Invariant manifolds

Here I want to say something about invariant manifolds of flows and diffeomorphisms. There are close connections between the two. I feel a closer attachment to the first (continuous evolution) than the second (discrete evolution) and so I will tend to emphasize it. I have known various things about invariant manifolds and used them in my work for many years. Just recently I was able to add some small things to my knowledge on the subject which has given me the feeling that I have a more global view. A treatment of this subject which I found very helpful is that given in lecture notes of Grant. I am interested here in the situation of a smooth dynamical system on with a stationary point. Let its linearization at that point be denoted by . Let be a real number such that no eigenvalue of has real part . Then the eigenvalues can be split into the sets with real parts less than and greater than respectively. The generalized eigenvectors corresponding to eigenvalues in the first set define a subspace called the pseudostable subspace. The pseudostable manifold theorem says that if the system is for some there is a submanifold passing through the stationary solution which is invariant and whose tangent space at the stationary point is . If the terminology is simplified by omitting the prefix ‘pseudo’ and this gives rise to the more widely known stable manifold theorem. By reversing the direction of time it is possible to get corresponding statements replacing ‘stable’ by ‘unstable’. If and there are no eigenvalues whose real parts lie in the interval then is called a centre-stable manifold . The intersection of a centre-stable and a centre-unstable manifold is called a centre manifold.The pseudostable manifold is uniquely determined in a small neighourhood of the stationary point if . The other invariant manifolds are in general not unique. These results for continuous time dynamical systems have analogues for a diffeomorphism (which by iteration defines a discrete dynamical system). It is merely necessary to replace the additive inequalities on the real part of the eigenvalues by multiplicative inequalities on the modulus of the eigenvalues.

There are two common methods to prove the stable manifold theorem. The first is called the Lyapunov-Perron method and is analytical in flavour while the second, called the graph transform and due to Hadamard, is more geometrical. The first method starts by writing down an integral equation. It is then proved that any solution of this integral equation is a solution of the dynamical system which lies on the unstable manifold. The stable manifold is obtained as a union of solutions of this type. I found the proof of these statements rather easy to follow. What disturbed me was that that I did not at all see where the integral equation comes from. Fortunately in his lecture notes Grant gives an elementary step by step description of how to get to that integral equation, starting from the solution formula for an inhomogeneous linear ODE (Duhamel’s formula). The second method represents the stable manifold as a graph over the stable subspace. It defines an iteration for the function describing this graph. The map from one iterate to the next is given by the time-one flow of the system. If you think about this by drawing a picture for a saddle point in the two-dimensional case it is very plausible that it works. The actual proof can be done by noting that thetime-one map is a diffeomorphism whose stable manifold is identical with the manifold being sought. So this proof reduces the continuous time case to the discrete time case.

Up to know I have been talking about a single dynamical system. There are useful extensions to systems which depend on a parameter . There is a trick which I had seen before but never really appreciated the importance of. Suppose we have a system for . Augment it by the equation . Then we have a dynamical system on . If for the system has a stationary point at then we can study invariant manifolds for the augmented system about the point . Considering for instance the centre-stable manifold in a situation of this type can give valuable information about the way in which solutions near that point change when passes through zero.

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The immortal Henrietta Lacks

I find immortal cell lines a fascinating topic and I mentioned the subject in a previous post on influenza vaccines. From time to time I had heard about HeLa cells. I knew that this was a cell line derived from tissue taken from a tumour in the 1950’s. I also knew that the name was an abbreviation of the name of the woman who was the donor of the cells. Now, while wandering through the internet, I discovered the blog ‘Culture Dish‘ of the science writer Rebecca Skloot. My attention was immediately attracted by an advertisement for her forthcoming book ‘The Immortal Life of Henrietta Lacks’. This is the story of the woman who ‘gave’ the world HeLa cells, and how this was not exactly voluntary. Now this is the most important cell line in medical research and has become notorious for contaminating other cell lines. I will say no more about this since of course I have not seen the book yet. If I read it, which I probably will, I may write about it in a future post. The abbreviation was not very successful in assuring the anonymity of Henrietta Lacks.

