Killing Vector Fields of Invariant Metrics

Abstract

We study the existence of Killing vector fields for right-invariant metrics on low-dimensional Lie groups. Specifically, Lie groups of dimension two, three and four are considered. Before attempting to implement the differential conditions that comprise Killing’s equations, the metric is reduced as much as possible by using the automorphism group of the Lie algebra. After revisiting the classification of the low-dimensional Lie algebras, we review some of the known results about Killing vector fields on Lie groups and add some new observations. Then we investigate indecomposable Lie algebras and attempt to solve Killing’s equations for each reduced metric. We introduce a matrix $MM$, that results from the integrability conditions of Killing’s equations. For $n=4$, the matrix $MM$ is of size $20\times 6$. In the case where $MM$ has maximal rank, for the Lie group problem considered in this article, only the left-invariant vector fields are Killing. The solution of Killing’s equations is performed by using MAPLE, and knowledge of the rank of $MM$ can help to confirm that the solutions found by MAPLE are the only linearly independent solutions. After finding a maximal set of linearly independent solutions, the Lie algebra that they generate is identified to one in a standard list.

1. Introduction

In this paper we are concerned with the existence of Killing vector fields for right-invariant metrics on Lie groups of dimension two, three, and four. It is known that any left-invariant vector field is automatically Killing for a right-invariant metric. The question is whether additional Killing vector fields exist. We start from the most general right-invariant such metric and use the automorphism group of the Lie algebra to reduce the metric. We shall also incorporate an overall scaling: it is important to reduce the metric as much as possible if there is to be any hope of implementing the theory developed in Section 4 below.

The three-dimensional case has already been done [1], but we redo it here, incorporating the reduction of the metric just mentioned. The larger the automorphism group is, the more possible it is to reduce the metric. Another issue, is that some of the Lie algebras concerned belong to continuous families depending on one or two parameters, which complicates the search for extra Killing vector fields. As the dimension of the Lie group increases, the dimension of its automorphism group is small in comparison to the number of parameters in the metric, so that a full normalization is possible only in the small dimensions studied in this paper. We succeed in uncovering a number of metrics that have one or more extra Killing vector fields. In each case the corresponding Lie algebra of Killing vector fields is found and identified, to the extent possible, as one on a standard list. After reducing a general right-invariant metric, we are left with what might be referred to as a “moduli” space of metrics, which could form a topic for further investigation.

A novel feature of this work, is that we are able to use the first level of integrability conditions to give a bound on the dimension of the space of Killing vector fields. We shall consistently denote this dimension by $\sigma$. The conditions involved in integrating Killing’s equations are far too complicated to do by hand, so we use the symbolic manipulation program MAPLE. However, in order to be sure that MAPLE has not missed any potential solutions, or perhaps has not given any solutions at all, it is vital to the use the extra integrability conditions.

The structure of the paper is outlined as follows. Section 2 of the paper gives a brief derivation of the indecomposable Lie algebras in dimensions two, three and four, particularly as they are used in this article. Concerning the Lie algebras occuring in this text, we have of course all the three and four dimensional Lie algebras themselves. All such algebras, in fact up to dimension six, are well documented in [2], for example. In [2], low-dimensional Lie algebras are denoted as $A_{i,j}$, where i is the dimension of the Lie algebra and j is the jth algebra in the list. Beyond the three and four dimensional cases, we shall also have occasion to refer to several simple Lie algebras for which the notation is more or less standard [3]. Section 3 provides some established theoretical results relevant to the existence of Killing vector fields and are not necessarily restricted to the context of invariant metrics. In this Section, a not necessarily right-invariant Riemannian metric is denoted by B, whereas we use g when we begin the case by case calculations. Section 4 starts from the definition of a Killing vector field for a Riemannian metric. In point of fact, it is much more convenient to work with the one-forms dual via the metric to the Killing vector fields. Some classical results and additional integrability conditions are derived. In Section 4, we use the classical tensor calculus including the summation convention on repeated indices. It is true that such notation may be considered to be archaic and to lack elegance, but a more sophisticated version would take us too far afield and unnecessarily lengthen the paper. In Section 5 we exhibit a worked example in dimension four where all the details are provided. We will usually be working with a constant positive definite matrix relative to a right-invariant coframe of one-forms and we shall refer frequently to “the matrix of the metric”. Section 6, Section 7 and Section 8 are devoted to each of the indecomposable Lie algebras in dimensions two, three and four, respectively. In each case we supply a Lie group representation and a basis for the left and right-invariant vector fields and one-forms. We also give the Lie algebra of derivations. We attempt, to the extent possible, to give cases in terms of increasing specialization, so that the more special the metric, in terms of restrictions on parameters in the Lie algebra or entries in the metric, the more Killing vector fields there are. Finally Section 9 offers a few concluding remarks that attempt to put some of the results obtained into context.

In Section 4 we shall introduce two matrices denoted by M and $MM$, respectively. The matrix M is the matrix of coefficients for unknowns $K_{i;j}, K_i$, the components of an unknown Killing form and its covariant derivatives. In n dimensions M will be a ${\displaystyle \frac{n^2(n^2-1)}{12}}\times {\displaystyle \frac{n(n+1)}{2}}$ matrix. A special feature that applies to a right-invariant metric on a Lie group is that each left-invariant vector fields is Killing. Accordingly, there must exist n linear relations that express columns ${\displaystyle \frac{n(n-1)}{2}}+1,\dots ,{\displaystyle \frac{n(n+1)}{2}}$ in terms of columns $1,\dots ,{\displaystyle \frac{n(n-1)}{2}}$ in the matrix M. Hence we can work with the submatrix $MM$ consisting of the first ${\displaystyle \frac{n(n-1)}{2}}$ columns of M instead of M itself. For our right-invariant metric, we can assert that if $MM$ has rank ${\displaystyle \frac{n(n-1)}{2}}$, then only the left-invariant vector fields will be Killing. The matrices M and $MM$ will continually be referred to in Section 8.

The idea then is to use matrix $MM$ to obtain an upper bound on $\sigma$. Although this idea sounds simple, it is usually far from simple to implement in practice. Even for the case $n=4$, the components of $MM$ are comprised of polynomials of degree three, four or five in the free entries of the metric as well as the parameters that occur in the Lie algebra under consideration. The hope is to be able to compute determinants of certain submatrices of $MM$ that must vanish if the rank of $MM$ is to be less than ${\displaystyle \frac{n(n-1)}{2}}$. Even for $n=4$, it is difficult to obtain a completely comprehensive analysis. In addition, it is challenging in terms of exposition even to provide details. In this article, we have attempted to follow a middle course, providing a few details to the extent possible. The reader will appreciate some of the difficulties involved, in, for example, cases $A_{4.3}$ and $A_{4.4}$ in Section 8. One might say that one has to work very hard only to achieve a null result. On the other hand, if we are willing to take the matrix of the metric in diagonal form, or even just a sum of squares of the right-invariant forms, then the matrices M and $MM$ are relatively easy to work with and lead to definitive results. Such a form of the metric may be expected to exhibit the maximal degree of symmetry but we have no general proof of such a result. For example, in the case of $A_{4.1}$, in Section 5, the Lie algebra contains no parameter, and we are able to reduce the metric so that it depends on just two parameters. In the case of $A_{4.2}$ we have to contend with three parameters in the metric and one in the Lie algebra, which makes a complete analysis very difficult.

The notion of a Killing symmetry lies at the nexus of a great many differential geometric structures and questions. One might contemplate trying to classify a particular class of homogeneous Riemannian manifolds or a class of Einstein spaces or Ricci solitons. Certainly a Lie group endowed with an invariant metric will be a locally homogeneous space. We shall not explore these issues further in this article but defer them as possible future directions of investigation. Finally, we mention several recent references. In [4], the author investigates the existence of “ad”-invariant, or what is the same thing, bi-invariant metrics on Lie groups. In [5,6] neutral, that is, signature $(2,2)$-invariant metrics on four-dimensional nilpotent Lie groups are studied. In [7], among other things, isometry groups of three-dimensional Lie groups are investigated and likewise for four-dimensional Lie groups in [8]. Regarding these latter two references, which were not available during the preparation of the current manuscript, they would seem to have a great deal of overlap with it. However, a superficial perusal suggests that it will require some effort to reconcile the content of [7,8] with the conclusions reached here, not least in terms of the use of completely different notation.

2. Lie Algebras of Dimension Three and Four

In this Section, we shall outline the classification of real, indecomposable Lie algebras in dimensions three and four. As regards three, there is only the root space $A_1$ that leads to the simple complex Lie algebra $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$, and its real forms are $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ and $\mathfrak{s}\mathfrak{o}\left(3\right)$, that are denoted by $A_{3.8}$ and $A_{3.9}$ in [2]. Next, it is routine to show that up to isomorphism there are just two three-dimensional nilpotent Lie algebras, that is, ${\mathbb{R}}^3$ considered as an abelian Lie algebra and Heisenberg, that we denote by H. Of course ${\mathbb{R}}^3$ is decomposable and a basis $\{e_1,e_2,e_3\}$ for H can be found such that the only non-zero Lie bracket is $[e_2,e_3]=e_1$. Any other three-dimensional Lie algebra must be solvable. It is a general fact that the nilradical $NR$ of a solvable Lie algebra N must satisfy $dim\left(NR\right)\ge {\displaystyle \frac{1}{2}}dim\left(N\right)$, so that a three-dimensional solvable, not nilpotent, Lie algebra must satisfy $dim\left(NR\right)$ = 2. However, the only two-dimensional nilpotent Lie algebra, up to isomorphism, is ${\mathbb{R}}^2$. It follows that there exists a basis $\{e_1,e_2,e_3\}$ for N such that the structure equations are:

$$[e_1,e_3]=a{e}_1+b{e}_2, [e_2,e_3]=c{e}_1+d{e}_2.$$

(1)

We still have the freedom to change basis in $NR$, spanned by $\{e_1,e_2\}$ and scale $e_3$ by a non-zero scalar. As such, we complete the classification by reducing the matrix $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ to its real Jordan normal form. The diagonal case with real roots corresponds to $A_{3.3}, A_{3.4}, A_{3.5}$ in [2], but we shall treat them just as one case. Similarly, the diagonal case with a complex conjugate pair of roots corresponds to $A_{3.6}, A_{3.7}$ in [2]. The remaining case, for which there is one Jordan block, gives $A_{3.2}$ in [2].

Concerning the real four-dimensional indecomposable Lie algebras, we note that because of the dimensions of the root systems, there is no real simple or semi-simple algebra. Furthermore, there is no Lie algebra that has a non-trivial Levi-decomposition, the smallest such being of dimension five. We leave the reader to classify the four-dimensional nilpotent Lie algebras as ${\mathbb{R}}^4, H\oplus \mathbb{R}$ and $A_{4.1}$, the only such indecomposable algebra, and for which a basis $\{e_1,e_2,e_3,e_4\}$ exists so that its structure equations are:

$$[e_2,e_4]=e_1, [e_3,e_4]=e_2.$$

(2)

The remaining real four-dimensional indecomposable Lie algebras N must be solvable and the nilradical $NR$ must be of dimension two or three. If $dim\left(NR\right)=3$, then $NR$ is either abelian or isomorphic to H. If $NR$ is abelian, we shall have structure equations of the form:

$$[e_1,e_4]=a{e}_1+b{e}_2+c{e}_3, [e_2,e_4]=d{e}_1+e{e}_2+f{e}_3, [e_3,e_4]=g{e}_1+h{e}_2+i{e}_3.$$

(3)

Again, the algebras are reduced by putting the matrix $\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$ into real Jordan normal form and using a scaling for $e_4$. We obtain $A_{4.2}-A_{4.6}$ in [2].

If we start from $NR$ isomorphic to H, we have the bracket $[e_2,e_3]=e_1$. It remains to add brackets involving the last basis vector $e_4$. Now the adjoint matrix $ad\left(e_4\right)$ is a derivation of H and such a derivation is of the form $\left[\begin{array}{ccc}a+d& e& f\\ 0& a& b\\ 0& c& d\end{array}\right]$, giving the brackets

$$[e_1,e_4]=(a+d)e_1, [e_2,e_3]=e_1, [e_2,e_4]=a{e}_2+b{e}_3, [e_3,e_4]=c{e}_3.$$

(4)

Of course $[e_4,e_4]=0$ and $NR$ is an ideal, so there are no more terms. We may make use of the inner derivations of H and the transformation that replaces $e_4$ by $e_4+f{e}_2-e{e}_3$, reduces e and f to zero. Now we may change basis in the subspace spanned by $\{e_2,e_3\}$ and put the matrix $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ into real Jordan normal form. Finally, by scaling we may set one of the diagonal entries to unity, thereby yielding the three cases $A_{4.7}, A_{4.8/9b}$ and $A_{4.10/11a}$ in [2].

Finally, turning to the case where $NR$ is of dimension two, we must have structure equations of the form:

$$[e_1,e_3]=a{e}_1+b{e}_2, [e_2,e_3]=c{e}_1+d{e}_2, [e_1,e_4]=e{e}_1+f{e}_2, [e_2,e_4]=g{e}_1+h{e}_2.$$

(5)

The Jacobi identity comes down to the commuting of the matrices $ad\left(e_3\right)$ and $ad\left(e_4\right)$, restricted to $NR$, which gives $bg-cf=0,$ $af-be+bh-df=0,$ $ag-ce+ch-dg=0$. This circumstance may be achieved in essentially two ways, that is, $a=h=1,$ $b=c=d=e=f=g=0$ and $a=d=g=1,$ $f=-1,$ $b=c=e=h=0$. However, the first possibility leads to a decomposable algebra and the second gives $A_{4.12}$.

Notice in each of the algebras $A_{4.7}, A_{4.8/9b}$ and $A_{4.10/11a}$ that $e_1$ spans a one-dimensional ideal and the quotient algebras are $A_{3.2}, A_{3.3/4/5a}$ and $A_{3.6/7a}$, respectively, and we have non-split one-dimensional extensions. We will simply remark, without details, that $A_{4.9(b=0)}$ or $A_{4.8}$, is the tangent algebra of $A_{2.1}$. Likewise, $A_{4.9(b=-1)}$ is the cotangent algebra of $A_{2.1}$. Finally, $A_{4.12}$ is the complexification of $A_{2.1}$. The algebra $A_{4.8}$ is also known as the “diamond Lie algebra” and $A_{4.11(a=0)}$ or $A_{4.10}$, as “osc”, see [4].

