2024/10/28 by Sigbjørn Hervik, Hervik, Sigbjørn
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2410.21161
We continue to study left-invariant pseudo-Riemannian metrics on Lie groups being in the null cone of the O(p,q)-action using the moving bracket approach. In particular, the Lie algebra being in the null cone implies that the pseudo-Riemannian metric have all vanishing scalar curvature invariants (VSI). We consider all Lie algebras of dimension ≤ 6 and we find that all solvable Lie algebras, and non-trivially Levi-decomposable Lie algebras, of dimension ≤ 6 are in the null cone, except the 3-dimensional solvable Lie algebra \mathfraks3,3. For \mathfrakg semi-simple, we also give a construction where \mathfrakg⊕ℝm is in the null cone and give examples of such spaces for all the real simple Lie algebras \mathfrakg. For example, for the exceptional split groups this construction places the split \mathfrake6⊕ℝ6, split \mathfrake7⊕ℝ7 and split \mathfrake8⊕ℝ8 in the null cone of the O(42,42), O(70.70) and O(128,128) action, respectively, and hence, their corresponding left-invariant pseudo-Riemannian metrics are VSI.