Critical wave maps

At the moment I am attending a research programme called ‘Quantitative studies of nonlinear wave phenomena’ at the Erwin Schrödinger Institute in Vienna. This has stimulated me to think again about a topic in the study of nonlinear wave equations which has seen remarkable developments just recently. This concerns critical wave maps. Wave maps are nonlinear generalizations of the wave equation associated to a Riemannian manifold called the target manifold. These days wave maps are one of the most important model problems in the study of nonlinear wave equations. If is a pseudo-Riemannian manifold and a mapping from to define a Lagrange function by the norm squared of the derivative of with the norm being determined by and . In local coordinates . Solutions of the corresponding Euler-Lagrange equations are called harmonic maps when is Riemannian and wave maps when is Lorentzian. Harmonic maps are elliptic while wave maps are hyperbolic. It is common to classify nonlinear evolution equations by the scaling of the energy into subcritical, critical and supercritical. The standard point of view is then that subcritical problems are relatively straightforward, supercritical problems are impossible at present and critical problems are on the borderline. It is therefore natural that critical problems are a hive of activity at the moment. For more information about this classification and its significance see this post of Terence Tao. Actually there are a couple of
results around on problems which could be called marginally supercritical. One, due to Tao, concerns the nonlinear wave equation in three space dimensions (arXiv:math/0606145). Note that the equation with nonlinearity is critical in three space dimensions. Another, due to Michael Struwe, concerns the equation in two space dimensions, where every power law nonlinearity is subcritical. Here criticality is decided by the value of the energy and Struwe’s result applies to the supercritical case, under the assumption of rotational symmetry. He gave a talk on this here on Monday.

Getting back to wave maps the criticality is decided by the dimension of the manifold . If its dimension is two (i.e. space dimension ) then the problem is subcritical. The case is critical while the problem is supercritical for . From now on I concentrate on the critical case. In this context there is a general guide for distinguishing between global smoothness of solutions and the development of singularities which is the curvature of the target manifold. Negative curvature tends to be good for regularity while positive curvature tends to encourage singularities. This is related to an analogous feature of harmonic maps which has been known for a long time. A good model problem for the negatively curved case is that where is the hyperbolic plane. The thesis of Shadi Tahvildar-Zadeh played an important role in making this problem known in the mathematics community. In this work the manifold is three-dimensional Minkowski space. It was done at the Courant Institute under the supervision of Jalal Shatah and Demetrios Christodoulou. To make the problem more tractable some symmetry assumptions were made. Consider the action of on by rotations about a point. There is also a natural action of the rotation group on the hyperbolic plane. One symmetry assumption is invariance – applying a rotation to a point leaves invariant. Another symmetry assumption is equivariance – applying a rotation by an angle to leads to a rotation of by an angle where is a positive integer. It can be seen that there are different types of equivariance depending on the parameter . The invariant case is related to the Einstein equations of general relativity. Cylindrically symmetric solutions of the vacuum Einstein equations lead to exactly this problem. On the other hand one of the key techniques used in the proofs originates in a paper of Christodoulou where he showed a stability result for an exterior region in a black hole spacetime in the presence of a scalar field. The results on the invariant case were published by Christodoulou and Tahvildar-Zadeh in 1993. They made no mention of the connection to the vacuum Einstein equations. The results on the equivariant case were published by Shatah and Tahvildar-Zadeh at about the same time. Interestingly the two papers use rather different terminology. While the work on the equivarant case uses the language of multipliers, traditional in PDE theory, the work on the invariant case uses the language of vector fields coming from the direction of general relativity.

Without the symmetry the problem becomes much harder and was the subject of a major project of Tao (project heatwave). For more information see this post of Tao and the comments on it. By the time the project was finished there were already alternative approaches to solving the same problem by Sterbenz and Tataru and Krieger and Schlag. Now this problem can be considered solved. The different papers use different methods and the detailed conclusions are different. Nevertheless they all include global regularity for wave maps from three-dimensional Minkowski space to the hyperbolic plane without any symmetry assumptions. The proof of Sterbenz and Tataru goes through an intermediate statement that if there were a singularity there would be a finite energy harmonic map into the given target. They actually concentrate on the case of a compact target manifold but the case of the hyperbolic plane can be handled by passing to the universal cover. There also give a more general statement for the case that there is a non-trivial harmonic map around. Then regularity is obtained as long as the energy is less than that of the harmonic map. These ideas are also related to results on the positive curvature case which show what some blow-up solutions look like when they occur. Consider the case that the target manifold is the two-sphere. Here it is possible, as in the case of the hyperbolic plane, to define invariant and equivariant wave maps. In recent work Raphaël and Rodnianski are able to describe the precise nature of the blow-up for an open set of initial data and for all values of the integer describing the type of equivariance. The case is exceptional. They have also proved similar results for an analogous critical problem, the Yang-Mills equations in dimensions. These results provide a remarkably satisfactory confirmation of conjectures made by Piotr Bizoń and collaborators on the basis of numerical and heuristic considerations which were an important stimulus for the field.

