Six-dimensional Lie-Einstein metrics
On an Einstein manifold the Ricci tensor is a multiple of the metric. In the present paper the authors study Einstein manifolds that arise for right-invariant Riemannian metrics on a six-dimensional Lie group \(G\). They impose severe restrictions on the Lie algebra it-self: indecomposable, six-dimensional and solvable. They have five-dimensional and four-dimensional nilradical algebras. In those cases they assume very specific forms of the metric. The metric was reduced by using the following method: the space of derivations was found and then exponentiated so as to obtain the connected component of the identity of the automorphism group of \(\mathfrak g\), the Lie algebra of \(G\). This automorphism group was used to reduce the matrix of the metric. At that point it became feasible to find the Ricci tensor of g and set it equal to a multiple of the metric. Remark that any constant non-zero multiple of a Lie-Einstein metric is again Lie-Einstein.
scientific article; zbMATH DE number 7362842
scientific article