3. Some Killing Structure Theorems

3.1. Classical Theorems

In this Section we state several general structure Theorems concerning the dimension of the Lie algebra of Killing vector fields. The first two results, which are classical, pertain to pseudo-Riemannian manifolds in general, and not just the narrower context of invariant metrics.

Theorem 1. 

On an n-dimensional pseudo-Riemannian manifold the maximal dimension of the Lie algebra of Killing vector fields is ${\displaystyle \frac{1}{2}}n(n+1)$ and this maximum is attained if and only if the space is of constant curvature.

• We shall outline the proof below and the reader could consult, for example, [9,10].

Theorem 2. 

An n-dimensional ($n>2$) pseudo-Riemannian manifold is of constant (sectional) curvature κ if and only if the the curvature tensor R of the metric B satisfies

$$R(X,Y)Z=\kappa \left(B\right(Z,Y)X-B(Z,X\left)Y\right)$$

where $X,Y,Z$ are arbitrary vector fields.

• For the proof see for example [10].

Theorem 3 

(Fubini [11]). On a Riemannian manifold of dimension n, $\sigma \ne {\displaystyle \frac{n(n+1)}{2}}-1$.

Theorem 4 

(Wang [12]). On a pseudo-Riemannian manifold of dimension n where $n\ne 4$, if $\sigma >{\displaystyle \frac{n(n-1)}{2}}+1$ then the manifold is of constant curvature and so $\sigma ={\displaystyle \frac{n(n+1)}{2}}$.

Theorem 5 

(Egorov [13]). On a Riemannian manifold of dimension n, which is not a space of constant curvature, $\sigma \le {\displaystyle \frac{n(n-1)}{2}}+2$ and equality holds only if the space is Einstein.

3.2. Invariant Metrics

Turning now to invariant metrics we have:

Proposition 1.

Any central element in the Lie algebra $\mathfrak{g}$ of a Lie group G is bi-invariant.

Proof. 

By default, we work with right-invariant objects on G. Hence by a “central element”, we mean a right-invariant vector field Z on G. Thus, Z commutes with all right-invariant vector fields on G. However, any right-invariant vector field commutes with any left-invariant vector field and hence Z is bi-invariant. □

Proposition 2.

Any central element in the Lie algebra $\mathfrak{g}$ of a Lie group G is a Killing vector field for any right or left-invariant Riemannian metric on G.

Proof. 

Let B be a right-invariant Riemannian metric on G and let K be a central element and $X,Y$ right-invariant vector fields. Then $B(X,Y)$ is constant and taking the Lie derivative gives

$$0=K\left(B(X,Y)\right)={\pounds }_{K}B(X,Y)+B([K,X],Y)+B(X,[K,Y]).$$

Thus ${\pounds }_{K}B(X,Y)=0$ is equivalent to $B\left(\right[K,X],Y)+B(X,[K,Y\left]\right)=0$ and the latter expression certainly is zero if K is central. Since central elements are bi-invariant, the proof for left-invariant metrics is similar. □

Proposition 3.

Any left-invariant vector field is Killing for any right-invariant Riemannian metric and similarly any right-invariant vector field is Killing for any left-invariant Riemannian metric.

Proof. 

The proof is immediate from the fact that any right-invariant vector field commutes with any left-invariant vector field and conversely. □

Proposition 4.

A right-invariant Riemannian metric B on a Lie group G is bi-invariant if and only if every right-invariant vector field is Killing for B. In fact a de Rham indecomposable Riemannian metric B on a Lie group G is bi-invariant if and only if B is a constant multiple of the Killing form and G is a compact simple Lie group.

• For the proof, see for example [14].

Theorem 6 

(Milnor [14]). A Lie group admits a flat invariant metric if and only if its Lie algebra splits as a vector space orthogonal direct sum $\mathfrak{b}⨁\mathfrak{u}$ in which $\mathfrak{b}$ is an abelian subalgebra, $\mathfrak{u}$ is an abelian ideal and finally every (invariant) vector field in $\mathfrak{b}$ is a Killing field.

Theorem 7. 

Necessary and sufficient conditions for $\{e_1,e_2,...,e_r,e_{r+1},e_{r+2},...,e_n\}$ to be an orthonormal basis for a flat right-invariant metric on a Lie group G are that the Lie algebra $\mathfrak{g}$ should be of the form $[e_i,e_a]=C_{ia}^j{e}_j$, where $1\le i\le r, r+1\le a\le n$, the matrices $C_{ia}^j$ should be skew-symmetric in i and j and that the $C_{ia}^j$ for $r+1\le a\le n$ should pairwise commute.

In terms of the classification of the low-dimensional Lie algebras, the flat metrics occur in case $A_{3.6/7,a=0}$ Other examples may be found by adding abelian factors and taking direct sums.

The following theorem forms a natural sequel to Theorem 6. See [15].

Theorem 8. 

An invariant metric on a Lie group G is of constant non-zero curvature if and only if the Lie algebra $\mathfrak{g}$ of G is solvable with a codimension one abelian nilradical $nil\left(\mathfrak{g}\right)$, and such that the linear transformation S, the self-adjoint part of ad$\left(\mathfrak{b}\right)$ restricted to $nil\left(\mathfrak{g}\right)$, where $\mathfrak{b}$ spans a complement to $nil\left(\mathfrak{g}\right)$, is a multiple of the identity.

In concrete terms the Lie algebra occurring in the Theorem can be written as follows, where a and each of the $b_i$’s is non-zero:

$$\begin{array}{ccc}\hfill & & [e_1,e_n]=a{e}_1+b_1{e}_2,[e_2,e_n]=a{e}_2-b_1{e}_2,[e_3,e_n]=a{e}_3+b_2{e}_4,\hfill \\ & & [e_4,e_n]=a{e}_4-b_2{e}_3,\dots ,[e_{2p-1},e_n]=a{e}_{2p-1}+b_p{e}_{2p},[e_{2p},e_n]=a{e}_{2p}-b_p{e}_{2p-1},\hfill \\ & & [e_{2p+1},e_n]=a{e}_{2p+1},[e_{2p+2},e_n]=a{e}_{2p},\dots ,[e_{n-1},e_n]=a{e}_{n-1}.\hfill \end{array}$$

(6)

• In fact, by scaling $e_n$, we can even assume that $a=1$.

The metric corresponding to (6) is given in adapted coordinates by

$$\begin{array}{ccc}\hfill & & \sum _{i=1}^p{(d{x}_{2i-1}-(a{x}_{2i-1}+b_i{x}_{2i})d{x}_n)}^2+{(d{x}_{2i}-(a{x}_{2i}-b_i{x}_{2i-1})d{x}_n)}^2\hfill \\ & & +\sum _{j=2p+1}^n{(d{x}_j-a{x}_{j}d{x}_n)}^2+{ϵ}^2{\left(d{x}_n\right)}^2.\hfill \end{array}$$

(7)

In (7) $ϵ$ can assume any non-zero value: the curvature of the metric in (7) is $-{\displaystyle \frac{a^2}{{ϵ}^2}}$. We may assume for simplicity that $ϵ=1$.

In terms of the classification obtained in [2], the three dimensional cases correspond to algebras $A_{3.3}, A_{3.6/7,a=1}$, the four dimensional cases to $A_{4.5,(a=b=1)}, A_{4.6,(a=b\ne 0)}$. We remark also that there are no flat metrics in dimension two and four and only $A_{3.6/7,a=0}$ in dimension three.

3.3. Lie-Einstein Spaces

An Einstein space is a Riemannian manifold whose Ricci tensor is a multiple of the metric. We refer to an Einstein space, where the underlying manifold is a Lie group, as a Lie-Einstein space. According to the Theorem of Egorov [13], Einstein spaces comprise the next level of symmetry after constant curvature spaces, with regard to the existence of Killing vector fields. We refer to [16] where it was shown, mutatis mutandis, that in dimension four there are just two Lie-Einstein spaces. They occur for the Lie algebra $A_{4.9(b=1)}$ for which the metric is

$${(dz-ydx+(xy-2z)dw)}^2+{(dx-xdw)}^2+{(dy-ydw)}^2+d{w}^2$$

(8)

and for the Lie algebra $A_{4.11a}$ for which the metric is

$$a{(dz-ydx+xdy+(x^2+y^2-2z)dw)}^2+{(dx-(ax+y)dw)}^2+{(dy+(x-ay)dw)}^2+d{w}^2.$$

(9)

In the second of these cases, a occurs genuinely as an essential parameter that cannot be removed by scaling, the only restriction being that $a>0$. We shall see in Section 8 that $\sigma =8$ only in the first of these cases, in agreement with Egorov’s Theorem.

Thus for the case $n=4$, in view of Egorov’s Theorem, it follows that for an invariant metric the only possible values for $\sigma$ are $4, 5, 6, 7, 8$ and 10. In fact $n=8$ occurs only for the metric (8).

4. Structure of Killing’s Equations

4.1. Classical Derivation of Integrability Conditions

In this Section we shall investigate the structure of Killing’s equations in degree one from a rather classical point of view. The analysis is applicable at the beginning to arbitrary Riemannian metrics and not just to invariant metrics on Lie groups.

Suppose then that ∇ is a symmetric linear connection on an n-manifold M. Using semi-colon to denote covariant derivative, Killing’s equation for a one-form can be written in a local coordinate system $\left(x^i\right)$ as

$$K_{i;j}+K_{j;i}=0.$$

(10)

In classical terminology $K_p$ is a “covariant vector field”. Starting from (10) we obtain the following equations, by permuting the indices:

$$K_{i;jk}+K_{j;ik}=0.$$

(11)

$$K_{j;ki}+K_{k;ji}=0.$$

(12)

$$K_{k;ij}+K_{i;kj}=0.$$

(13)

Now add (11) and (12) and then subtract (13), and use the Ricci identity and its permutations

$$K_{i;jk}-K_{i;kj}=K_p{R}_{ijk}^p$$

(14)

to obtain

$$2K_{j;ik}+K_p(R_{ijk}^p+R_{kji}^p-R_{jik}^p)=0.$$

(15)

Finally use the first Bianchi identity of $R_{ijk}^p$ to derive

$$K_{i;jk}+K_p{R}_{kij}^p=0.$$

(16)

Notice that (16) expresses all second order derivatives in terms of zeroth and first order derivatives. Starting from (16), we differentiate again to find

$$K_{i;jkm}+K_{p;m}{R}_{kij}^p+K_p{R}_{kij;m}^p=0$$

(17)

and by interchanging indices we also obtain

$$K_{i;jmk}+K_{p;k}{R}_{mij}^p+K_p{R}_{mij;k}^p=0.$$

(18)

Subtracting (18) from (17) and using Ricci’s identities to eliminate third order derivatives, we obtain

$$K_{p;j}{R}_{ikl}^p+K_{i;p}{R}_{jkl}^p=K_{p;k}{R}_{lij}^p+K_p{R}_{lij;k}^p-K_{p;l}{R}_{kij}^p-K_p{R}_{kij;l}^p.$$

(19)

We rewrite (19) as

$$({\delta }_i^m{R}_{jkl}^p-{\delta }_j^m{R}_{ikl}^p-{\delta }_l^m{R}_{kij}^p+{\delta }_k^m{R}_{lij}^p)K_{m;p}+K_p(R_{kij;l}^p-R_{lij;k}^p)=0.$$

(20)

It should be noted that the discussion so far pertains to a general symmetric connection. In particular it is still valid if ∇ is the Levi-Civita connection of a metric B: we then obtain the (contravariant) Killing vector field by “raising an index with g”. In any case we may regard (19) as a new set of first order conditions that should be appended to (10) unless of course they happen to be identities, a possibility that we shall now examine. If (19) are identically satisfied, then the coefficients of $K_{m;p}$ must be skew in m and p; but now contracting this condition, implies that the curvature satisfies the equation appearing in Theorem 2. Hence also the curvature tensor is covariantly constant and the coefficients of $K_p$ in (19) also vanish. Apart from some details, we have now established the proof of Theorem 1.

4.2. Structure of the Integrability Conditions

The structure of the “new” Killing conditions in (19) can be understood with the help of the following Lemma whose proof depends on the first and second Bianchi identities.

Lemma 1.

Equation (20) is invariant under the simultaneous interchange of indices $i\leftrightarrow k, j\leftrightarrow l$. Furthermore taking the skew-symmetric part of the left hand side of (20) with respect to the indices $j, k, l$, gives an identity. Hence the the number of independent conditions in (20) is at most ${\displaystyle \frac{n^2(n^2-1)}{12}}$.

Proof. 

Clearly the first group of terms in (20) that depend on the first derivatives are already invariant under the interchange $i\leftrightarrow k, j\leftrightarrow l$. For the second group of terms, consider

$$R_{kij;l}^p-R_{lij;k}^p-R_{ikl;j}^p+R_{jkl;i}^p$$

$$=R_{kij;l}^p-R_{lij;k}^p+R_{lik;j}^p+R_{kli;j}^p+R_{jkl;i}^p$$

$$=R_{kij;l}^p+R_{ljk;i}^p+R_{kli;j}^p+R_{jkl;i}^p$$

$$=-R_{kjl;i}^p+R_{ljk;i}^p+R_{jkl;i}^p$$

$$=R_{klj;i}^p+R_{ljk;i}^p+R_{jkl;i}^p=0.$$

Now consider $K_{i;p}{R}_{jkl}^p+K_{p;j}{R}_{ikl}^p+K_{k;p}{R}_{lij}^p+K_{p;l}{R}_{kij}^p$ which comprise the first group of terms of the left hand side of (20). Then skew-symmetrize over the indices $j, k, l$ to obtain

$$\begin{array}{ccc}& & K_{p;[j}{R}_{\widehat{i}kl]}^p-K_{p;[k}{R}_{l\widehat{i}j]}^p-K_{p;[l}{R}_{kj]i}^p\hfill \\ & & =K_{p;[j}{R}_{\widehat{i}kl]}^p+K_{p;[k}{R}_{lj]i}^p-K_{p;[l}{R}_{kj]i}^p\hfill \\ & & =-K_{p;[j}{R}_{kl]i}^p+K_{p;[j}{R}_{lk]i}^p+2K_{p;[k}{R}_{lj]i}^p=0.\hfill \end{array}$$

For the second group of terms in (20), using the first and second Bianchi identities gives

$$\begin{array}{ccc}& & 0=6R_{[k\widehat{i}j;l]}^p=R_{kij;l}^p+R_{lik;j}^p+R_{jil;k}^p+R_{lji;k}^p+R_{kli;j}^p+R_{jki;l}^p\hfill \\ & & =-R_{ilj;k}^p-R_{ijk;l}^p-R_{ikl;j}^p=0.\hfill \end{array}$$

□

• Briefly stated, the Lemma says that Equation (20) has exactly the same symmetries as the fully covariant Riemann tensor. It has an important practical consequence: for example, for $n=3$, we need only consider a system of six, rather than nine, linear equations and for $n=4$, only 20 linear equations rather than 36.