Induced pluripotent stem cells

The usual career of a living cell proceeds from its beginning as a stem cell in the embryo through a process of differentiation where it becomes more and more specialized until (in most cases) it finally takes its place in some tissue as a terminally differentiated cell. This process involves various genes being switched on or off. Usually in the past this process has been thought of as being more or less irreversible. This leads to the great interest in embryonic stem cells as a potential basis of the treatment of various illnesses by regeneration of certain types of cells. Unfortunately embryonic stem (ES) cells have two big problems associated with them. The first is that their use raises ethical concerns in many people which act as a powerful inhibitor of the development of the technology. The other is that they may involve medical dangers. If the cells develop in the wrong direction they may lead to tumours, especially the type called a teratoma where cells are found which are of the wrong type of tissue (and often of many types) for the place they are in.

It was discovered in 2006 by Shinya Yamanaka and his associates that the usual development can be run backwards, producing stem cells from terminally differentiated cells, for instance skin cells. They named these cells induced pluripotent stem cells (iPS cells). On the web page of the National Institutes of Health where they have videos of lectures (http://videocast.nih.gov/) there is a talk given by Yamanaka on January 14th, 2010 which is inspiring and at the same time presented in an entertaining style. The introduction by Francis Collins, director of the NIH suggests that Yamanaka will not have to wait long for his Nobel Prize. iPS cells are an ethically safe alternative to ES cells. Their medical safety does not look so good at the moment. Under some circumstances the safety profile of iPS cells is similar to that of EC cells. Under other circumstances a subset of the cells seem to be refractory to differentiation and can then produce teratomas at a later time. It is necessary to learn to control their development better before they can be used in regenerative medicine. Of course it would be important to know what characterizes this subset. Yamanaka suggests that this may have to do with epigenetic factors and this ideas is being tested in his laboratory now. An application of iPS cells less risky than tissue regeneration is to use cells produced from iPS cells to test drugs which are toxic, or even lethal, for certain patients but not for the majority. The idea is to take skin cells from the patient, turn them into stem cells and test the drug on those cells. Unfortunately this process requires a lot of time and money.

The normal cells are turned into iPS cells by the application of certain transcription factors. This may be done by tranferring genetic material or by using the proteins themselves directly. Originally four different transcription factors had to be combined. Recent work by Hans Schöler and collaborators indicates that one of these, Oct4, is enough in humans. The article is in Nature, 461 (2009) 649.

Asymptotic discrete self-similarity in cosmological models

In the previous post I discussed some ideas from the theory of dynamical systems and mentioned some applications of these ideas to models arising from chemistry and biology. Here I will explain an example where they can be applied to mathematical cosmology. In cosmology it is common to study spatially homogeneous solutions of the Einstein equations with matter modelled by a perfect fluid with linear equation of state . In a spatially homogeneous solution there is a preferred foliation by spacelike hypersurfaces defined by the orbits of the symmetry group. An important simplifying assumption is that the four-velocity of the fluid is orthogonal to these hypersurfaces. If this assumption is not satisfied the fluid is said to be tilted and the dynamics of the models becomes significantly more complicated. Physically tilt means that in a frame of reference where the energy density is spatially constant there is a bulk motion of the fluid.

Starting with a paper of Coley and Hervik it has been observed that solutions of the Einstein equations with tilted fluid can show dynamical behaviour which is qualitatively different from anything which is possible in the absence of tilt. The approach used combines analytical and numerical techniques. I have been aware of this work and found it interesting for a long time but I now see that I had not gone into it deeply enough to get a real understanding of what is going on there. This dynamics involves periodic solutions and Hopf bifurcations, which is why it is related to the previous post. The interesting new phenomena occur for certain Bianchi types of class B, namely types , and . Some background information on this terminology can be found here. The relevant dynamical regime is that where the spacetime expands forever. Many of the basic quantities tend to zero at late time and to get a less degenerate description it is useful to introduce a dynamical system for certain dimensionless quantities. When this is done a stationary solution of the dimensionless system corresponds to a continuously self-similar solution of the original Einstein-Euler system and a periodic solution of the dimensionless system corresponds to a discretely self-similar solution of the original system. The latter type of solution will be of particular interest in what follows.

