Ricci-flat Invariant Geometry in Four Dimensions

On 4-dimensional Lie-Groups that bear a left-invariant metric so that the Ricci-curvature tensor vanishes but the Riemannian curvature is non-zero.

Introduction

In the article Invariants of real low dimensional Lie-algebras, the authors J. Patera, R. Sharp, P. Winternitz, and H. Zassenhaus state a classification of 4-dimensional Lie-algebras. The classification comprises of 12 types of pairwise non-isomorphic algebras. Several of those types depend on one or more real parameters.

A left-invariant metric on a Lie-group is determined by a single scalar product defined on the tangent space of a single point of the Lie-group. We define the scalar product on the tangent space G with respect to the same basis used to define the Lie-algebra on G. The coefficients we choose for the scalar product are arranged as

B=(                )

We aim to find all pairs (g,B), where g is a 4-dimensional Lie-algebra that corresponds to a Lie-group with left-invariant metric g, and in the neutral element e∈G the metric =B coincides with the scalar product B.

Our strategy is to investigate all 12 types of Lie-algebras in the classification of J. Patera, R. Sharp, and P.Winternitz reproduced in their paper Invariants of real low dimension Lie algebras. For each such Lie-algebra we initially assume a general scalar product B on G. Then, the entries of the Ricci- and Riemannian-curvature tensors and on G are polynomials in the coefficients of the scalar product B. With respect to the left-invariant metric on G, we demand the following algebraic equations

 det B≠0,  We employ Mathematica to solve the equations, i.e. to obtain restrictions on the coefficients . However, depending on the Lie-algebra structure, the equations are quite difficult to solve. Among the 12 types of Lie-algebras there remain 4 types that we are able to solve at all.

A useful reference is the paper Four-dimensional Pseudo-Riemannian Homogeneous Spaces by B. Komrakov. The author explains how =B relates to . Alternatively, have a look at Section 1.4 of my Thesis On Lorentzian Ricci-flat homogeneous manifolds by Jan P. Hakenberg.

Investigations of pairs

We investigate individually all 12 types of Lie-algebras { :i=1,2,...,12} in the classification of J. Patera, R. Sharp, and P.Winternitz reproduced in their paper Invariants of real low dimension Lie algebras. For each such Lie-algebra g= we initially assume a general scalar product B on G. All geometric considerations are local, and with respect to the left-invariant metric on G.

At this point the scalar product B has 10 coefficients. Whenever possible, we simplify B by applying suitable automorphisms of the Lie-algebra g to B. An automorphisms grants the isomorphy (isometry) of two pairs ( , ) and ( , ). In some of the twelve cases this reduction helps considerably from the computational point of view.

Technically, we ask Mathematica to grant Ricci-flatness and a non-degenerated scalar-product, =0, and det B≠0. The homogeneous pairs (g,B) that are necessarily Riemannian-flat are dropped in the next chapter.

Summary of ricci-flat pairs

In the previous section, for each pair {( ,B):i=1,2,...,12} we have imposed =0, and naturally det B≠0. Below, we summarize all pairs (g,B) featuring this geometry that we have collected over the course of investigation.

Specifically, our results are exhaustive for the cases i∈I:={1,2,3,4,7,8,10,12}. That means, any Ricci-flat non-Riemannian-flat pair {( ,B):i∈I} is isomorphic to one pair listed below.

Isomorphy types of ricci-flat pairs

This final section demonstrates how to further classify the ricci-flat pairs for the case ( ,B). The other pairs are much more complicated.

Future work

One investigates the remaining pairs ( ,B) with j∈J:={5,6,9,11} exhaustively.

One classifies the ricci-flat pairs up to isomorphy to simplify the visual representation.

One distinguishes the signatures of the scalar-products. For instance, any Ricci-flat non-Riemannian-flat pairs of type ( ,B) has even signature, because |B|>0. This observation gives rise to the conjecture, that all existing pairs are already covered by my diploma thesis.

I hope to collaborate with Thomas Neukirchner on these issues.

 Created by Mathematica  (September 15, 2007) 