4.3. Further Integrability Conditions

Before the introduction of (20), $K_{i;j}$ plays the role merely of a skew-symmetric matrix. Suppose, however, that (20) can be solved for the unknowns $K_{i;j}$; then (20) assert that the $K_{i;j}$ are indeed the covariant derivatives of the $K_i$. Furthermore, analysis of the PDE system (10) is concluded and the only linearly independent solutions are the ones known form the outset, that is, the left-invariant vector fields.

What happens if the metric is neither of constant curvature nor (20) permits all of the $K_{i;j}$ to be solved for? In that case there is no recourse, but to differentiate (20) covariantly again; then, second order derivatives of $K_i$ may be eliminated by using (16) and the result gives us yet more linear conditions on $K_i$ and $K_{i;j}$, viz:

$$\begin{array}{ccc}\hfill & & (({\delta }_i^m{R}_{jkl;q}^p-{\delta }_j^m{R}_{ikl;q}^p-{\delta }_l^m{R}_{kij;q}^p+{\delta }_k^m{R}_{lij;q}^p)+{\delta }_q^m(R_{lij;k}^p-R_{kij;l}^p))K_{m;p}\hfill \\ & & +(R_{kij;lq}^r-R_{lij;kq}^r-R_{qmp}^r({\delta }_i^m{R}_{jkl}^p-{\delta }_j^m{R}_{ikl}^p-{\delta }_l^m{R}_{kij}^p+{\delta }_k^m{R}_{lij}^p))K_r=0.\hfill \end{array}$$

(21)

This procedure may be iterated. Eventually, the totality of such linear conditions will stabilize, in the sense that at some stage, the new linear conditions that are added will not decrease the dimension of the solution space already found at the previous stage; it can then be shown that if one differentiates yet one more time, the solution space will remain the same as was found in the two previous stages and hence for all higher stages. However, it has to be admitted that even relatively simple systems can lead to extreme computational difficulties.

Even when the linear system discussed in the previous paragraph does stabilize, the purely algebraic solutions obtained are not necessarily solutions of the original PDE system (10). In general one has to take a function linear combination of the algebraic solutions and apply Frobenius integrability to obtain bona fide solutions of (10). Finally, we should remark that there is really nothing special about the form of the linear conditions (20). A similar analysis applies to quite general linear systems arising from a linear PDE system of finite type, that is to say, where all possible derivatives of a certain order can be solved in terms of lower and zeroth order derivatives.

4.4. The Case $n=3$

As a result of the Lemma we have the following system of six rather than nine equations for the six unknowns $K_1, K_2, K_3, K_{1;2}, K_{2;3}, K_{3;1}$ in the case $n=3$:

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill 2(R_{112}^1+R_{212}^2)K_{12}-2R_{112}^3{K}_{23}-2R_{212}^3{K}_{31}+K_p(R_{112;2}^p-R_{212;1}^p)& =\hfill & 0\\ \hfill 2(R_{223}^2+R_{323}^3)K_{23}-2R_{223}^1{K}_{31}-2R_{323}^1{K}_{12}+K_p(R_{223;3}^p-R_{323;2}^p)& =\hfill & 0\\ \hfill 2(R_{331}^3+R_{131}^1)K_{31}-2R_{331}^2{K}_{12}-2R_{131}^2{K}_{23}+K_p(R_{331;1}^p-R_{131;3}^p)& =\hfill & 0\\ \hfill (R_{231}^2+R_{331}^3-R_{123}^2)K_{23}+(R_{323}^3+R_{123}^1-R_{231}^1)K_{31}& & \\ \hfill -(R_{323}^2+R_{331}^1)K_{12}+K_p(R_{323;1}^p-R_{123;3}^p)& =\hfill & 0\\ \hfill (R_{312}^3+R_{112}^1-R_{231}^3)K_{31}+(R_{131}^1+R_{231}^2-R_{312}^2)K_{12}& & \\ \hfill -(R_{131}^3+R_{112}^2)K_{23}+K_p(R_{131;2}^p-R_{231;1}^p)& =\hfill & 0\\ \hfill (R_{123}^1+R_{223}^2-R_{312}^1)K_{12}+(R_{212}^2+R_{312}^3-R_{123}^3)K_{23}& & \\ \hfill -(R_{212}^1+R_{223}^3)K_{31}+K_p(R_{212;3}^p-R_{312;2}^p)& =\hfill & 0.\end{array}\end{array}$$

Let the resulting $6\times 6$ matrix of coefficients be denoted by M. If M is non-singular, then there will be no non-zero solutions to Killing’s equations and at the opposite extreme, if it vanishes, then there will be six linearly independent solutions and the space must be of constant curvature. By examining the algebraic form of M, it is easy to see that $\sigma \ne 5$. It also follows from Fubini’s Theorem [11]. On the other hand, we know that $\sigma \ge 3$. Hence:

Proposition 5.

For an invariant metric on a three-dimension al Lie group we can only have $\sigma =3, 4$ or 6.

4.5. General Dimension

Let us come back to the main issue, that is, the dimension of the Lie algebra of Killing vector fields for a given right-invariant metric g on an n-dimensional Lie group G. We can use (20) to construct an ${\displaystyle \frac{n^2(n^2-1)}{12}}\times {\displaystyle \frac{n(n+1)}{2}}$ matrix of coefficients, denoted by M, for the unknowns $K_{i;j}, K_i$. Now we know that the each of the left-invariant vector fields is a Killing vector field; as such there must exist n linear relations that express columns ${\displaystyle \frac{n(n-1)}{2}}+1,...,{\displaystyle \frac{n(n+1)}{2}}$ in terms of columns $1,...,{\displaystyle \frac{n(n-1)}{2}}$. Hence we can work with the submatrix $MM$ consisting of the first ${\displaystyle \frac{n(n-1)}{2}}$ columns of M instead of M itself. For our right-invariant metric, we can assert that if $MM$ has rank ${\displaystyle \frac{n(n-1)}{2}}$, then only the left-invariant vector fields will be Killing. We shall give some examples below.

5. Worked Example ${\mathit{A}}_{\mathbf{4}.\mathbf{1}}$

In this Section we illustrate the theory of Section 4 with an example in which we give complete details. The example that we shall consider is the Lie algebra $A_{4.1}$, which has non-zero Lie brackets

$$[e_2,e_4]=e_1,[e_3,e_4]=e_2.$$

(22)

A representation for the associated Lie group is given by

$$S=\left[\begin{array}{cccc}1& w& {\displaystyle \frac{w^2}{2}}& x\\ 0& 1& w& y\\ 0& 0& 1& z\\ 0& 0& 0& 1\end{array}\right],$$

and the corresponding invariant vector fields and one forms are:

• Left-invariant vector fields: $D_x, D_y+w{D}_x, D_z+w{D}_y+{\displaystyle \frac{w^2}{2}}{D}_x, D_w$
• Left-invariant one forms: $dx-wdy+{\displaystyle \frac{1}{2}}{w}^{2}dz, dy-wdz, dz, dw$
• Right-invariant vector fields: $D_x, D_y, D_z, D_w+y{D}_x+z{D}_y$.
• Right-invariant one-forms $dx-ydw, dy-zdw, dz, dw$.

We shall consider a general right-invariant Riemannian metric which we may write as:

$$\left[\begin{array}{cccc}dx-ydw& dy-zdw& dz& dw\end{array}\right]\left[\begin{array}{cccc}a& b& c& d\\ b& e& f& g\\ c& f& h& i\\ d& g& i& j\end{array}\right]\left[\begin{array}{c}dx-ydw\\ dy-zdw\\ dz\\ dw.\end{array}\right]$$

(23)

and the matrix G occurring in the middle of the sandwich, is positive definite. Because we are considering a right-invariant Riemannian metric, any left-invariant vector field is automatically a Killing vector field.

We shall use automorphisms of the Lie algebra to reduce the metric (23). The Lie algebra of the automorphism group is the space of derivations of $A_{4.1}$ and consists of the following $4\times 4$ matrices:

$$\left[\begin{array}{cccc}{s}_1& s_2& s_3& s_4\\ 0& s_5& s_2& s_6\\ 0& 0& 2s_5-s_1& s_7\\ 0& 0& 0& s_1-s_5\end{array}\right].$$

(24)

We begin by exponentiating the nilpotent part of the derivation algebra, so as to give the matrix Lie group

$$P=\left[\begin{array}{cccc}1& \alpha & \beta & \gamma \\ 0& 1& \alpha & \delta \\ 0& 0& 1& ϵ\\ 0& 0& 0& 1\end{array}\right].$$

Then the symmetric matrix $GG=P^{t}GP$ is given by:

$$\left[\begin{array}{cccc}a& \alpha a+b& \beta a+\alpha b+c& \gamma a+\delta b+ϵc+d\\ *& \left(\alpha a+b\right)\alpha +\alpha b+e& \left(\alpha a+b\right)\beta +\left(\alpha b+e\right)\alpha +\alpha c+f& G{G}_{24}\\ *& *& \left(\beta a+\alpha b+c\right)\beta +\left(\alpha e+\beta b+f\right)\alpha +\alpha f+\beta c+h& G{G}_{34}\\ *& *& *& G{G}_{44}\end{array}\right].$$

The entries $G{G}_{24}, G{G}_{34}, G{G}_{44}$ are as follows:

$$\begin{array}{ccc}& & G{G}_{24}=\left(\alpha a+b\right)\gamma +\left(\alpha b+e\right)\delta +\left(\alpha c+f\right)ϵ+\alpha d+g\hfill \\ & & G{G}_{34}=\left(\beta a+\alpha b+c\right)\gamma +\left(\alpha e+\beta b+f\right)\delta +\left(\alpha f+\beta c+h\right)ϵ+\alpha g+\beta d+i\hfill \\ & & G{G}_{44}=\left(\gamma a+\delta b+ϵc+d\right)\gamma +\left(\gamma b+\delta e+ϵf+g\right)\delta \hfill \\ & & +\left(\gamma c+\delta f+ϵh+i\right)ϵ+\gamma d+\delta g+ϵi+j.\hfill \end{array}$$

Now we proceed as follows: first of all, solve $\alpha a+b=0$ for $\alpha$. Then replace b and $\alpha$ by zero. Next, solve $a\beta +c=0$ for $\beta$, then replace c and $\beta$ by zero. Then solve $a\gamma +d=0$ for $\gamma$ and replace d and $\gamma$ by zero. At the next step, we have to solve the system $\delta e+ϵf+g=0, \delta f+ϵh+i=0$, and we note that the matrix of coefficients is non-zero, since G is assumed to be positive definite.

Now we turn attention to the semi-simple part of the derivation algebra, as well as an overall scaling. As such we can use the following matrix:

$$T=\left[\begin{array}{cccc}{\mathrm{e}}^{\lambda +\nu }& 0& 0& 0\\ 0& {\mathrm{e}}^{\mu +\nu }& 0& 0\\ 0& 0& {\mathrm{e}}^{2\mu -\lambda +\nu }& 0\\ 0& 0& 0& {\mathrm{e}}^{\lambda -\mu +\nu }\end{array}\right]$$

and find $T^{t}GT$ as:

$$\left[\begin{array}{cccc}{\mathrm{e}}^{2\lambda +2\nu }a& 0& 0& 0\\ 0& {\mathrm{e}}^{2\mu +2\nu }e& f{\mathrm{e}}^{3\mu +2\nu -\lambda }& 0\\ 0& f{\mathrm{e}}^{3\mu +2\nu -\lambda }& {\mathrm{e}}^{4\mu -2\lambda +2\nu }h& 0\\ 0& 0& 0& {\mathrm{e}}^{2\lambda -2\mu +2\nu }j\end{array}\right].$$

Now we choose

$$\lambda =-{\displaystyle \frac{1}{2}}ln\left({\displaystyle \frac{a^2}{ej}}\right),\mu =-{\displaystyle \frac{1}{2}}ln\left({\displaystyle \frac{a}{j}}\right),\nu ={\displaystyle \frac{1}{2}}ln\left({\displaystyle \frac{a}{ej}}\right).$$

Recycling some letters, the matrix of the metric has been reduced to the form

$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& b& 0\\ 0& b& a& 0\\ 0& 0& 0& 1\end{array}\right]$$

(25)

where $b^2<a$. Thus the reduced form of the metric is

$$g={(dx-ydw)}^2+{(dy-zdw)}^2+2b(dy-zdw)dz+ad{z}^2+d{w}^2.$$

(26)

Now we shall write down the $20\times 10$ matrix M that arises from the linear conditions (20).

$$M=\left[\begin{array}{cccccccccc}2b& 6b& 0& 0& 0& 0& 0& 0& 0& 4b\\ a& 3a-4& -b& 0& 0& 0& 0& 0& 0& 2a-2\\ 0& -8b& -2a& 0& 0& 0& 0& 0& 0& -4b\\ -2b& -2b& -2& 0& 0& 0& 0& 0& 0& -2b\\ -2b^2-a& -a& -3b& 0& 0& 0& 0& 0& 0& -b^2-a\\ b& -b& b^2-a+1& 0& 0& 0& 0& 0& 0& 0\\ -4ab& 0& -4b^2& 0& 0& 0& 0& 0& 0& -2ab\\ a& -2b^2+a& b& 0& 0& 0& 0& 0& 0& -b^2+a\\ 0& 0& 2& 0& 0& 0& 0& 0& 0& 0\\ -4a+1& -1& -3b& 0& 0& 0& 0& 0& 0& -2a\\ 4b& 0& a+3& 0& 0& 0& 0& 0& 0& 2b\\ \left(-4b^2+4a\right)b& 0& -2b^2+2a& 0& 0& 0& 0& 0& 0& \left(-2b^2+2a\right)b\\ 0& 0& 0& b^2-a& 0& 3b^2-3a& 0& -3b^2+2a& b& 0\\ 0& 0& 0& -\left(-2b^2+2a\right)b& -4b^2+4a& 0& 2b^2-2a& \left(a+2\right)b& -b^2-2& 0\\ 0& 0& 0& b& 0& b& 0& -b& 0& 0\\ 0& 0& 0& -4b^2+4a-1& -2b& 1& b& -2a& 2b& 0\\ 0& 0& 0& -b& -2a& b& a& -b& 1& 0\\ 0& 0& 0& a& 0& a& 0& -a& 0& 0\\ 0& 0& 0& -b& -2b^2& b& b^2& 0& 0& 0\\ 0& 0& 0& -4b^2+3a& -2ab& 2b^2-a& ab& -2b^2-a& 3b& 0\end{array}\right]$$

(27)

The original matrix has been modified by multiplying throughout by a non-zero factor and reordering the rows, in such a way as to exhibit block structure.