In the classes of solutions considered here there are stationary solutions of the dimensionless systems which play the role of late time attractors in most cases. The solutions are asymptotically self-similar at late times. There is, however, a small exceptional region called the loophole where no stable self-similar solutions are available. In that case the asymptotic behaviour is more complicated. The dimensionless variable corresponding to the energy density of the fluid tends to zero at late times. Thus in a sense the solution is converging to a vacuum solution. However certain variables describing the fluid retain a non-trivial dynamics so that in a sense the fluid leaves a trace on the vacuum solution. More precisely, in the limit the spacetime geometry converges in a suitable sense to that of a solution of the vacuum Einstein equations while the fluid variables describing the tilt converge to a test fluid on that background. A test fluid means that while the fluid satisfies the Euler equations its energy-momentum tensor does not make a contribution to the Einstein equations. The limiting dynamical system is defined on the unit ball in with the unit sphere being an invariant submanifold. On the unit sphere the test fluid can be thought of as null dust, where the four-velocity is null. The parameter in the equation of state vanishes completely from the restrictions of the equations for the tilt variables to the boundary sphere. In terms of the dimensionless dynamical system the solutions with test fluid are not solutions of that system but solutions of a smooth extension of that system to part of the boundary of the physical region. The restriction of the extended system to that part of the boundary will be called the asymptotic system.

For some values of the parameters the asymptotic system has periodic solutions. These can be of different types. Probably the simplest case is that described in Theorem 4.3 of gr-qc/0409106. It concerns Bianchi type . There the system on the boundary sphere has precisely two stationary points which are hyperbolic sources. Thus it follows immediately by Poincaré-Bendixson theory that there exists a periodic solution. The uniqueness of this periodic solution which is also part of the assertion of the theorem requires a more complicated proof. A Hopf bifurcation occurs explicitly in the treatment of Bianchi type in arXiv:0706.3184 where the authors give the expression for the first Lyapunov number in this case.

Hopf bifurcations and Lyapunov numbers

The simplest type of stationary solutions of a dynamical system are those which are hyperbolic, which means that all eigenvalues of the linearization about the given point have non-zero real parts. If smooth one-parameter families of dynamical systems are considered the simplest type of loss of hyperbolicity is when some eigenvalue hits the imaginary axis at an isolated value of the parameter, say . The most generic examples of this are when a real eigenvalue passes through the origin or when a pair of complex eigenvalues meet at a point of the imaginary axis away from the origin. The latter case is the scenario of the Hopf bifurcation and is the one I will discuss in what follows. For the moment only the two-dimensional case will be considered. The parameter will be chosen such that the real part of the eigenvalues has the same sign as . Thus as the parameter passes through zero while increasing the stationary point loses stability. It will be assumed that the eigenvalue crosses the axis with non-zero velocity. My primary source of information on this subject is the book of Kuznetsov already mentioned in a previous post. The figures 3.5 and 3.7 of that book are useful for visualizing what is going on. If a further genericity assumption is made the phase portrait of the bifurcation can be shown to be topologically equivalent to that of a simple explicit model. This assumption is the non-vanishing of a quantity called the first Lyapunov number . When this number is non-zero its sign can be used to distinguish two different kinds of Hopf bifurcation called super- and subcritical. In the supercritical case () periodic solutions exist for all small positive values of and they are stable. In the subcritical case small periodic solutions exist for all small negative values of and are unstable. Kuznetsov gives an intuitive interpretation of this difference, calling the first case a soft loss of stability and the second a catastrophic one.

The first Lyapunov number depends not only on the linearization of the system about the bifurcation point but also on second and third order derivatives there. Calculating this quantity is elementary but usually lengthy, even for simple systems. A trick used to simplify this calculation is the introduction of a suitable complex coordinate. When I first saw this I did not like it but if it reduces cumbersome calculations there is a clear motivation for proceeding in this way. It is not even that unnatural, given the fact that a pair of purely imaginary eigenvalues are at the heart of the Hopf bifurcation. In the book the example of the Brusselator is discussed in an exercise. Although this is a very simple example the reader is encouraged to use computer algebra to do the calculations. I did some of them by hand and saw that it is not impossible but it is tedious. An alternative approach is to use a formula which has been derived once and for all by someone. A formula of this type is given on p. 353 of Perko’s book “Differential equations and dynamical systems”. I tried using this in the case of the Brusselator and it seemed easier than the alternative mentioned above. If it is possible to go further by assuming that another quantity , the second Lyapunov number, is non-zero. This gives rise to what is called a Bautin bifurcation. There is a natural generalization of the Hopf bifurcation to higher dimensions. It is just necessary to assume that all the eigenvalues except for the pair responsible for the bifurcation have non-zero real parts. Under this assumption the problem can be reduced to a centre manifold. This is a centre manifold adapted to the bifurcation problem rather than just a centre manifold for the system with fixed values of the parameter.In this way the two-dimensional setting can be recovered.

