
where c1,c2 are constants and g(z) is a smooth nowhere zero function.

If c≠0,-1, then the differential equations (3.17) become
yf′′-cf′=0,2h′-(1+c)h=0.
The solutions to (3.22) are
f(y)=c1yc+1+c2,h(w)=c3e(1+c)w/2,
and so
a(y,z,w)=(c1yc+1+c2)e(1+c)w/2g(z),
where g(z) is a no-where zero smooth function in z.

If c=-1, then the differential equations (3.17) become
yf′′+f′=0,h′=0.
The solutions to (3.25) are
f(y)=c1ln(y)+c2,h(w)=c3,
and so
a(y,z,w)=(c1ln(y)+c2)c3g(z)=(c1ln(y)+c2)g(z),
where c1,c2 are constants and g(z) is a no-where zero smooth function in z.

4. The Holonomy of the New Metrics

In this section we compute the infinitesimal holonomy algebra and make sure that it produces a two-dimensional algebra. To do so, we consider our metric g given by

g=[0010000110a(y,z,w)0010y].
The only nonzero components of the curvature are

R2323=-12ayy,R2334=12awy-14ay,R3434=-12aww+y4ay-14aw.
The holonomy matrices are

Rij23=[000000-12ayy00-12ayy000000],Rij34[00000012awy-14ay0012awy-14ay0-12aww+y4ay-14aw00-12aww+y4ay-14aw0],
Now, in order for the metric to produce two-dimensional holonomy, we must have

ayy≠0,-12aww+y4ay-14aw≠0.

We have to check these equations for the three cases discussed in Section 3.

We consider
a(y,z,w)=(c1y+c2)ew/2g(z).
Then
ayy=0,
and we obtain a one-dimensional holonomy. Therefore we must exclude this case.

We consider
a(y,z,w)=(c1yc+1+c2)e(1+c)w/2g(z),
where c≠0,-1. In this case
ayy=c1c(c+1)yc-1e(1+c)w/2g(z)≠0.
The second condition is that (4.5) gives
e(c+1)w/28(2c1yc+1c2+5c1yc+1c+3c1yc+1+2c2c2+3c2c+c2)≠0.
This shows that the metric we constructed in Section 3 is an Einstein metric with a two-dimensional holonomy. In fact, its holonomy Lie algebra is A17.

We consider
a(y,z,w)=(c1ln(y)+c2)g(z).
In this case the first condition is that (4.4) gives
ayy=-c1g(z)y2≠0,
and the second condition is that (4.5) gives
c1g(z)4≠0.
And once again we obtain an Einstein metric with A17 as its two-dimensional holonomy Lie algebra.

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