We claim that the matrix M in (27) always has rank six when $b^2<a$. To that end, note that we have the following dependence relations between the columns:

$$\begin{array}{ccc}\hfill & & C_5+2C_7=0\hfill \\ & & C_1+C_4-2C_{10}=0\hfill \\ & & C_3+C_6+C_8+b{C}_9=0\hfill \\ & & C_5+2b{C}_6+2b{C}_8+2a{C}_9=0.\hfill \end{array}$$

These conditions arise from the one-forms dual via the metric (26) to the left-invariant vector fields. Accordingly, in trying to ascertain the rank of matrix (27), we may delete the last four columns.

Now consider the $3\times 3$ submatrix of M consisting of rows $13,14,18$ and columns 4 to 6. It has determinant $-4$. On the other hand, look at the $3\times 3$ submatrix of M consisting of rows $7,9,10$ and columns 1 to 3. It has determinant $-8ab$ and so rank less than 3 only if $b=0$, since $a\ne 0$. If $b=0$ then we look at the $3\times 3$ submatrices of M consisting of rows $8,9,10$ and columns 1 to 3 and rows $2,8,9$ and columns 1 to 3. These submatrices have determinant zero only if $a={\displaystyle \frac{1}{2}}$ and $a=2$, respectively. To summarize, we have shown that unconditionally, the matrix (27) has rank six and hence there are just the four linearly independent Killing vector fields, that is, the left-invariant vector fields, that are known from the outset.

6. Dimension Two

• $A_{2.1}$$[e_1,e_2]=e_2$:

$$S=\left[\begin{array}{cc}{e}^x& y\\ 0& 1\end{array}\right].$$

• Left-invariant vector fields: $D_x, e^x{D}_y$
• Left-invariant one forms: $dx, e^{-x}dy$
• Right-invariant vector fields: $-(D_x+y{D}_y), D_y$.
• Right-invariant one forms: $-dx, dy-ydx$.
• All derivations are inner and by scaling a right-invariant metric may be reduced to

  $$d{x}^2+{(dy-ydx)}^2.$$
• This case is covered by Theorem 8. We have a metric of constant curvature $-1$ and another model for hyperbolic two-space.
• Extra Killing vector field: $e^{-x}(2y{D}_x+(y^2+1)D_y)$
• Killing Lie algebra: $\mathfrak{s}\mathfrak{l}(2,R),[e_1,e_2]=e_2,[e_1,e_3]=-e_3,[e_2,e_3]=2e_1$.
• Geodesics:

  $$\ddot{x}={(y\dot{x}-\dot{y})}^2,\ddot{y}=y(y^2+1){\dot{x}}^2-2y^2\dot{x}\dot{y}+y{\dot{y}}^2.$$

7. Dimension Three

• $A_{3.1}$$[e_2,e_3]=e_1$:

$$S=\left[\begin{array}{ccc}1& x& z\\ 0& 1& y\\ 0& 0& 1\end{array}\right].$$

• Left-invariant vector fields: $D_z,D_x, D_y+x{D}_z$
• Left-invariant one forms: $dz-xdy, dy, dx$
• Right-invariant vector fields: $D_z, D_y, D_x+y{D}_z$.
• Right-invariant one forms: $dz-ydx, dx, dy$.
• The space of derivations is given by $\left[\begin{array}{ccc}\alpha +\delta & ϵ& \varphi \\ 0& \alpha & \beta \\ 0& \gamma & \delta \end{array}\right]$ $(\alpha , \beta , \gamma , \delta , ϵ, \varphi \in \mathbb{R})$ and the component of the identity of the group of automorphisms, by $P=\left[\begin{array}{cc}det\left(Q\right)& u^t\\ 0& Q\end{array}\right]$, where $Q\in GL(2,\mathbb{R})$ and $u\in {\mathbb{R}}^2$ is a column vector.

Consider a positive definite symmetric $3\times 3$ matrix g given by $g=\left[\begin{array}{cc}a& b^t\\ b& C\end{array}\right]$. The action of the automorphism group of $A_{3.1}$ on g is given by

$$\left[\begin{array}{cc}adet{Q}^2& detQ(a{u}^t+b^{t}Q)\\ detQ(au+Q^{t}b)& au{u}^t+u{b}^{t}Q+Q^{t}b{u}^t+Q^{t}CQ\end{array}\right].$$

(28)

Since the metric is assumed to be Riemannian, $a>0$. As such we may define $u=-{\displaystyle \frac{1}{a}}b$, which has the effect of removing b in g. Now we continue to normalize g, assuming that $b=0$ and $u=0$. Thus the matrix of the metric can be reduced to $\left[\begin{array}{ccc}a& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ giving as right-invariant metric, where $a>0$:

$$a{(dz-ydx)}^2+d{x}^2+d{y}^2.$$

When we impose (20), we find that there are two independent conditions, viz.,

$$a{K}_{2;1}+K_3=0,a{K}_{3;1}-K_2=0.$$

(29)

• It follows from (29) that $\sigma \le 4$. On the other hand, we find that there are indeed four independent Killing vector fields, the extra one being $e_4=y{D}_x-x{D}_y-{\displaystyle \frac{1}{2}}(x^2-y^2)D_z$.
• Killing Lie algebra: $A_{4.10}$ after $(e_1,e_2,e_3,e_4)\mapsto (-e_4,e_2,e_3,e_1)$, with brackets:
  $[e_2,e_3]=e_1,[e_2,e_4]=-e_3,[e_3,e_4]=e_2$.

• $A_{3.2}$$[e_1,e_3]=e_1,[e_2,e_3]=e_1+e_2$:

$$S=\left[\begin{array}{ccc}{e}^z& z{e}^z& x\\ 0& e^z& y\\ 0& 0& 1\end{array}\right].$$

• Left-invariant vector fields: $e^z{D}_x, e^z(D_y+z{D}_x), -D_z$
• Left-invariant one forms: $e^{-z}(dx-zdy), e^{-z}dy, dz$
• Right-invariant vector fields: $D_x, D_y, D_z+(x+y)D_x+y{D}_y$.
• Right-invariant one forms: $dx-(x+y)dz, dy-ydz, dz$.
• The space of derivations is given by $\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ 0& \alpha & \delta \\ 0& 0& 0\end{array}\right]$ and the right-invariant metric may be reduced to, where $a>0$:

  $$a{(dx-(x+y)dz)}^2+{(dy-ydz)}^2+d{z}^2.$$
• Conditions (20) give the matrix M as

  $$M=\left[\begin{array}{cccccc}0& 0& 2& 2& a& 0\\ 0& 2a& 2a& a& a-2& 0\\ 2& 0& 0& 0& 0& a\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$$

  and hence $\sigma =3$ since the matrix $MM$ always has rank three.

• $A_{3.3}(a=1), A_{3.4}(a=-1), A_{3.5a}(0<|a|<1)$$[e_1,e_3]=e_1, [e_2,e_3]=a{e}_2$:

$$S=\left[\begin{array}{ccc}{e}^z& 0& x\\ 0& e^{az}& y\\ 0& 0& 1\end{array}\right].$$

• Left-invariant vector fields: $e^z{D}_x, e^{az}{D}_y, D_z$
• Left-invariant one forms: $e^{-z}dx, e^{-az}dy, dz$
• Right-invariant vector fields: $D_x, D_y, D_z+x{D}_x+ay{D}_y$.
• Right-invariant one forms: $dx-xdz, dy-aydz, dz$.
• In the generic case $a\ne 1$, the space of derivations is given by $\left[\begin{array}{ccc}\alpha & 0& \beta \\ 0& \gamma & \delta \\ 0& 0& 0\end{array}\right]$ and the matrix of the metric can be reduced to $\left[\begin{array}{ccc}1& b& 0\\ b& 1& 0\\ 0& 0& 1\end{array}\right]$, where $b^2<1$, with metric:
• Right-invariant metric: ${(dx-xdz)}^2+2b(dx-xdz)(dy-aydz)+{(dy-aydz)}^2+d{z}^2$:

The $6\times 6$ matrix M coming from (20) is

$$(a-1)\left[\begin{array}{cccccc}2& 0& 0& 0& 0& \left(-1+a\right)b\\ 0& 2b^2-2& 0& -\left(-1+a\right)b& 2a-\left(a+1\right)b^2& 0\\ 0& 0& -2a\left(b^2-1\right)& -a\left(\left(a+1\right)b^2-2\right)& \left(-1+a\right)ab& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$$

• If $a\ne 1$ then $\sigma =3$.
• The case $(a=1)$ is covered by Theorem 8.
• Extra Killing vector fields:
• $e^{-z}((x^2+y^2+1)D_x+2x{D}_z), e^{-z}((x^2+y^2+1)D_y+2y{D}_z), y{D}_x-x{D}_y$
• Killing Lie algebra: $\mathfrak{s}\mathfrak{o}(3,1)$: $[e_1,e_3]=-e_1, [e_1,e_4]=2e_3, [e_1,e_5]=-2e_6, [e_1,e_6]=-e_2, [e_2,e_3]=-e_2, [e_2,e_4]=2e_6, [e_2,e_5]=2e_3, [e_2,e_6]=e_1, [e_3,e_4]=-e_4, [e_3,e_5]=-e_5, [e_4,e_6]=-e_5, [e_5,e_6]=e_4$.
• We recover the well known fact that $SO(3,1)$ is the isometry group of hyperbolic 3-space.

• $A_{3.6}(a=0)$, $A_{3.7a}(a>0)$, $[e_1,e_3]=a{e}_1-e_2$, $[e_2,e_3]=e_1+a{e}_2$:

$$S=\left[\begin{array}{ccc}{e}^{az}cosz& e^{az}sinz& x\\ -e^{az}sinz& e^{az}cosz& y\\ 0& 0& 1\end{array}\right].$$

• Left-invariant vector fields: $e^{az}(cosz{D}_x-sinz{D}_y), e^{az}(sinz{D}_x+cosz{D}_y), D_z$
• Left-invariant one-forms: $coszdx-sinzdy, sinzdx+coszdy, dz$
• Right-invariant vector fields: $D_x, D_y, D_z+(ax+y)D_x+(ay-x)D_y$.
• Right-invariant one forms: $dx-(ax+y)dz, dy-(ay-x)dz, dz$.
• Right-invariant metric: ${(dx-(ax+y)dz)}^2+{(dy-(ay-x)dz)}^2+d{z}^2$.
• $(a=0)$: See Theorems 6 and 7. Flat metric.
• Extra Killing vector fields:
• $y{D}_x-x{D}_y,-(y^{2}cosz+(xy-z)sinz)D_x+((xy+z)cosz+x^{2}sinz)D_y+(ycosz+xsinz)D_z, (y^{2}sinz-(xy-z)cosz)D_x+(x^{2}cosz-(xy+z)sinz)D_y-(xcosz-ysinz)D_z$.
• Killing Lie algebra: Lie algebra of the Euclidean group of ${\mathbb{R}}^3$ after
• $(e_1,e_2,e_3,e_4,e_5,e_6)\mapsto (-e_4,e_5,e_6,e_2,e_1,e_3+e_4)$ with brackets:
• $[e_1,e_4]=-e_2, [e_1,e_5]=-e_3, [e_2,e_4]=e_1, [e_2,e_6]=-e_3, [e_3,e_5]=e_1, [e_3,e_6]=e_2, [e_4,e_5]=e_6, [e_4,e_6]=-e_5, [e_5,e_6]=e_4$.
• This Lie algebra is also referred to as $L_{6.1}$ in [17].
• $(a>0)$: Extra Killing vector fields:
• $e^{-az}((-(a(a{x}^2-2xy+a{y}^2)+1)cosz-2a{y}^{2}sinz)D_x+(2a{x}^{2}cosz+(a(a{x}^2-2xy+a{y}^2)+1)sinz)D_y+2a(ysinz-xcosz)D_z), e^{-az}(-((2a{y}^{2}cosz+a(a{x}^2+2xy+a{y}^2)+1)sinz)D_x+(-(a(a{x}^2-2xy+a{y}^2)+1)cosz+2a{x}^{2}sinz)D_y-2a(ycosz+xsinz)D_z),$
• $y{D}_x-x{D}_y$.
• We see, after the change of basis
• $(e_1,e_2,e_3,e_4,e_5,e_6)\mapsto ({\displaystyle \frac{e_1+e_4}{2a}},{\displaystyle \frac{e_2+e_5}{2a}},e_6,{\displaystyle \frac{e_1-e_4}{2a}},{\displaystyle \frac{e_2-e_5}{2a}},-{\displaystyle \frac{e_3+e_6}{a}})$:
• Killing Lie algebra: $\mathfrak{s}\mathfrak{o}(3,1)$ with brackets:
• $[e_1,e_2]=e_3, [e_1,e_3]=-e_2, [e_1,e_4]=-e_6, [e_1,e_6]=e_4, [e_2,e_3]=e_1, [e_2,e_5]=-e_6, [e_2,e_6]=e_5, [e_3,e_4]=e_5, [e_3,e_5]=-e_4, [e_4,e_5]=-e_3, [e_4,e_6]=e_1, [e_5,e_6]=e_2$.