Up to this point I have just presented the Lyapunov numbers as the result of messy calculations which may be diagnostic for certain things. This gives no intuition about what they mean. To obtain this kind of intuition, note first that the Lyapunov numbers are characteristics of the dynamical system for . They do not involve any derivatives with respect to . In fact they arise in the study of two-dimensional dynamical systems which have a focus at some point, say the origin. This means that solutions starting near the origin spiral towards the origin as or , depending on whether the focus is stable or unstable. I now restrict to the stable case for definiteness. In this situation it is possible to define a Poincaré map which is similar to that in the more familiar setting of periodic solutions. Consider a radial line segment of length . If it is short enough then following solutions through one rotation defines a mapping from the line segment to itself. Call it and define . It is always true that . If then its sign determines the stability with the negative sign corresponding to the stable case. This is the case where the eigenvalues have non-zero real part and the stationary point is hyperbolic. The case of interest in connection with the Hopf bifurcation is that where . It can be shown that in general the first non-vanishing derivative of at the origin is of odd order. If its order is three then it corresponds to the first Lyapunov number. So in a sense that number measures the leading order deviation of the solution from a circle. If the first Lyapunov number vanishes then the fifth derivative of gives the second Lyapunov number. An account of this can be found in section 3.4 of Perko’s book.

There are very many applications where the Hopf bifurcation plays a role. A first example is the Brusselator mentioned above. This is a schematic two-dimensional model for a chemical reactor. When I hear the name I get a mental picture of Brussels sprouts. This is of course nonsense. The name comes from the fact that the model was developed in Brussels and is a simplification of a three-dimensional model called the Oregonator which was developed in Oregon. The latter name was influenced by the fact that it is a kind of oscillator. The Oregonator is nothing other then the Field-Noyes model discussed in a recent post. As mentioned there the Field-Noyes model also exhibits Hopf bifurcations. Hopf bifurcations occur in the FitzHugh-Nagumo and Hogdkin-Huxley systems. Thus they are potentially relevant for electrical signalling by neurons. They may also come up in another kind of biological signalling, namely that by calcium. For an extensive review of this subject I refer to a paper of Martin Falcke (Adv. Phys. 53, 255). In section 5 of that paper the author discusses experimental evidence indicating that certain calcium oscillations cannot be modelled using Hopf bifurcations and that it might be better to use other types of bifurcation. On the other hand he suggests that the evidence for this is not conclusive. Oscillations in glycolysis are modelled by the Higgins-Selkov oscillator, a two-dimensional system bearing a superficial resemblance to the Brusselator. The unknowns are the concentrations of ADP and the enzyme phosphofructokinase. This simple system describing a part of glycolysis exhibits a Hopf bifurcation. More information on this and related systems can be found in the book of Klipp et. al. on systems biology quoted in a previous post.

Four-dimensional Lie algebras

In mathematical general relativity it is common to study solutions of the Einstein equations with symmetry. In other words, solutions are considered which are invariant under the action of a Lie group . (In what follows I will restrict consideration to the vacuum case to avoid having to talk about matter. So a solution means a Lorentzian metric satisfying .) It is usual to concentrate on the four-dimensional case, corresponding to the fact that in everyday life we encounter one time and three space dimensions. One type of solutions with symmetry are the spatially homogeneous ones where the orbits of the group action are three-dimensional and spacelike. Then the Einstein equations reduce from partial differential equations to ordinary differential equations. This is a huge simplification although the solutions of the ODEs obtained are pretty complicated. Here I will make the further assumptions that the Lie group is of dimension three and that it is simply connected. The first of these assumptions is a real restriction but the second is not from my point of view since it does not change the dynamics of the solutions, which is what I am mainly interested in. With these assumptions the unknown can naturally be considered as a one-parameter family of left-invariant Riemannian metrics on a three-dimensional Lie group. These Riemannian metrics are obtained as the metrics induced by the spacetime metric on the orbits of the group action. Any connected three-dimensional Lie group can occur. Connected and simply connected Lie groups are in one to one correspondence with their Lie algebras.Thus it is important to understand what three-dimensional Lie algebras there are. Fortunately there exists a classification which was found by Bianchi in 1898. People working in general relativity call the spatially homogenous solutions of the Einstein equations with symmetry property defined by Lie groups in this way Bianchi models. They use the terminology of Bianchi, who distinguished types I-IX. A lot of work has been done on the dynamics of these solutions. Some more information on this can be found in a previous post on the Mixmaster model.