• $A_{3.8}$$[e_1,e_2]=2e_2, [e_1,e_3]=-2e_3, [e_2,e_3]=e_1$:

$$S=\left[\begin{array}{ccc}{e}^{2z}{(1+xy)}^2& 2x{e}^{2z}(1+xy)& e^{2z}{x}^2\\ y(1+xy)& 1+2xy& x\\ e^{-2z}{y}^2& 2e^{-2z}y& e^{-2z}\end{array}\right].$$

• Left-invariant vector fields: $D_z-2x{D}_x+2y{D}_y, (1+2xy)D_x-y^2{D}_y-y{D}_z, D_y$
• Left-invariant one forms: $ydx+(1+2xy)dz, dx+2xdz, dy-y^{2}dx-2y(1+xy)dz$
• Right-invariant vector fields $D_z, e^{2z}(x^2{D}_x+D_y-x{D}_z), e^{-2z}{D}_x$
• Right-invariant one forms $dz+xdy, e^{-2z}dy, e^{2z}(dx-x^{2}dy)$.
• Right-invariant metric: $a{(dz+xdy)}^2+b{e}^{-4z}d{y}^2+c{e}^{4z}{(dx-x^{2}dy)}^2$.
• We have given the metric in diagonal form. We know that the automorphism group of the Lie algebra, at least its connected component of the identity, is isomorphic to the Lie group $SL(2,\mathbb{R})$. All derivations are inner, since $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ is simple, and it becomes difficult to remove parameters in the general metric. In the diagonal case, (20) gives three independent conditions:

$$\begin{array}{ccc}& & K_3+2K_{12}=0\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & a^3{K}_1-2c(3a^2-16bc)K_2+4abc{K}_{31}-2c(a^2-8bc)K_{23}=0\hfill \end{array}$$

(31)

$$\begin{array}{ccc}& & a^3{K}_2-2b(3a^2-16bc)K_1+4abc{K}_{23}-2b(a^2-8bc)K_{31}=0.\hfill \end{array}$$

(32)

• So for the diagonal case we deduce that $\sigma =3$. We remark finally, that it is perhaps more natural in this case, to consider a multiple of the Killing form $d{z}^2+2xdydz+dxdy$, which yields a bi-invariant Lorentzian metric of constant curvature $-1$.

• $A_{3.9}, \mathfrak{so}\left(3\right)$, $[e_1,e_2]=e_3, [e_2,e_3]=e_1, [e_3,e_1]=e_2$:

$$S=\left[\begin{array}{ccc}cosxcosycosz-sinxsinz& sinxcosycosz+cosxsinz& -sinycosz\\ -cosxcosysinz-sinxcosz& -sinxsinzcosy+cosxcosz& sinysinz\\ cosxsiny& sinxsiny& cosy\end{array}\right].$$

• Left-invariant vector fields: $D_x, cosx{D}_y-{\displaystyle \frac{sinxcosy}{siny}}{D}_x+{\displaystyle \frac{sinx}{siny}}{D}_z, sinx{D}_y+{\displaystyle \frac{cosxcosy}{siny}}{D}_x$$-{\displaystyle \frac{cosx}{siny}}{D}_z$
• Left-invariant one-forms: $dx+cosydz, cosxdy+sinxsinydz, -sinxdy+cosxsinydz$
• Right-invariant vector fields: $D_z, {\displaystyle \frac{sinz}{siny}}{D}_x+cosz{D}_y-{\displaystyle \frac{cosysinz}{siny}}{D}_z, {\displaystyle \frac{cosz}{siny}}{D}_x-sinz{D}_y-{\displaystyle \frac{cosycosz}{siny}}{D}_z$
• Right-invariant one-forms: $dz+cosydx, sinysinzdx+coszdy, sinycoszdx-sinzdy$
• Right-invariant metric: $a{(dz+cosydx)}^2+b{(sinysinzdx+coszdy)}^2+c{(sinycoszdx-sinzdy)}^2$
• $(0<c<b<a\le 1), \sigma =3$
• $(0<a\ne b=c=1), \sigma =4$
• Extra Killing vector fields: $D_z$: Killing Lie algebra $\mathfrak{so}\left(3\right)\oplus \mathbb{R}$
• $(a=b=c=1)$
• Right-invariant metric: $d{x}^2+d{y}^2+2cosydxdz+d{z}^2$
• Extra Killing vector fields: Right-invariant vector fields
• Killing Lie algebra: $\mathfrak{so}\left(3\right)\oplus \mathfrak{so}\left(3\right)$
• $[e_1,e_2]=e_3, [e_1,e_3]=-e_2, [e_2,e_3]=e_1, [e_3,e_1]=e_2, [e_4,e_5]=e_6, [e_4,e_6]=-e_5, [e_5,e_6]=e_4$.
• This example has been well studied in connection with the motion of a free rigid body. See [18] for a comprehensive discussion from both the classical and modern view points and the relationship to Lie-Poisson and Euler-Poincare reduction. Any invariant metric may be diagonalized by means of an orthogonal transformation. The eigenvalues correspond to the reciprocals of the moments of inertia. If all three coincide, the body rotates with constant angular velocity about a fixed axis and we obtain a bi-invariant Riemannian metric. If all three are distinct, there are no extra Killing fields and integration of the equations of motion necessitates the use of Jacobi elliptic functions. Finally, if precisely two of the moments of inertia are equal, the motion can be interpreted in terms of a cone in the body rolling on a cone in space. See [18] for further details.

8. Dimension Four

• $A_{4.2a}(a\ne 0)$, $[e_1,e_4]=a{e}_1, [e_2,e_4]=e_2, [e_3,e_4]=e_2+e_3$:

$$S=\left[\begin{array}{cccc}{e}^{aw}& 0& 0& x\\ 0& e^w& w{e}^w& y\\ 0& 0& e^w& z\\ 0& 0& 0& 1\end{array}\right].$$

• Left-invariant vector fields: $e^{aw}{D}_x, e^w{D}_y, e^w(D_z+w{D}_y), D_w$
• Left-invariant one forms: $e^{-aw}dx, e^{-w}(dy-wdz), e^{-w}dz, dw$
• Right-invariant vector fields: $D_x, D_y, D_z, D_w+ax{D}_x+(y+z)D_y+z{D}_z$
• Right-invariant one forms: $dx-axdw, dy-(y+z)dw, dz-zdw, dw$.
• $(a\ne 1)$ Derivations:

  $$\left[\begin{array}{cccc}{s}_1& 0& 0& s_2\\ 0& s_3& s_4& s_5\\ 0& 0& s_3& s_6\\ 0& 0& 0& 0\end{array}\right].$$

Reduced metric ($b^2<1, c^2<d, d(1-b^2)>c^2$):

$$\left[\begin{array}{cccc}dx-axdw& dy-(y+z)dw& dz-zdw& dw\end{array}\right]\left[\begin{array}{cccc}1& b& c& 0\\ b& 1& 0& 0\\ c& 0& d& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}dx-axdw\\ dy-(y+z)dw\\ dz-zdw\\ dw\end{array}\right].$$

(33)

$(a=1)$ Derivations:

$$\left[\begin{array}{cccc}{s}_1& 0& s_7& s_2\\ s_8& s_3& s_4& s_5\\ 0& 0& s_3& s_6\\ 0& 0& 0& 0\end{array}\right].$$

Reduced metric ($d>0$):

$$\left[\begin{array}{cccc}dx-xdw& dy-(y+z)dw& dz-zdw& dw\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& d& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}dx-xdw\\ dy-(y+z)dw\\ dz-zdw\\ dw\end{array}\right].$$

(34)

The following conditions express the linear dependence among the columns of the matrix M, that enable one to reduce from M to the matrix $MM$:

$$\begin{array}{ccc}\hfill & & 2a{C}_4+(a+1)b{C}_5+(ac+b+c)C_6+2C_7+2b{C}_8+2c{C}_9=0\hfill \\ & & (a+1)b{C}_4+2C_5+C_6+2b{C}_7+2C_8=0\hfill \\ & & (a-1)b{C}_1-C_2+(b+c-ac)C_3+2C_{10}=0\hfill \\ & & (b+(a+1)c)C_4+C_5+2d{C}_6+2c{C}_7+2d{C}_9=0.\hfill \end{array}$$

Because of the complexity of the matrix $MM$ we shall only consider the case:

$(b=c=0)$

$$MM=\left[\begin{array}{cccccc}0& -4a& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -4da\left(a-1\right)& 2a\\ 2da& 0& 4da-4d+1& 0& 0& 0\\ 0& 0& 0& -4da+4d-1& 0& 0\\ 0& 0& 0& -2d\left(a-2\right)& 0& 0\\ 0& 4da& 0& 0& 0& 0\\ 0& 0& 0& 0& 2da& -4da\left(a-1\right)\\ d\left(4da-4d+1\right)& 0& 2da& 0& 0& 0\\ 0& 0& 0& -2d\left(a-2\right)& 0& 0\\ 0& 0& 0& -d\left(4da-4d-3\right)& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 4a^{2}d-4d+1& 0& 4& 0& 0& 0\\ -4d& 0& -4a^{2}d+4d+3& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 4d& 0\\ 0& 0& 0& 0& 4d& -4d\\ 0& -8& 0& 0& 0& 0\\ 0& -4& 0& 0& 0& 0\\ 0& 8d& 0& 0& 0& 0\end{array}\right]$$

• If we use rows $1, 3, 4, 5, 8, 9$ to construct a $6\times 6$ matrix, we find that its determinant is
• $32d^3{a}^3(1-4d){(4a^{2}d-8ad+4d-1)}^2(a-2)$. If we replace row 9 by row 10 we obtain a determinant of $16d^3{a}^3(1-4d){(4a^{2}d-8ad+4d-1)}^2(4ad-4d-3)$. Since $ad\ne 0$, we have three cases to consider: $a=2, d={\displaystyle \frac{3}{4}}; d={\displaystyle \frac{1}{4}}; dd={\displaystyle \frac{1}{4{(a-1)}^2}}$. In each of these cases, it is routine, albeit tedious, to check that $MM$ has rank six and hence $\sigma =4$ in all cases.

• $A_{4.3}$$[e_1,e_4]=e_1, [e_3,e_4]=e_2$

$$S=\left[\begin{array}{cccc}{e}^w& 0& 0& x\\ 0& 1& w& y\\ 0& 0& 1& z\\ 0& 0& 0& 1\end{array}\right].$$

• Left-invariant vector fields: $e^w{D}_x, D_y, D_z+w{D}_y, D_w$
• Left-invariant one forms: $e^{-w}dx, dy-wdz, dz, dw$
• Right-invariant vector fields: $D_x, D_y, D_z, D_w+x{D}_x+z{D}_y$.
• Right-invariant one forms: $dx-xdw, dy-zdw, dz, dw$.
• Derivations:

  $$\left[\begin{array}{cccc}{s}_1& 0& 0& s_2\\ 0& s_3& s_4& s_5\\ 0& 0& s_3& s_6\\ 0& 0& 0& 0\end{array}\right].$$
• Reduced metric ($b^2<1, c^2<d, d(1-b^2)>c^2$):

$$\left[\begin{array}{cccc}dx-xdw& dy-zdw& dz& dw\end{array}\right]\left[\begin{array}{cccc}1& b& c& 0\\ b& 1& 0& 0\\ c& 0& d& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}dx-xdw\\ dy-zdw\\ dz\\ dw\end{array}\right].$$

(35)

The matrix $MM$ is of the following form:

$$\left[\begin{array}{cc}A& 0_{12\times 3}\\ 0_{8\times 3}& B\end{array}\right]$$

(36)

where

$$A=\left[\begin{array}{ccc}-2c\left(2b^2+bc-2\right)& 2b^4-6b^2-2bc+4& -2\left(b^2+bc-1\right)b\\ d{b}^3+dc{b}^2-2b{c}^2-c^3-bd& b^4+2b^{3}c+b^{2}d-b^2-2bc-c^2& d{b}^3+b^3-b^{2}c-b{c}^2-b+c\\ 2b^{2}d+dcb-c^2-2d& d{b}^3+b^3+3b^{2}c+b{c}^2-b-3c& b^{2}d+b^2-1\\ 2\left(c+b\right)\left(b^{2}d+dcb+c^2-d\right)& 2b^{3}c+2b^2{c}^2+2b^{2}d+2dcb+4c^2-4d& 2b\left(b^{2}d+dcb+bc+c^2-2d\right)\\ b^{2}d+2dcb+c^{2}d+c^2-d& d{b}^3+dc{b}^2+2b^{2}c+4b{c}^2+c^3-3bd& dcb+2bc+c^2-2d\\ 2b^3+6b^{2}c-8bd-2b+2c& 0& 6b^3-6b+8c\\ b^2+3bc+c^2-4d-1& -3b^3+3b-4c& bc\\ -\left(bd-c\right)\left(c+3b\right)& -b^3-3b^{2}c+4bd+b-c& -b^{2}d+3b^2+bc+4d-3\\ 2dc& -2b^{2}d+2bc+2c^2& -2bd+2c\\ -2c& -2bc& 0\\ -3bd+3c& b^{2}d-4b^2-4bc-c^2+4& c\\ 0& 2\left(bd-c\right)\left(c+3b\right)& 6bd-6c\end{array}\right]$$

and

$$B=\left[\begin{array}{ccc}bd-c& -\left(b-c\right)\left(bd-c\right)& -b\left(bd-c\right)\\ -dc& 4b^{2}d+dcb+c^{2}d+4c^2-4d& -c\left(bd-2c\right)\\ -b^{2}d-bc-2c^2+2d& b\left(b^{2}d+dcb+bc+c^2-2d\right)& -b^2\left(bd-c\right)\\ 3b^{2}d-dcb+3c^2-3d& \left(c+b\right)\left(b^{2}d+dcb+c^2-d\right)& -b\left(3b^{2}d+dcb+c^2-3d\right)\\ b\left(2b^{2}d-bc-d\right)& b^{3}c+b^2{c}^2+b^{2}d+dcb+2c^2-2d& -b^{3}c-5b^{2}d+2bc-4c^2+4d\\ -b^2-bc+1& \left(b^2+bc-1\right)\left(b-c\right)& \left(b^2+bc-1\right)b\\ -2b^{2}d-c^2+2d& -2d{b}^3-3b{c}^2-c^3+2bd& c\left(2b^2+bc-2\right)\\ -b^3-2b^{2}c+b+c& b^4+b^{3}c-4b^{2}d-b^2-5c^2+4d& -b^4+3b^2+bc-2\end{array}\right].$$

In order that the matrix should have rank less than six, the matrix A or B must have rank less than three.

We shall consider first of all B. Since our only concern is the rank of B, we are at liberty to use row and column transformations. As such introduce P and Q as

$$P=\left[\begin{array}{ccc}1& -c& b\\ 0& 1& 0\\ 0& -1& 1\end{array}\right]$$

and

$$Q=\left[\begin{array}{cccccccc}{b}^2+bc-1& 0& 0& 0& 0& bd-c& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& {\displaystyle \frac{1}{b^{2}d+c^2-d}}& 0& 0& 0& 0& 0\\ 0& -{\displaystyle \frac{b}{b^{2}d+c^2-d}}& 0& {\displaystyle \frac{1}{b^{2}d+c^2-d}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 2d\left(b^2+bc-1\right)& -b^2-bc+1& bd-c\\ 0& 0& 0& 0& 0& -b& 0& 1\end{array}\right].$$

Then $BB=QBP$ is given by

$$BB=\left[\begin{array}{ccc}0& 0& 0\\ -cd& 2(b^{2}d+bcd+c^{2}d+c^2-2d)& 2c\left(c-bd\right)\\ -{\displaystyle \frac{b^{2}d+bc+2c^2-2d}{b^{2}d+c^2-d}}& 2(b+c)& -2b\\ 3& -2c& 0\\ b\left(2b^{2}d-bc-d\right)& 2(b^{3}c-b^{3}cd+b^2{c}^2+3b^{2}d+bcd-bc+3c^2-3d)& 2(b^{4}d-b^{3}c-3b^{2}d+bc-2c^2+2d)\\ -b^2-bc+1& 0& 0\\ B{B}_{7,1}& 2\left(b^{2}d+c^2-d\right)\left(2b^3+b^{2}c-2bd-2b+3c\right)& 0\\ c\left(1-b\right)\left(1+b\right)& 2b^4+2b^{3}c+2b^2{c}^2-4b^{2}d-4b^2-2bc-6c^2+4d+2& 2\left(1-b\right)\left(1+b\right)\left(b^2+bc-1\right)\end{array}\right].$$

Here the entry $B{B}_{7,1}$ is as follows:

$$B{B}_{7,1}=3b^2{c}^2-b^{4}d-4b^{3}cd-2b^2{c}^{2}d+b^{3}c+b{c}^3+b^{2}d+3bcd-bc-2c^2.$$

If $BB$ has rank three, we find from the determinants of the third, fourth and sixth rows, and last three rows, respectively:

$$bc(b^2+bc-1)=0$$

(37)

and

$$2b^3+b^{2}c-2bd-2b+3c=0.$$

(38)

Here, several factors have been removed which are known to be non-zero, because the metric is positive definite. If $c=0$ and $b\ne 0$, then (38) gives $d=b^2-1<0$, which is impossible. On the other hand, if $b=0$, then (38) gives $c=0$. If $b=c=0$ the second, third and fifth rows in $BB$ are linearly independent, since $d>0$.

The final possibility from (37) is $c={\displaystyle \frac{1-b^2}{b}}$ and then (38) gives $d={\displaystyle \frac{1+b^2-2b^4}{2b^4}}$. Now, however, the determinant of the first three rows of $BB$ is $-{\displaystyle \frac{(1-b^4){(b^4-b^2+1)}^2}{b^2}}$, which can never be zero if b is real and the metric is positive definite. In conclusion, for all allowable values of $b, c, d$ the matrix B has rank three.

Next, we shall examine if it is possible for the rank of A to be two. To that end, multiply row nine by 3 and add row 12 to it. Then take the determinant of rows nine, ten and eleven, which gives, $8c^3(c-bd)$. Thus, if the rank of A is two, either $c=0$ or $c=bd$. Suppose first that $c=0$, then the determinant of rows one, two and three in A is $4bd{(1-b^2)}^4$. Since $b^2<1$ we can only have $b=0$. However, in that case, we see from rows $1, 3, 5, 8$ that A has rank three for all values of d.

It remains to discuss $c=bd$. In this case, we find that the determinant of rows two, three and four in A is $4b{d}^2(1-b^2)(b^2+2d-1){(b^{2}d+b^2-1)}^2$. Since $b^2<1$ we can only have $b=0, b^2+2d-1=0$ or $b^{2}d+b^2-1=0$. However, if $b=0$ then $c=0$, which case has already been studied. Next, if $d={\displaystyle \frac{1-b^2}{2}}$ we use the determinant of rows eight, ten and eleven in the modified form of A, which is $-{\displaystyle \frac{1}{2}}b{(1-b^2)}^3(b^4+b^2-8)=0$. The only new case that would be possible here is given by $b^4+b^2-8=0$, which violates $b^2<1$.

Finally we suppose that $d={\displaystyle \frac{1-b^2}{b^2}}$. However, the determinant of rows eight, ten and eleven in the modified form of A is ${\displaystyle \frac{8{(1-b^2)}^5}{b^5}}$, which cannot be zero. In conclusion, for all allowable values of $b, c, d$ the matrix A has rank three and M rank six so that in all cases we must have $\sigma =4$.

• $A_{4.4}$$[e_1,e_4]=e_1, [e_2,e_4]=e_1+e_2, [e_3,e_4]=e_2+e_3$:

$$S=\left[\begin{array}{cccc}{e}^w& w{e}^w& {\displaystyle \frac{w^2}{2}}{e}^w& x\\ 0& e^w& w{e}^w& y\\ 0& 0& e^w& z\\ 0& 0& 0& 1\end{array}\right].$$

• Left-invariant vector fields: $e^w{D}_x, e^w(D_y+w{D}_x), e^w(D_z+w{D}_y+{\displaystyle \frac{w^2}{2}}{D}_x), D_w$
• Left-invariant one forms: $e^{-w}(dx-wdy+{\displaystyle \frac{w^2}{2}}dz), e^{-w}(dy-wdz), e^{-w}dz, dw$
• Right-invariant vector fields: $D_x, D_y, D_z, D_w+(x+y)D_x+(y+z)D_y+z{D}_z$
• Right-invariant one forms: $dx-(x+y)dw, dy-(y+z)dw, dz-zdw, dw$.
• Derivations:

  $$\left[\begin{array}{cccc}{s}_1& s_2& s_3& s_4\\ 0& s_1& s_2& s_5\\ 0& 0& s_1& s_6\\ 0& 0& 0& 0\end{array}\right].$$
• Reduced matrix of metric ($a>0, b>0, ab-c^2>0$):

  $$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& a& c& 0\\ 0& c& b& 0\\ 0& 0& 0& 1\end{array}\right].$$
• Again, we find that the matrix $MM$ has the same form as in (36). As such, after performing some row and column transformations, the matrix B can be modified as follows:

  $$\left[\begin{array}{ccc}4ab+ac-4c^2& 2a^2-2ac+c& 2a^2+c\\ -a^3+4ab-4c^2& a^2-4ab+2c^2& \left(2c+a\right)a\\ 0& 0& 2\\ ab& 2ab+2ac-4c^2+b& 2ac+b\\ a\left(2ab-ac-2c^2\right)& -2ab+ac-2bc+2c^2& -2ab+ac+4c^2\\ 3a^{2}b+4abc-4a{c}^2-4c^3& -ab-2b^2+2c^2& -ab+2bc+2c^2\\ -2c-a& 4a-3& -3\\ -b-c& 4c& -2a\end{array}\right].$$
• We wish to find conditions for this matrix to be of rank two. All that is required is for the the first and second columns to be proportional, which yields the six conditions, the last one of which is $4ab-8c^2-3b-3c=0$. Since $b>0$, we can easily solve for a and feed the result back into the five remaining conditions giving:

  $$\begin{array}{ccc}\hfill A& =& 8b^3+3b^2-24c(2c+1)b-64c^4-96c^3-27c^2=0\hfill \\ \hfill B& =& (8c+6)b^4+(16c^2-9c)b^3-c^2(72c+27)b^2-(64c^5+32c^4+3c^3)b\hfill \\ & & +64c^6+48c^5+9c^4=0\hfill \\ \hfill C& =& 5b^3+17b^{2}c+b{c}^2(24c+15)+8c^4+3c^3=0\hfill \\ \hfill E& =& 48b^5+(96c^2-96c-9)b^4-(160c+192)b^3{c}^2+(-64c^2+120c+54)b^2{c}^2\hfill \\ & & +(512c^2+384c+72)b{c}^3+512c^7+576c^6+216c^5+27c^4=0\hfill \\ \hfill G& =& (92c+9)b^3+(96c^3+152c^2+27c)b^2+(96c^2+108c+27)b{c}^2\hfill \\ & & +64c^5+48c^4+9c^3=0.\hfill \end{array}$$

  We now have a system of five irreducible polynomial equations for the two unknowns $b, c$. It is important to understand that there is no general reason to suppose that such a system should have a non-trivial solution: in fact it does not. We will sketch the argument that depends on the application of Euclid’s algorithm. We divide C into $A, B, E, G$, respectively, and obtain from the remainders, new, equivalent polynomials that are quadratic in b. We choose the quadratic coming from A and C and divide it into the three other quadratics and again, from the remainders, obtain three polynomials that are linear in b. We repeat the procedure, finally ending up with two functions that are rational in c:

  $$\begin{array}{ccc}& & {\displaystyle \frac{(8c+3)(580608c^5+936960c^4+58064c^3+217968c^2+98271c+11340)c^5}{4608c^4+17088c^3+26920c^2-129c-4887}}=0\hfill \\ & & {\displaystyle \frac{(8c+3)(16704c^4+42240c^3+26651c^2+4653c-612)c^4}{4608c^4+17088c^3+26920c^2-129c-4887}}=0.\hfill \end{array}$$

  The values $c=0$ and $c=-{\displaystyle \frac{3}{8}}$ are easily discounted by substituting back into the original five polynomials. The denominator $4608c^4+17088c^3+26920c^2-129c-4887=0$ needs more careful analysis as well as the value $c=-{\displaystyle \frac{15}{136}}$, that had to be assumed non-zero at a previous stage. This latter value again is easily handled.

Now we apply the Euclidean algorithm to the polynomials

$$\begin{array}{ccc}& & 580608c^5+936960c^4+58064c^3+217968c^2+98271c+11340\hfill \\ & & 16704c^4+42240c^3+26651c^2+4653c-612.\hfill \end{array}$$

After four interations, we conclude that they have no non-trivial common factor, and hence the matrix B cannot have rank two.

Using the first, third and fifth conditions, it is possible to solve for b as

$$b=-{\displaystyle \frac{c(2304c^4-128c^3-4236c^2-1521c-27)}{3(384c^4+1056c^3-490c^2-495c-9)}},$$

provided the denominator is not zero.

Finally, in case $4608c^4+17088c^3+26920c^2-129c-4887=0$ we obtain two quartic polynomials in c, as well as the two other linear polynomials. Again, in this case we may apply the Euclidean algorithm and reach a similar conclusion that B cannot have rank two.

We now have to analyze the $12\times 3$ matrix A coming from (36), to see if it possible for $rank\left(A\right)<3$. By performing suitable row and column operations, A may be reduced to:

$$\left[\begin{array}{ccc}4ab+2ac-4c^2& -4a^2-2c& 0\\ -ab-4bc-2c^2& 4ab+2ac+4c^2-b& 0\\ a\left(2ab+ac-2c^2\right)& -a\left(2a^2+c\right)& 0\\ 0& 2& 1\\ b{a}^2+4bca-4c^3& -4b{a}^2-2c{a}^2+4c^{2}a+ab-2c^2& 0\\ -2b& -a+4c& 0\\ a^2+4ac-4b& -8a^2-a+10c& 0\\ 0& -22a^2-24ac+14b-4c& -16ab-4ac+16c^2\\ 2b+2c& -4c-2a& 0\\ 8ab+2ac-8c^2& -8a^2+6c& 0\\ 0& -4a^2+8ab-6ac-8c^2+3b& {\displaystyle \frac{1}{4}}a\left(a-4c\right)\\ 0& 12ab-8ac-16c^2& 0\end{array}.\right]$$

The determinant coming from rows $3, 4$ and 5 is $-2a^2(a^2-4ab+4c^2+b+c)(ab-c^2)$. Since G is positive definite, we must have $a^2-4ab+4c^2+b+c=0$, a condition that is easy to solve for b, provided $a\ne {\displaystyle \frac{1}{4}}$. The determinant coming from rows $4, 5$ and 6 is ${\displaystyle \frac{-(a^2+c)(2c+a)(3a^2-2ac+2c)}{(4a-1)}}$, which leads to three alternatives each of which is easily checked and found not to give $rank\left(A\right)=3$. Finally the case $a={\displaystyle \frac{1}{4}}$ must also be tested. In conclusion, for all allowable values of $a, b, c$ the matrix A has rank three and M rank six so that in all cases we must have $\sigma =4$.

A similar conclusion may be reached by several different methods, for example by using Gröbner bases, or graphically by plotting some of the five polynomials and looking for intersection points, or by using an equation linear in b and substituting into the original five polynomials. The downside of the last approach is that one encounters very complicated rational functions of c.

• $A_{4.5ab}$$(ab\ne 0,-1\le a\le b\le 1),$$[e_1,e_4]=e_1, [e_2,e_4]=a{e}_2, [e_3,e_4]=b{e}_3$:

$$S=\left[\begin{array}{cccc}{e}^w& 0& 0& x\\ 0& e^{aw}& 0& y\\ 0& 0& e^{bw}& z\\ 0& 0& 0& 1\end{array}\right].$$

Left-invariant vector fields: $e^w{D}_x, e^{aw}{D}_y, e^{bw}{D}_z, D_w$

Left-invariant one forms: $e^{-w}dx, e^{-aw}dy, e^{-bw}dz, dw$

Right-invariant vector fields: $D_x, D_y, D_z, D_w+x{D}_x+ay{D}_y+bz{D}_z$

Right-invariant one forms: $dx-xdw, dy-aydw, dz-bdw$.