For reasons of pure mathematical curiosity, or otherwise, it is interesting to ask what happens to all this in space dimensions greater than three. Recently Arne Gödeke has written a diploma thesis on some aspects of this question under my supervision and this has led me to go into the issue in some depth. One thing which naturally comes up is the question of classifying Lie algebras in dimensions. As far as I can see there is not a useful complete classification in general dimensions but there is quite a bit of information available in low dimensions. Here I will concentrate on the case of four dimensions. In that case there is a classification which was found by Fubini in 1904 and since then other people have produced other versions. Having worked with Bianchi models for many years I feel very much at home with the three-dimensional Lie algebras. In contrast the four-dimensional classification appeared to me quite inhospitable and so I have invested some time in trying to fit the four-dimensional Lie algebras into a framework which I find more appealing. I record some of what I found here. The best guide I found was the work of Sigbjørn Hervik, in particular his paper in Class. Quantum Grav. 19, 5409 (cf. arXiv:gr-qc/0207079).

From the point of view of the dynamics of the Einstein equations one Bianchi type which is notably different from all others is type IX.The reason for this is that the Lie group (which is ) admits left-invariant metrics of positive scalar curvature. Is there a natural analogue for four-dimensional Lie algebras? A useful tool here is the Levi-Malcev theorem which provides a way of splitting a general Lie algebra into two simpler pieces. More precisely it says that each Lie algebra is the semidirect sum of a semisimple and a solvable Lie algebra. The semisimple part is called a Levi subalgebra and is unique up to isomorphism. It turns out that the information about whether there exists a metric of positive scalar curvature is contained in the Levi subalgebra. There are not many semisimple Lie algebras in low dimensions. In fact in dimension no greater than four there are only two, and . These correspond to Bianchi types IX and VIII respectively. The only possible non-trivial Levi decompositions are the semidirect sum of one of the two Lie algebras just mentioned and the real numbers. In fact it turns out that the semidirect sum of a semisimple Lie algebra and the real numbers is automatically a direct sum because any derivation of a semisimple Lie algebra is an inner derivation. The corresponding simply connected Lie group is a direct product. It can be concluded from this that the only simply connected and connected four-dimensional Lie group which admits a metric of positive scalar curvature is . This is the analogue of Bianchi type IX for .

It is common in general relativity to divide the three-dimensional Lie algebras into two disjoint classes, Class A and Class B. The first of these consist of the unimodular Lie algebras, i.e. those whose structure constants have vanishing trace. They are closely associated with the class of Lie groups whose left-invariant metrics can be compactified by taking the quotient by a discrete group of isometries. They also have the pleasant property that their dynamics can be reduced to the case where the matrix of components of the metric in a suitable basis of left-invariant one-forms is diagonal. This is important for the Wainwright-Hsu system, a dynamical system formulation of the Einstein equations for Class A Bianchi models which is the basis for most of the rigorous results on the dynamics of these solutions obtained up to now. If type IX is omitted there are five different Lie algebras in Class A. One way of getting unimodular Lie algebras of dimension four is to take the direct sum of the three dimensional Lie algebras with the real numbers. Call the others indecomposable. The indecomposable unimodular four-dimensional Lie algebras can be classified into six types. Four of these are individual Lie algebras while the other types are one-parameter families of non-isomorphic algebras. One way of putting these into a larger framework is to note that each of them has a three-dimensional Abelian subalgebra. They can therefore be considered as special cases of solutions with three commuting spacelike Killing vector fields. This generalizes the fact that all the Class A Bianchi types except VIII and IX can be considered as solutions with two commuting Killing vector fields. I do not have an overview of the questions of compactification and diagonalization for these metrics. It seems that calculations done by Isenberg, Jackson and Lu in their study of the Ricci flow on homogeneous four-dimensional manifolds (Commun. Anal. Geom. 14, 345) might be helpful in this context.

More details on some of the things mentioned in this post will be given in a forthcoming preprint by Gödeke and myself.