$(a\ne b, a\ne 1, b\ne 1)$ Derivations:

$$\left[\begin{array}{cccc}{s}_1& 0& 0& s_4\\ 0& s_2& 0& s_5\\ 0& 0& s_3& s_6\\ 0& 0& 0& 0\end{array}\right].$$

Reduced matrix of metric ($c^2<1, d^2<1, e^2<1, 1-c^2-d^2-e^2+2cde>0$):

$$\left[\begin{array}{cccc}1& c& d& 0\\ c& 1& e& 0\\ d& e& 1& 0\\ 0& 0& 0& 1\end{array}\right].$$

In the case $c=d=e=0$, the matrix $MM$ is given by

$$\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 4(a-b)& 0& 0& 0& 0\\ 0& 0& 0& 0& 4a-4& 0\\ 0& 0& -4\left(b-1\right)a& 0& 0& 0\\ 0& 0& 0& 4a\left(a-1\right)& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 4(b-1)\\ -4\left(a-1\right)b& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 4b\left(b-1\right)& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 4\left(1-a^2\right)& 0& 0& 0& 0& 0\\ 0& 0& 4\left(b^2-1\right)& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -4a\left(a-b\right)\\ 0& 0& 0& 0& -4b\left(a-b\right)& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 4\left(a^2-b^2\right)& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right].$$

If $a=b=1$ then $MM$ is zero and if $a=1, b\ne 1$ then $MM$ has rank five.

$(a=1, b\ne 1)$ Derivations:

$$\left[\begin{array}{cccc}{s}_1& s_2& 0& s_3\\ s_4& s_5& 0& s_6\\ 0& 0& s_7& s_8\\ 0& 0& 0& 0\end{array}\right].$$

Reduced matrix of metric:

$$\left[\begin{array}{cccc}1& 0& d& 0\\ 0& 1& 0& 0\\ d& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$$

$(d=0)$: Extra Killing vector field: $x{D}_y-y{D}_x$

Killing Lie Algebra: $A_{5.35(a=0,b)}$, $[e^{bw}{D}_z, e^w{D}_y, e^w{D}_x, -D_w, x{D}_y-y{D}_x]$ with Lie brackets:

$[e_1,e_4]=b{e}_1, [e_2,e_4]=e_2, [e_3,e_4]=e_3, [e_2,e_5]=-e_3, [e_3,e_5]=e_2$.

$(a=b=1)$ Derivations:

$$\left[\begin{array}{cccc}{s}_1& s_2& s_3& s_4\\ s_5& s_6& s_7& s_8\\ s_9& s_{10}& s_{11}& s_{12}\\ 0& 0& 0& 0\end{array}\right].$$

Reduced matrix of metric:

$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$$

Killing Lie Algebra: $\mathfrak{s}\mathfrak{o}(4,1)$, $[e^w{D}_x,e^w{D}_y,e^w{D}_z,x{D}_y-y{D}_x,y{D}_z-z{D}_y,z{D}_x-x{D}_z,e^{-w}((x^2+y^2+z^2+1)D_x+2x{D}_w),e^{-w}((x^2+y^2+z^2+1)D_y+2y{D}_w),e^{-w}((x^2+y^2+z^2+1)D_z+2z{D}_w),D_w]$ with Lie brackets:

$[e_1,e_4]=e_2, [e_1,e_6]=-e_3, [e_1,e_7]=2e_{10}, [e_1,e_8]=2e_4, [e_1,e_9]=-2e_6, [e_1,e_{10}]=-e_1, [e_2,e_4]=-e_1, [e_2,e_5]=e_3, [e_2,e_7]=-2e_4, [e_2,e_8]=2e_{10}, [e_2,e_9]=2e_5, [e_2,e_{10}]=-e_2, [e_3,e_5]=-e_2, [e_3,e_6]=e_1, [e_3,e_7]=2e_6, [e_3,e_8]=-2e_5, [e_3,e_9]=2e_{10}, [e_3,e_{10}]=-e_3, [e_4,e_5]=-e_6, [e_4,e_6]=e_5, [e_4,e_7]=-e_8, [e_4,e_8]=e_7, [e_5,e_6]=-e_4, [e_5,e_8]=-e_9, [e_5,e_9]=e_8, [e_6,e_7]=e_9, [e_6,e_9]=-e_7, [e_7,e_{10}]=e_7, [e_8,e_{10}]=e_8, [e_9,e_{10}]=e_9$.

Refer to Theorem 8.

• $A_{4.6ab}$$(a\ne 0, b\ge 0)$$[e_1,e_4]=a{e}_1, [e_2,e_4]=b{e}_2-e_3, [e_3,e_4]=e_2+b{e}_3$:

$$S=\left[\begin{array}{cccc}{e}^{aw}& 0& 0& z\\ 0& e^{bw}cosw& e^{bw}sinw& x\\ 0& -e^{bw}sinw& e^{bw}cosw& y\\ 0& 0& 0& 1\end{array}\right].$$

Left-invariant vector fields: $e^{aw}{D}_z, e^{bw}(cosw{D}_x-sinw{D}_y), e^{bw}(sinw{D}_x+cosw{D}_y), D_w$

Left-invariant one forms: $e^{-aw}dz, e^{-bw}(coswdx-sinw,dy), e^{-bw}(sinwdx+coswdy), dw$

Right-invariant vector fields: $D_z, D_x, D_y, D_w+az{D}_z+(bx+y)D_x+(by-x)D_y$

Right-invariant one forms: $dz-azdw, dx-(bx+y)dw, dy-(by-x)dw, dw$.

Derivations:

$$\left[\begin{array}{cccc}{s}_1& 0& 0& s_2\\ 0& s_3& s_4& s_5\\ 0& -s_4& s_3& s_6\\ 0& 0& 0& 0\end{array}\right].$$

Reduced matrix of metric ($e>c^2+d^2$):

$$\left[\begin{array}{cccc}1& 0& c& 0\\ 0& 1& d& 0\\ c& d& e& 0\\ 0& 0& 0& 1\end{array}\right].$$

$(c=d=0, e=1)$: Extra Killing vector field: $x{D}_y-y{D}_x$

Killing Lie Algebra: $A_{3.6}\oplus A_{2.1}$, $[e_1,e_3]=e_2, [e_2,e_3]=-e_1, [e_4,e_5]=e_4]$,

$[e^{bw}(cosw{D}_x-sinw{D}_y),e^{bw}(sinw{D}_x+cosw{D}_y),x{D}_y-y{D}_x,e^w{D}_z,-D_w+x{D}_y-y{D}_x].$

$(a=b=1)$:

Killing Lie Algebra: $\mathfrak{s}\mathfrak{o}(4,1)$: $[D_w,y{D}_x-x{D}_y,e^w{D}_z,e^{-w}((x^2+y^2+z^2+1)D_z+2yz{D}_x-2xz{D}_y+2z{D}_w),e^w(sinw{D}_x+cosw{D}_y),e^w(cosw{D}_x+sinw{D}_y),zsinw{D}_x+zcosw{D}_y-(xsinw+ycosw)D_z,zcosw{D}_x-zsinw{D}_y+ysinw-xcosw)D_z,e^{-w}(((x^2+y^2+z^2+2xy+1)sinw+2y^{2}cosw)D_x+((x^2+y^2+z^2-2xy+1)cosw-2x^{2}sinw)D_y+2(xsinw+ycosw)D_w),e^{-w}(((x^2+y^2+2xy+z^2+1)cosw-2y^{2}sinw)D_x-((x^2+y^2+z^2-2xy+1)sinw+2x^{2}cosw)D_y+2(xcosw-ysinw)D_w)]$.

Lie Brackets:

$[e_1,e_3]=e_3, [e_1,e_4]=-e_4, [e_1,e_5]=e_5+e_6, [e_1,e_6]=e_6-e_5, [e_1,e_7]=e_8, [e_1,e_8]=-e_7, [e_1,e_9]=e_{10}-e_9, [e_1,e_{10}]=-e_{10}-e_9, [e_2,e_5]=-e_6, [e_2,e_6]=e_5, [e_2,e_7]=-e_8, [e_2,e_8]=e_7, [e_2,e_9]=-e_{10}, [e_2,e_{10}]=e_9, [e_3,e_4]=2e_1+2e_2, [e_3,e_7]=e_5, [e_3,e_8]=e_6, [e_3,e_9]=2e_7, [e_3,e_{10}]=2e_8, [e_4,e_5]=2e_7, [e_4,e_6]=2e_8, [e_4,e_7]=e_9, [e_4,e_8]=e_{10}, [e_5,e_7]=-e_3, [e_5,e_9]=2e_1+2e_2, [e_5,e_{10}]=2e_2, [e_6,e_8]=-e_3, [e_6,e_9]=-2e_2, [e_6,e_{10}]=2e_1+2e_2, [e_7,e_8]=-e_2, [e_7,e_9]=e_4, [e_8,e_{10}]=e_4$.

Refer to Theorem 8.

• $A_{4.7}$$[e_2,e_3]=e_1, [e_1,e_4]=2e_1, [e_2,e_4]=e_2, [e_3,e_4]=e_2+e_3$:

$$S=\left[\begin{array}{cccc}{e}^{2w}& -z{e}^w& (y-zw)e^w& x\\ 0& e^w& w{e}^w& y\\ 0& 0& e^w& z\\ 0& 0& 0& 1\end{array}\right].$$

Left-invariant vector fields: ${\displaystyle \frac{1}{2}}{e}^{2w}{D}_x, e^w(D_y-z{D}_x), e^w(D_z+w{D}_y+(y-zw)D_x), -D_w$

Left-invariant one forms: $e^{-2w}(dx+zdy-ydz), e^{-w}(dy-wdz), e^{-w}dz, dw$

Right-invariant vector fields: $-{\displaystyle \frac{1}{2}}{D}_x, z{D}_x+D_y, D_z-y{D}_x, D_w+2x{D}_x+(y+z)D_y+z{D}_z$

Right-invariant one forms: $dx-zdy+ydz+(z^2-2x)dw, dy-(y+z)dw, dz-zdw, dw$.

Derivations:

$$\left[\begin{array}{cccc}2s_1& s_2& s_3& s_4\\ 0& s_1& s_5& s_3-s_2\\ 0& 0& s_1& -s_2\\ 0& 0& 0& 0\end{array}\right].$$

Reduced matrix of metric ($a>0, b>0, ab-c^2>b{d}^2$):

$$\left[\begin{array}{cccc}a& c& d& 0\\ c& b& 0& 0\\ d& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$$

$(a=b=1, c=d=0)$. Matrix $MM$ is purely numerical of rank six.

$\sigma =4$.

• $A_{4.8}, A_{4.9b}(-1\le b\le 1)$$[e_2,e_3]=e_1, [e_1,e_4]=(b+1)e_1, [e_2,e_4]=e_2, [e_3,e_4]=b{e}_3$:

$$S=\left[\begin{array}{ccc}{\mathrm{e}}^{\left(b+1\right)w}& x{\mathrm{e}}^{bw}& z\\ 0& {\mathrm{e}}^{bw}& y\\ 0& 0& 1\end{array}\right].$$

Left-invariant vector fields: $e^{(b+1)w}{D}_z, e^w{D}_x, e^{bw}(D_y+x{D}_z), -D_w$

Left-invariant one forms: $e^{-(b+1)w}(dz-xdy), e^{-w}dx, e^{-bw}dy, -dw$

Right-invariant vector fields: $D_z, y{D}_z+D_x, -D_y, D_w+(b+1)z{D}_z+x{D}_x+by{D}_y$

Right-invariant one forms: $dz-ydx+(xy-(b+1\left)z\right)dw, dx-xdw, -(dy-bydw), dw$.

Derivations:

$$\left[\begin{array}{cccc}{s}_1& s_2& s_3& s_4\\ 0& s_5& 0& s_3\\ 0& 0& s_1-s_5& -b{s}_2\\ 0& 0& 0& 0\end{array}\right]$$

Reduced matrix of metric ($a>0, c^2<1, 1-c^2-d^2-e^2+2cde>0$):

$$g=\left[\begin{array}{cccc}a& 0& 0& 0\\ 0& 1& c& d\\ 0& c& 1& e\\ 0& d& e& 1\end{array}\right].$$

$\sigma =4$.

$(b=1)$

Derivations:

$$\left[\begin{array}{cccc}{s}_1& s_2& s_3& s_4\\ 0& s_5& s_6& s_3\\ 0& s_7& s_1-s_5& -s_2\\ 0& 0& 0& 0\end{array}\right].$$

Reduced matrix of metric ($a>0, d^2<1$):

$$g=\left[\begin{array}{cccc}a& 0& 0& 0\\ 0& 1& 0& d\\ 0& 0& 1& 0\\ 0& d& 0& 1\end{array}\right].$$

$(b=1, d=0)$ Extra Killing vector field: $(x^2-y^2)D_z-2y{D}_x+2x{D}_y$

• Killing Lie algebra: $A_{5.37}$: $[e^{2w}{D}_z, -e^w{D}_x, -e^w(D_y+x{D}_z), -D_w, -{\displaystyle \frac{1}{2}}(x^2-y^2)D_z+y{D}_x-x{D}_y]$, with brackets:
• $[e_1,e_4]=2e_1,[e_2,e_3]=e_1,[e_2,e_4]=e_2,[e_3,e_4]=e_3,[e_2,e_5]=-e_3,[e_3,e_5]=e_2$.
• $(a=4,b=1,d=0)$: Einstein space.
• Right-invariant Einstein metric:
• $4{(dz-ydx+(xy-2z)dw)}^2+{(dx-xdw)}^2+{(dy-ydw)}^2+d{w}^2$
• Killing Lie algebra: $\mathfrak{s}\mathfrak{u}(2,1)$

  $$\begin{array}{ccc}& & [e^{2w}{D}_z,e^w{D}_x,e^w(D_y+x{D}_z),-D_w,(x^2-y^2)D_z-2y{D}_x+2x{D}_y,\hfill \\ & & e^{-w}(2y{D}_w-4(xy+z)D_x+(3x^2+y^2+1)D_y+(x^3-3x{y}^2+4yz)D_z),\hfill \\ & & e^{-w}(-2x{D}_w-(x^2+3y^2+1)D_x+4z{D}_y-(y+2y^3)D_z),\hfill \\ & & e^{-2w}(8(xy-2z)D_w-4y(x^2+y^2+1)D_x+4x(x^2+y^2+1)D_y\hfill \\ & & +(x^4-6x^2{y}^2-3y^4+16xyz-4y^2-16z^2-1)D_z\left)\right].\hfill \end{array}$$
• Lie brackets:

  $$\begin{array}{ccc}& & [e_1,e_4]=-2e_1, [e_1,e_6]=4e_2, [e_1,e_7]=4e_3, [e_1,e_8]=-16e_4, [e_2,e_3]=e_1, [e_2,e_4]=-e_2, \hfill \\ & & [e_2,e_5]=2e_3, [e_2,e_6]=3e_5, [e_2,e_7]=-2e_4, [e_2,e_8]=4e_6, [e_3,e_4]=-e_3, [e_3,e_5]=-2e_2, \hfill \\ & & [e_3,e_6]=2e_4, [e_3,e_7]=3e_5, [e_3,e_8]=4e_7, [e_4,e_6]=-e_6, [e_4,e_7]=-e_7, [e_4,e_8]=-2e_8, \hfill \\ & & [e_5,e_6]=-2e_7, [e_5,e_7]=2e_6, [e_6,e_7]=e_8.\hfill \end{array}$$
• In this case we shall provide the geodesics of the metric to give an idea of the complexity that is involved.
• Geodesics:

  $$\begin{array}{ccc}& & \ddot{x}=(8x{y}^2+x){\dot{x}}^2-4y\dot{x}\dot{y}-16xy\dot{x}\dot{z}+(-16x^2{y}^2+32xyz-2x^2+4y^2)\dot{x}\dot{w}+{\dot{y}}^2\hfill \\ & & +4\dot{y}\dot{z}+(2xy-8z)\dot{y}\dot{w}+8{\dot{z}}^{2}x+(16x^{2}y-32xz-4y)\dot{z}\dot{w}\hfill \\ & & +(8x^3{y}^2-32x^{2}yz+x^3-3x{y}^2+32x{z}^2+8yz+x){\dot{w}}^2\hfill \\ & & \ddot{y}=(8y^3+5y){\dot{x}}^2-(16y^2+4)\dot{x}\dot{z}+(-16x{y}^3+32y^{2}z-10xy+8z)\dot{x}\dot{w}\hfill \\ & & +y{\dot{y}}^2-2y^2\dot{y}\dot{w}+8y{\dot{z}}^2+(16x{y}^2-32yz+4x)\dot{z}\dot{w}\hfill \\ & & +(8x^2{y}^3-32x{y}^{2}z+5x^{2}y+y^3+32y{z}^2-8xz+y){\dot{w}}^2\hfill \\ & & \ddot{z}=(16y^{2}z+2z){\dot{x}}^2+(1-4y^2)\dot{x}\dot{y}-32yz\dot{x}\dot{z}-(32x{y}^{2}z-4y^3-64y{z}^2+4xz-y)\dot{x}\dot{w}\hfill \\ & & +2z{\dot{y}}^2+4y\dot{y}\dot{z}+(4x{y}^2-12yz-x)\dot{y}\dot{w}+16z{\dot{z}}^2+(32xyz-4y^2-64z^2)\dot{z}\dot{w}\hfill \\ & & +(16x^2{y}^{2}z-4x{y}^3-64xy{z}^2+2x^{2}z+10y^{2}z+64z^3-xy+4z){\dot{w}}^2\hfill \\ & & \ddot{w}=(8y^2+1){\dot{x}}^2-16y\dot{x}\dot{z}+(-16x{y}^2+32yz-2x)\dot{x}\dot{w}+{\dot{y}}^2-2y\dot{y}\dot{w}+8{\dot{z}}^2\hfill \\ & & +(16xy-32z)\dot{z}\dot{w}+(8x^2{y}^2-32xyz+x^2+y^2+32z^2){\dot{w}}^2.\hfill \end{array}$$

• $A_{4.10}, A_{4.11}(a\ge 0)$$[e_2,e_3]=e_1, [e_1,e_4]=2a{e}_1, [e_2,e_4]=a{e}_2-e_3, [e_3,e_4]=e_2+a{e}_3$:

$$S=\left[\begin{array}{cccc}{e}^{2aw}& e^{aw}(xsinw+ycosw)& e^{aw}(ysinw-xcosw)& z\\ 0& e^{aw}cosw& e^{aw}sinw& x\\ 0& -e^{aw}sinw& e^{aw}cosw& y\\ 0& 0& 0& 1\end{array}\right].$$

• Left-invariant vector fields: $-2e^{2aw}{D}_z, e^{aw}(cosw{D}_x-sinw{D}_y+(ycosw+xsinw)D_z), e^{aw}(sinw{D}_x+cosw{D}_y-(xcosw-ysinw)D_z), -D_w$
• Left-invariant one forms: ${\displaystyle \frac{1}{2}}{e}^{-2aw}(-dz+ydx-xdy), e^{-aw}(coswdx-sinwdy), e^{-aw}(sinwdx+coswdy), -dw$
• Right-invariant vector fields: $2D_z, D_x-y{D}_z, D_y+x{D}_z, D_w+(ax+y)D_x+(ay-x)D_y+2az{D}_z$
• Right-invariant one forms: ${\displaystyle \frac{1}{2}}(dz+ydx-xdy-(x^2+y^2+2az)dw), dx-(ax+y)dw,dy+(x-ay)dw,dw$.
• Derivations:

  $$\left[\begin{array}{cccc}2s_1& s_2& s_3& s_4\\ 0& s_1& s_5& a{s}_3-s_2\\ 0& -s_5& s_1& -a{s}_2-s_3\\ 0& 0& 0& 0\end{array}\right].$$
• Reduced matrix of metric ($b>0,e>c^2+d^2$):

  $$\left[\begin{array}{cccc}b& 0& 0& 0\\ 0& 1& 0& c\\ 0& 0& 1& d\\ 0& c& d& e\end{array}\right].$$
• $(a=c=d=0,e=1)$
• Extra Killing vector field: $D_w+y{D}_x-x{D}_y$
• Killing Lie algebra: $A_{4.10/11(a=0)}\oplus \mathbb{R}$
• $(a\ne 0, c=d=0, e=1)$
• Extra Killing vector field: $y{D}_x-x{D}_y$.
• Killing Lie algebra: $A_{5.37}$, $[-2e^{2aw}{D}_z,e^{aw}(cosw{D}_x-sinw{D}_y+(ycosw+xsinw)D_z),e^{aw}(sinw{D}_x+cosw{D}_y-(xcosw-ysinw)D_z),{\displaystyle \frac{1}{a}}(-D_w-y{D}_x+x{D}_y),y{D}_x-x{D}_y]:$
• $[e_2,e_3]=e_1, [e_1,e_4]=2e_1, [e_2,e_4]=e_2, [e_3,e_4]=e_3, [e_2,e_5]=-e_3, [e_3,e_5]=e_2$.

• $A_{4.12},[e_1,e_3]=e_1, [e_2,e_3]=e_2, [e_1,e_4]=-e_2, [e_2,e_4]=e_1$:

$$S=\left[\begin{array}{ccc}{e}^{z}cosw& e^{z}sinw& x\\ -e^{z}sinw& e^{z}cosw& y\\ 0& 0& 1\end{array}\right].$$

• Left-invariant vector fields: $e^z(cosw{D}_x-sinw{D}_y), e^z(sinw{D}_x+cosw{D}_y), D_z, D_w$
• Left-invariant one forms: $e^{-z}(coswdx-sinwdy), e^{-z}(sinwdx+coswdy), dz, dw$
• Right-invariant vector fields: $D_x, D_y, D_z+x{D}_x+y{D}_y, D_w+y{D}_x-x{D}_y$.
• Right-invariant one forms: $dx-xdz-ydw, dy-ydz+xdw, dz, dw$.
• This Lie algebra is the complexification of the non-abelian two-dimensional Lie algebra $A_{2.1}$. Like $A_{2.1}$, its derivation algebra consists of just inner derivations and is isomorphic to $A_{4.12}$ itself. As such the matrix of the metric can only be reduced to the form $(b>0, b(1-c^2)>d^2, -b{c}^{2}f+2bcde+c^4+2c^2{d}^2+d^4-b{d}^2-b{e}^2-2cde-d^{2}f+bf-c^2>0$):

  $$\left[\begin{array}{cccc}1& 0& c& d\\ 0& b& -d& c\\ c& -d& 1& e\\ d& c& e& f\end{array}\right].$$
• The reason for choosing this form, is to try to make the metric as close to an Hermitian metric as possible, which would require, in addition, $b=f=1, e=0$.

The matrix $MM$ is very complicated in this case. Nonetheless, the determinant of a certain $6\times 6$ submatrix is $2cd(f+1){(b+1)}^{7}b{(b{c}^{2}e-bcd+d^{2}e+cd)}^2$ and so to have $\sigma >4$, we are going to need either $c=0$ or $d=0$ or $b{c}^{2}e-bcd+d^{2}e+cd=0$. Assuming for example, that $c=0$, another $6\times 6$ submatrix has determinant ${(b+1)}^7{d}^6{e}^3$, so that either $d=0$ or $e=0$. Continuing in this manner, every possibility may be chased down and we find:

• $(b=1, c=d=0)$ Extra Killing vector fields: $D_w+y{D}_x-x{D}_y$.
• Killing Lie algebra: $A_{4.12}\oplus \mathbb{R}$
• $(b=1, c=d=e=0)$ Extra Killing vector fields:
• ${\mathrm{e}}^{-z}(\left(x^2+y^2+1\right)cosw{D}_x-\left(x^2+y^2+1\right)sinw{D}_y+2\left(xcosw-ysinw\right)D_z),{\mathrm{e}}^{-z}(\left(x^2+y^2+1\right)sinw{D}_x+\left(x^2+y^2+1\right)cosw{D}_y+2\left(xsinw+ycosw\right)D_z),D_w+y{D}_x-x{D}_y$.
• Killing Lie algebra: $\mathfrak{so}(3,1)\oplus \mathbb{R}$
• $[e_1,e_3]=e_1, [e_1,e_4]=-e_2, [e_1,e_5]=e_3, [e_1,e_6]=e_4, [e_2,e_3]=e_2, [e_2,e_4]=e_1, [e_2,e_5]=-e_4, [e_2,e_6]=e_3, [e_3,e_5]=e_5, [e_3,e_6]=e_6, [e_4,e_5]=e_6, [e_4,e_6]=-e_5$.

9. Concluding Remarks

9.1. Possible Values of $\sigma$

In the various cases corresponding to the four-dimensional indecomposable Lie algebras, we have seen that $\sigma$ can take the values $4, 5, 7, 8, 10$. We know that $\sigma =9$ is excluded. If we are willing to accept decomposable four-dimensional Lie algebras as well, we can also achieve the value $\sigma =6$. If we take a direct product of two copies of $A_{2.1}$, each one has a three-dimensional Lie algebra of Killing vector fields, and so the product gives $\sigma \ge 6$. In fact, if one constructs the matrix $MM$, it turns out to have rank four, so that one can indeed assert that actually $\sigma =6$.

It is worth noting in passing, that a product of (several) not necessarily isomorphic spaces of constant non-zero curvature, is not a space of constant curvature, but it will be an Einstein space provided the sectional curvature constants are all the same number. Again, the example of two copies of $A_{2.1}$, shows that Egorov’s Theorem [13] goes only in one direction.

9.2. Block Structure

The block structure of the matrix $MM$ occurring in Section 6, results from the fact that certain of the Lie algebras have abelian nilradicals. We shall revisit this phenomenon in a future venue.

9.3. Lie-Einstein Cases

Among the four-dimensional indecomposable Lie algebras, there are two cases where one obtains a Lie-Einstein metric, that is, is a right-invariant metric whose type $(0,2)$ Ricci tensor is proportional to the metric. These cases are $A_{4.9(b=1)}$ and $A_{4.11a}$, for $a>0$. Now, loosely speaking, these classes of Lie algebra are “complex-equivalent”, that is to say, if one treats them as Lie algebras over $\mathbb{C}$, they are isomorphic. However, one needs to exercise some caution here. It is true that the change of basis

$$e_1^{\prime }=-2i{e}_1,e_2^{\prime }=e_2+i{e}_3,e_3^{\prime }=e_2-i{e}_3,e_4^{\prime }={\displaystyle \frac{b-1}{2}}{e}_4,$$

together with $a={\displaystyle \frac{i(b+1)}{b-1}}$ will change $A_{4.9b}$ into $A_{4.11a}$, provided$b\ne 1$. However, $b=1$ was precisely the critical value for which $A_{4.9b}$ gives an Einstein space. Moreover, for $b=1$, $A_{4.9b}$ has a seven-dimensional derivation algebra, whereas the derivation algebra of $A_{4.11a}$ is five-dimensional for all values of a, so that $A_{4.9(b=1)}$ can never be isomorphic to any $A_{4.11a}$. Therefore, in retrospect, it is not so surprising that $A_{4.11a}$ has only a five-dimensional Killing Lie algebra.

9.4. Future Directions

We intend to look for Killing vector fields for right-invariant metrics in higher-dimensional Lie groups in another venue. We hope to be able to report on the five-dimensional and six-dimensional nilpotent cases in the near future.

9.5. Towards a Rationale for the Existence of Extra Killing Fields

It is a natural question to ask what general principles, if any, lie behind the existence of extra Killing vector fields. Such a question is of course complicated. One small step that may be taken in this direction, is that such extra Killing vector fields often take the form of infinitesimal rotations. Such rotations are frequently associated to the fact that the Lie algebra or Lie group under consideration, possesses a partial complex structure. The Lie algebra $A_{4.6ab}$ gives an example of this phenomenon in a special case. Many more similar examples will appear in dimension five.